Singularities of Mappings: The Local Behaviour of Smooth and Complex Analytic Mappings 3030344398, 9783030344399

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Grundlehren der mathematischen Wissenschaften  357 A Series of Comprehensive Studies in Mathematics

David Mond Juan J. Nuño-Ballesteros

Singularities of Mappings The Local Behaviour of Smooth and Complex Analytic Mappings

Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics Volume 357

Editors-in-Chief Alain Chenciner, IMCCE - Observatoire de Paris, Paris, France John Coates, Emmanuel College, Cambridge, UK S.R.S. Varadhan, Courant Institute of Mathematical Sciences, New York, NY, USA Series Editors Pierre de la Harpe, Université de Genève, Genève, Switzerland Nigel J. Hitchin, University of Oxford, Oxford, UK Antti Kupiainen, University of Helsinki, Helsinki, Finland Gilles Lebeau, Université de Nice Sophia-Antipolis, Nice, France Fang-Hua Lin, New York University, New York, NY, USA Shigefumi Mori, Kyoto University, Kyoto, Japan Bao Chau Ngô, University of Chicago, Chicago, IL, USA Denis Serre, UMPA, École Normale Supérieure de Lyon, Lyon, France Neil J. A. Sloane, OEIS Foundation, Highland Park, NJ, USA Anatoly Vershik, Russian Academy of Sciences, St. Petersburg, Russia Michel Waldschmidt, Université Pierre et Marie Curie Paris, Paris, France

Grundlehren der mathematischen Wissenschaften (subtitled Comprehensive Studies in Mathematics), Springer’s first series in higher mathematics, was founded by Richard Courant in 1920. It was conceived as a series of modern textbooks. A number of significant changes appear after World War II. Outwardly, the change was in language: whereas most of the first 100 volumes were published in German, the following volumes are almost all in English. A more important change concerns the contents of the books. The original objective of the Grundlehren had been to lead readers to the principal results and to recent research questions in a single relatively elementary and accessible book. Good examples are van der Waerden’s 2-volume Introduction to Algebra or the two famous volumes of Courant and Hilbert on Methods of Mathematical Physics. Today, it is seldom possible to start at the basics and, in one volume or even two, reach the frontiers of current research. Thus many later volumes are both more specialized and more advanced. Nevertheless, most books in the series are meant to be textbooks of a kind, with occasional reference works or pure research monographs. Each book should lead up to current research, without over-emphasizing the author’s own interests. There should be proofs of the major statements enunciated, however, the presentation should remain expository. Examples of books that fit this description are Maclane’s Homology, Siegel & Moser on Celestial Mechanics, Gilbarg & Trudinger on Elliptic PDE of Second Order, Dafermos’s Hyperbolic Conservation Laws in Continuum Physics ... Longevity is an important criterion: a GL volume should continue to have an impact over many years. Topics should be of current mathematical relevance, and not too narrow. The tastes of the editors play a pivotal role in the selection of topics. Authors are encouraged to follow their individual style, but keep the interests of the reader in mind when presenting their subject. The inclusion of exercises and historical background is encouraged. The GL series does not strive for systematic coverage of all of mathematics. There are both overlaps between books and gaps. However, a systematic effort is made to cover important areas of current interest in a GL volume when they become ripe for GL-type treatment. As far as the development of mathematics permits, the direction of GL remains true to the original spirit of Courant. Many of the oldest volumes are popular to this day and some have not been superseded. One should perhaps never advertise a contemporary book as a classic but many recent volumes and many forthcoming volumes will surely earn this attribute through their use by generations of mathematicians.

More information about this series at http://www.springer.com/series/138

David Mond • Juan J. Nu˜no-Ballesteros

Singularities of Mappings The Local Behaviour of Smooth and Complex Analytic Mappings

David Mond Mathematics Institute University of Warwick Coventry, UK

Juan J. Nu˜no-Ballesteros Departament de Matem`atiques Universitat de Val`encia Burjassot, Spain

ISSN 0072-7830 ISSN 2196-9701 (electronic) Grundlehren der mathematischen Wissenschaften ISBN 978-3-030-34439-9 ISBN 978-3-030-34440-5 (eBook) https://doi.org/10.1007/978-3-030-34440-5 Mathematics Subject Classification (2010): 14A10, 14B05, 14B07, 14J17, 14B10, 13H10 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Gabi, to my sons Gabriel and Isaac, and to Maíra and Ruben who were entrusted to me D.M. To Conxín, and our children Carme and Joan J.J.N.B.

Preface

This book is concerned with the local behaviour of mappings. The local behaviour at an immersive point, where the derivative is injective, and at a submersive point, where the derivative is surjective, is uninteresting; in the first case, the mapping is equivalent, under right-left equivalence (smooth changes of coordinates in source and target), to a linear inclusion. In the second, it is equivalent to a linear projection. Singularity theory begins where neither of these two conditions holds. Singularities of algebraic varieties have been studied by algebraic geometers since the beginning of the subject and continue to play a central role in the minimal model programme for the birational classification of varieties of dimension ≥ 3. This book has a different focus. It is not concerned with singularities of varieties though they appear and play an important role, so much as with singularities of mappings, and its focus is not algebraic geometry, but smooth (C ∞ ) and complex analytic geometry though the gap is not all that wide. There are many sources for this part of the subject. • The earliest is probably the Morse Lemma (Example 3.1 below), proved by Marston Morse in the 1920s (see [Mor96]): any smooth function of n variables with a non-degenerate critical point at x0 can be transformed, by a change of coordinates in some neighbourhood of x0 , into a quadratic form. The Morse Lemma is an example of “finite determinacy” (cf Chap. 6), in which a finite portion of the Taylor series of a smooth or analytic map determines it up to some kind of equivalence—in this case, equivalence under change of coordinates in the domain, which we refer to from now on as right equivalence. Finite determinacy allows the algebraisation of the spaces and maps it covers, and this can have important consequences. The simple normal form for a function in the neighbourhood of a non-degenerate critical point is used in Morse Theory to prove important statements about the global topology of manifolds. • In the 1940s and 1950s, Whitney was concerned with immersing manifolds in Euclidean spaces. Immersions are dense in the space of proper maps M n → R2n (see below, Exercise A.7). An n-manifold can also be immersed in R2n−1 , even though immersions are no longer dense: some non-immersive points cannot be vii

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Preface

removed by small deformations. To remove them (by large deformations), one needs to understand their geometry. In [Whi44], Whitney found and described the singularities which persist, up to right-left equivalence—see Exercise 7.2.4, and in [Whi55] he did the same in the case of maps from surfaces to the plane— see Exercises 3.3.6 and 3.3.7. These are examples of what we now call stable singularities. • Knot theorists defined invariants by means of “generic planar projections”: planar projections with only nodes as singularities. K. Reidemeister, seeking to understand how one generic planar projection could be deformed into another, described the three other singularities which were unavoidable in the process (see [Rei27b]).They are now known as the three Reidemeister moves; in the language of singularity theory they are the codimension 1 singularities of parameterised plane curves—see Example 3.4 below. • Moving on from questions about the representability of homology classes by embedded submanifolds, René Thom developed a general theory of smooth maps, and in the process proved his celebrated transversality theorem (see Theorem A.1 in Appendix A), which puts the “generic” into every subsequent theorem about the generic behaviour of maps. Maps f, g : N → P are leftright equivalent if there are diffeomorphisms ϕ on N and ψ of P such that f = ψ ◦ g ◦ ϕ. A smooth mapping f : N → P is stable if its orbit under left-right equivalence is open in the space C ∞ (N, P ) of smooth mappings with its Whitney topology. Thom asked whether the set of stable mappings is dense in C ∞ (N, P ) and quickly found a counterexample (See Sect. 5.2 below). His question became when are stable mappings dense. The answer was provided by John Mather, in a remarkable series of six papers collectively called “Stability of C ∞ mappings”, [Mat68a, Mat68b, Mat69a, Mat69b, Mat70, Mat71]. Part I of this book is concerned with these ideas. In particular, we cover most of the material in [Mat68a, Mat68b, Mat69a, Mat69b, Mat70, Mat71] though our focus is always local rather than global. Mather went on to develop a theory of topological stability, in which ϕ and ψ are required merely to be homeomorphisms; we comment only very briefly on this, in Sect. 5.3. John Milnor started another strand in the subject with his 1968 book “Singular points of complex hypersurfaces”, [Mil68], by studying the level sets of a complex polynomial f (x1 , . . ., xn+1 ) in the neighbourhood of an isolated critical point. Intersected with a suitably small ball around the critical point, these level sets have the homotopy type of a wedge of n-spheres. The resulting fibration over the complement of the critical value, the so-called Milnor fibration, has been the object of intensive study since then, notably by the Russian school led by Vladimir Arnold and the German school led by Egbert Brieskorn. One of the salient features of the subject is the fact that the Milnor number of the singularity, the number of spheres in the wedge, is equal to its deformation-theoretic codimension with respect to right equivalence (see Definition 3.3 and Sect. 8.2 below). If we regard the Milnor fibre as the stable object near to the (unstable) singular hypersurface, then Milnor’s study

Preface

ix

sets the theme for Part II of the book, where we study the stable mapping (on a suitably defined domain) which results from perturbing an unstable map-germ. It is important to point out a bifurcation in the subject here. Milnor was concerned with preimages, and the largest part of singularity theory in this direction, studying singularities of varieties—see e.g. [GLS07, Dim92, Loo84, AGZV12a, AGZV12b]. As mentioned earlier, there is also a large literature on singularities of varieties in the context of birational geometry and the minimal model programme in algebraic geometry. In this book, we are interested in singularities of mappings, and those properties which are preserved by right-left equivalence. As Felix Klein remarked, the shape of the subject is determined by the equivalence relation one uses, and for right-left equivalence, it turns out that the topological counterpart to the deformation theory of an unstable singularity is found not in the fibres of the map, but in the discriminant of a stable perturbation, in case the dimension of the source, n, is greater than or equal to the dimension of the target, p, and the image otherwise. When n ≥ p − 1, the discriminants and images of stable perturbations have the homotopy type of wedges of spheres, just like Milnor fibres, and the number of spheres in the wedge is closely related to the deformation-theoretic codimension of the germ itself, echoing Milnor’s findings unexpectedly closely. Part II of the book is concerned with discriminants and images and with the relation between their topology and the deformation theory. The book is a monograph and not a textbook—its shape reflects the subject, or rather our knowledge of it, rather than the structure of a course. Even so we hope that it could be used as the basis for a graduate course. We have included many exercises and some preliminary material that will make it approachable for a beginning graduate student. A graduate course consists of • • • • • •

Chapter 3: Sects. 3.2, 3.3, 3.4, 3.5 Chapter 4: Sects. 4.1, 4.2, 4.3, 4.4 Chapter 5: Sects. 5.1, 5.5 Chapter 6: Statement of finite determinacy theorems Chapter 7: Sects. 7.1, 7.2 Chapter 8: Sects. 8.1, 8.2, 8.3, 8.4, 8.5.

Many people have contributed to our understanding of the subject. DM is especially grateful to Terry Wall, Jim Damon, Terry Gaffney, Duco van Straten, Vladimir Arnol’d and Victor Goryunov for explanations, conversations and collaborations. He also thanks Jairo Charris and Carlos Ruiz for encouragement when it most counted. JJNB thanks Lê D˜ung Tráng for helpful conversations and Raúl Oset Sinha, Guillermo Peñafort Sanchis and Roberto Giménez Conejero for their help with the revision of some parts of this book. And we both apologise to the people—surely there are many—whose important work we have not been able to cover. Coventry, UK Burjassot, Spain July 2019

David Mond Juan-José Nuño-Ballesteros

Contents

1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Real or Complex? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 The Nearby Stable Object. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Exercises and Open Questions .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Part I

1 2 3 6 10 10

Thom-Mather Theory: Right-Left Equivalence, Stability, Versal Unfoldings, Finite Determinacy

2

Manifolds and Smooth Mappings. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Manifolds and Their Tangent Spaces .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Inverse Mapping Theorem and Consequences ... . . . . . . . . . . . . . . . . . . . 2.4 Submanifolds .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Vector Fields and Flows. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Transversality .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7 Local Conical Structure .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

13 13 15 24 29 32 40 44

3

Left-Right Equivalence and Stability.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Classification of Functions by Right Equivalence . . . . . . . . . . . . . . . . . . 3.2 Left-Right Equivalence and Stability . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Right Equivalence and Left Equivalence . . . . . . . . . . . . . . . . . . 3.3 First Calculations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Multi-Germs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.1 Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Infinitesimal Stability Implies Stability . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Stability of Multi-Germs .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

45 46 58 71 74 81 81 88 92

4

Contact Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 97 4.1 The Contact Tangent Space . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 97 4.2 Using T Ke f to Calculate T Ae f . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 102 xi

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4.3 4.4 4.5

Construction of Stable Germs as Unfoldings . . .. . . . . . . . . . . . . . . . . . . . Contact Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Geometric Criterion for Finite Ae -Codimension . . . . . . . . . . . . . . . . . . . 4.5.1 Sheafification . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Transversality .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Thom–Boardman Singularities . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

105 108 116 117 122 128

5

Versal Unfoldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Versality.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Global Stability of C ∞ Mappings .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.1 Stable Maps Are Not Always Dense . .. . . . . . . . . . . . . . . . . . . . 5.2.2 Mather’s Nice Dimensions . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Topological Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Bifurcation Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 The Notion of Stable Perturbation of a Map-Germ .. . . . . . . . . . . . . . . .

141 142 156 158 160 162 163 176

6

Finite Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Proof of the Finite Determinacy Theorem . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Estimates for the Determinacy Degree . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Determinacy and Unipotency .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.1 Unipotent Affine Algebraic Groups . . .. . . . . . . . . . . . . . . . . . . . 6.3.2 Unipotent Groups of k-Jets of Diffeomorphisms . . . . . . . . . 6.3.3 When Is a Closed Affine Space of Germs Contained in a G -Orbit? .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.4 Complexification and Determinacy Degrees . . . . . . . . . . . . . . 6.3.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Complete Transversals . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Notes and Further Developments .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

181 184 191 200 204 206

Classification of Stable Germs by Their Local Algebras .. . . . . . . . . . . . . . 7.1 Stable Germs Are Classified by Their Local Algebras . . . . . . . . . . . . . 7.2 Construction of Stable Germs as Unfoldings . . .. . . . . . . . . . . . . . . . . . . . 7.3 The Isosingular Locus.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.1 Weighted Homogeneity and Local Quasihomogeneity . . . 7.4 Quasihomogeneity and the Nice Dimensions .. .. . . . . . . . . . . . . . . . . . . . 7.4.1 Multi-Germs . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.2 The Case n ≥ p . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4.3 The Case n < p . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

217 217 223 227 231 232 235 236 238

4.6 4.7

7

Part II 8

209 209 209 210 215

Images and Discriminants: The Topology of Stable Perturbations

Stable Images and Discriminants . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.1 Complex Not Real . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Review of the Milnor Fibre . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

253 253 258 259

Contents

8.3 8.4

8.5

8.6 8.7 8.8 8.9

9

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The Homotopy Type of the Discriminant of a Stable Perturbation: Discriminant and Image Milnor Numbers . . . . . . . . . . . Finding TA1 e f in the Geometry of f : Maps from n-Space to n + 1-Space .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.1 The Conductor Ideal .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Finding TA1 e f in the Geometry of f : Sections of Stable Discriminants and Images . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.5.1 Critical Space and Discriminant . . . . . . .. . . . . . . . . . . . . . . . . . . . Bifurcation Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Calculating the Discriminant Milnor Number . .. . . . . . . . . . . . . . . . . . . . Image Milnor Number and Ae -Codimension .. .. . . . . . . . . . . . . . . . . . . . Further Developments.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.9.1 Almost Free Divisors . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.9.2 Thom Polynomial Techniques . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.9.3 Does μ Constant Imply Topological Triviality? .. . . . . . . . 8.9.4 The Milnor–Tjurina Relation . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.9.5 Augmentation and Concatenation: New Germs from Old . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

261 269 275 279 283 289 292 298 299 299 300 300 300 301

Multiple Points .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Choosing the Right Definition .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.1 Semi-Simplicial Spaces . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . k 9.2.2 When Is Dcl (f ) Reduced? . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.3 Irritating Notation, Occasionally Necessary.. . . . . . . . . . . . . . 9.2.4 Equations or Procedures? . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Expected Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Equations for D 2 (f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.5 Equations for D k (f ) When f Is a Corank 1 Germ .. . . . . . . . . . . . . . . . 9.5.1 Generalities on Functions of One Variable . . . . . . . . . . . . . . . . 9.5.2 Application to Multiple Points . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6 Bifurcation Sets for Germs of Corank 1. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.7 Disentangling a Singularity: The Geometry of a Stable Perturbation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.8 Blowing-Up Multiple Points .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.8.1 Construction of an Ambient Space for Kk . . . . . . . . . . . . . . . . 9.8.2 Construction of Kk (f ) as Subspace of Bk (X) . . . . . . . . . . . . 9.9 What Remains To Be Done .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

303 303 304 311 312 312 315 315 320 327 327 333 341 345 351 352 355 366

10 Calculating the Homology of the Image. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 The Alternating Chain Complex .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 The Image Computing Spectral Sequence . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.1 Towards the ICSS . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.2 The Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

369 370 373 380 384 385

xiv

Contents

10.3

10.4 10.5 10.6 10.7

10.8

10.2.3 The Spectral Sequence of a Filtered Complex . . . . . . . . . . . . 10.2.4 The Spectral Sequences Arising from the Two Filtrations on the Total Complex of the Double Complex.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Finite Simplicial Maps .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.1 Triangulating D k (f  ) . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .  Alt • 10.3.2 Cn (D (f )), #• Is a Resolution of Cn (Y ) .. . . . . . . . . . . . . Finite Complex Maps Are Triangulable .. . . . . . . .. . . . . . . . . . . . . . . . . . . . Other Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Cohomology .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Examples and Applications of the ICSS . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7.1 The Reidemeister Moves .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7.2 Reidemeister I . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7.3 Reidemeister II.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7.4 Reidemeister III.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7.5 Map-Germs of Multiplicity 2 . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7.6 Codimension 1 Corank 1 Germs . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7.7 Generalised Mayer–Vietoris . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7.8 Relation Between AH∗ and H∗ . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.7.9 Exercises for Sect. 10.7 .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Open Questions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

11 Multiple Points in the Target: The Case of Parameterised Hypersurfaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1 Finding a Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1.1 Using Macaulay2 to Find a Presentation . . . . . . . . . . . . . . . . . . 11.2 Fitting Ideals and Multiple Points in the Target.. . . . . . . . . . . . . . . . . . . . 11.2.1 Are the Fitting Ideal Spaces Mk (f ) Cohen–Macaulay? .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.3 Double Points in the Target . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.4 Ae -Codimension and Image Milnor Number of Map-Germs(Cn , S) → (Cn+1 , 0) . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.5 The Rank Condition .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.6 Corank 1 Mappings: Cyclic Extensions .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.7 Duality and Symmetric Presentations . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.7.1 Gorenstein Rings and Symmetric Presentations . . . . . . . . . . 11.7.2 Geometrical Interpretation of the Trace Homomorphism . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.8 Triple Points in the Target . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

385

386 389 391 394 398 399 399 402 403 403 404 405 406 409 410 411 411 412 413 414 418 421 428 431 436 442 447 451 455 458 462

A

Jet Spaces and Jet Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 469

B

Stratifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . B.1 Stratification of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . B.2 Stratification of Mappings . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . B.3 Semialgebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

477 477 481 483

Contents

xv

C

Background in Commutative Algebra . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.1 Spaces and Functions on Spaces. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.2 Associated Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.3 Dimension, Depth and Cohen–Macaulay Modules .. . . . . . . . . . . . . . . . C.3.1 Krull Dimension .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.3.2 Slicing Dimension .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.3.3 Hilbert–Samuel Dimension . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.3.4 Weierstrass Dimension . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.3.5 The Hauptidealsatz . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.3.6 Depth and Cohen–Macaulay Modules.. . . . . . . . . . . . . . . . . . . . C.4 Free Resolutions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.4.1 Cohen–Macaulay Modules and Freeness .. . . . . . . . . . . . . . . . . C.4.2 Examples of Cohen–Macaulay Spaces . . . . . . . . . . . . . . . . . . . . C.5 Pulling Back Algebraic Structures . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.6 Samuel Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

489 489 493 495 495 496 498 498 498 500 503 504 505 508 513

D

Local Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . D.1 The Preparation Theorem .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . D.2 Local Properties of Analytic Sets and Finite Mappings . . . . . . . . . . . . D.3 Degree and Multiplicity .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . D.4 Normalisation of Analytic Set-Germs .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . D.4.1 Extension Theorems .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . D.4.2 Normalisation .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

517 517 521 525 528 531 532

E

Sheaves .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . E.1 Presheaves and Sheaves .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . E.2 Coherence.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . E.3 Conservation of Multiplicity .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . E.3.1 Representatives . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . E.4 Conservation of Multiplicity II . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

537 537 541 545 545 549

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 553 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 563

Chapter 1

Introduction

What is singularity theory? The crucial notion is the derivative of a smooth or analytic map: if f : X → Y is a map of manifolds and x ∈ X then dx f : Tx X → Tf (x) Y is the derivative, defined by dx f (x) ˆ = lim

h→0

f (x + hx) ˆ − f (x) h

if X and Y are open sets in linear spaces. If X and Y are contained, but not open, in linear spaces, dx f can be defined by restricting to Tx X the derivative of a suitable extension of f to an open set in the ambient space; otherwise one uses charts. It is also worth recalling that every tangent vector xˆ ∈ Tx X is the tangent vector γ  (0) to a parameterised curve γ : (R, 0) → (X, x) (or γ : (C, 0) → (X, x) in the complex analytic category), and that dx f satisfies dx f (γ  (0)) = (f ◦ γ ) (0).

(1.1)

This may be taken as the definition. It gives a useful heuristic in infinite-dimensional cases, such as where X is a group of diffeomorphisms. A point x0 ∈ X is a regular point of f if dx0 f is surjective, in which case f is a submersion at x0 , and a critical point otherwise, and f is an immersion at x0 if dx0 f is injective. The image of a critical point is a critical value of f ; any point in Y which is not a critical value is a regular value (even if it has no preimages). The set of all critical values is often called the discriminant of the map f . If f is a submersion at x0 , then from the inverse function theorem follows Theorem 1.1 (Normal Form for Submersions) Suppose that dim X = n ≥ k = dim Y and x0 is a regular point of f : X → Y . Then one can choose coordinates x1 , . . ., xn on X around x0 , and y1 , . . ., yk on Y around f (x0 ), such that f takes the form f (x1 , . . ., xn ) = (x1 , . . ., xk ). © Springer Nature Switzerland AG 2020 D. Mond, J. J. Nuño-Ballesteros, Singularities of Mappings, Grundlehren der mathematischen Wissenschaften 357, https://doi.org/10.1007/978-3-030-34440-5_1

1

2

1 Introduction

These notions are only of interest when dim X ≥ dim Y ; when dim X < dim Y , all points of X are critical points, and the set of critical values of f is the whole image of f . In this case one is interested in whether or not dx f is injective. If it is, one has Theorem 1.2 (Normal Form for Immersions) Suppose that dim X = n ≤ k = dim Y and that f : X → Y is an immersion at x0 . Then one can choose coordinates around x0 and f (x0 ) such that f takes the form f (x1 , . . ., xn ) = (x1 , . . ., xn , 0, . . ., 0). Both statements follow from the inverse function theorem, by incorporating f into a suitable auxiliary mapping whose derivative is invertible; we give the standard proof, as Corollary 2.1, in the introductory chapter on smooth manifolds and maps which follows this introduction. Both statements can be summarised by saying that any smooth mapping can be “linearised” in a neighbourhood of a regular point (that is, the local behaviour of a mapping near a regular point is like a linear mapping). Both are examples of the phenomenon of finite determinacy, which we study in Chaps. 3 and 6, in which the behaviour of a mapping on some neighbourhood of a point x0 is determined up to diffeomorphisms in source and/or target by a finite portion of its Taylor series at x0 . A finitely determined mapping is thus equivalent to the polynomial mapping defined by this portion of its Taylor series. In the case of immersions and submersions, first order information alone is enough, and thus they are equivalent to linear injections and projections, respectively. Singularity theory begins where f is neither a submersion nor an immersion— though it is also concerned with the way that multiple immersions can interact. In this book we concentrate on the local behaviour of mappings. This is best described with the notion germ of mapping, which we describe in Sect. 2.1.

1.1 Real or Complex? A large part of the theory we will cover holds both in the complex analytic category, where one considers complex analytic (holomorphic) maps between complex manifolds, and in the C ∞ category, where one considers real C ∞ manifolds and mappings. In order to give a single account, where possible we will refer to both real affine space Rn and complex affine space Cn as Fn . In the parts where the two are treated jointly, we will refer to both C ∞ manifolds and mappings, and complex analytic manifolds and mappings, simply as smooth. However, we are not in any sense trying to work over a general unspecified field. Our F is at all times either R or C or both. Both R and C have a Euclidean metric (which for C renders it the same as R2 ) and we make free use of this.

1.2 Structure of the Book

3

1.2 Structure of the Book This book is divided into two parts. Part I is concerned primarily with the purely local theory of germs of mappings, and of their unfoldings. All but a few minor details here is the work of other authors, beginning with René Thom and especially with John Mather, whose six papers jointly called Stability of C ∞ mappings [Mat68a, Mat68b, Mat69a, Mat69b, Mat70, Mat71] set the groundwork for the subject. It also contains later work of Bruce, Damon, Gaffney, Looijenga, Martinet, du Plessis, K. Saito, Wall and others. Part II is of more recent origin, though it takes its inspiration from Milnor’s book [Mil68]. It is concerned with what we can call the semi-local theory: what happens in a suitably small neighbourhood of an unstable point of a mapping when the mapping is deformed. That this makes sense at all is a slightly subtle point, though our intuition easily furnishes a working definition of what it means, for example, for a repeated root of a complex polynomial to split into a certain number of ordinary roots. The formal definitions here rely on the locally conical structure of analytic sets. Locally conical structure is easy to prove in spaces with isolated singularities: one integrates a radial vector field, which is not hard to construct. In spaces with non-isolated singularities, considerably more technical sophistication is needed, in particular the theory of Whitney stratifications and controlled vector fields. Incorporating every technical development as it is needed would give a very uneven narrative, so we consign a significant number, including those just mentioned, to appendices, where they are summarised. We hope that this may also make the book useful as a reference. Three aspects to the local theory are covered in Part I. The fundamental invariant in the theory is the left-right codimension. For now one can use the following relatively non-technical working definition: the codimension is the minimal number of parameters for a family of mappings in which a singularity left-right equivalent to the one in question occurs ‘stably’ or “irremovably”. In other words, it is the smallest numbers of parameters for a parameterised family in which one can expect to see singularities of the given type. The three key notions are, first, left-right stability itself, studied in Chap. 3, which then extends to the second notion, of (left-right) versal unfolding, studied in Chap. 5. We prove Martinet’s theorem [Mar82] that every germ of finite left-right codimension has a versal unfolding, by means of his infinitesimal criterion for versality. We introduce the bifurcation set in the base of a versal unfolding, and discuss Mather’s nice dimensions, originally defined as those ∞ (N, P ), dimension-pairs (n, p) for which globally stable mappings are dense in Cpr whenever dim N = n and dim P = p. We show that this property is equivalent to the property that every germ of finite left-right codimension (Fn , 0) → (Fp , 0) has a stable perturbation. In Chap. 6 we study the third of the key notions, finite determinacy. We prove Mather’s theorem [Mat68b] that germs of finite left-right codimension are finitely determined, with the improvements to Mather’s original estimates for the determinacy degree due to Gaffney and du Plessis, and then discuss

4

1 Introduction

the work of Bruce, du Plessis and Wall in [BdPW87] on unipotent groups and their role in finite determinacy. In almost every aspect of the local theory it is unnecessary to distinguish between R and C. One of the reasons for this is finite determinacy mentioned above. Every real polynomial, and every convergent real power series, has a complexification, defined by the same formula. Of course there are C ∞ functions which do not have a complexification, such as the smooth functions one uses to construct partitions of unity, where there are points where all derivatives vanish. But in a sense which will become clear later, such functions and singularities are “infinitely degenerate”. And a finitely determined mapping is equivalent to a polynomial mapping, which does have a natural complexification. All of the singularities of finite codimension, and in particular, all stable ones, have this property. And the codimension of a real singularity is the same as the codimension of its complexification: the algebraic object with which one calculates the codimension is in general a vector space, real in the C ∞ case and complex in the complex analytic case, and in every case the complex vector space corresponding to the complexification of a real singularity is the tensor product with C of the real vector space for the real singularity (Proposition 3.8, below). Thus the calculations involved in determining it are formally the same. Note that behind the proof of finite determinacy of germs of finite codimension for left-right equivalence is the Weierstrass preparation theorem, proved in the late nineteenth century, in the complex analytic case, and, in the C ∞ case, the preparation theorem proved by Malgrange in the early 1960s. Malgrange’s remarkable result is an essential underpinning for the close link between the C ∞ and the complex analytic theory. However, in general classification over R is more complicated than over C. A single complex equivalence class may have many inequivalent real forms, as witness the non-degenerate critical point (Morse singularity) of a function of n variables. Here, in the complex case, the Morse Lemma tells us that with  respect to suitable coordinates centred on the critical point, f is equal to f (0)+ j xj2 . In the real case there are n + 1 different possibilities, allow coordinate changes only in the  if we  source: f is equivalent to f (0) − kj =1 xj2 + nj=k+1 xj2 for some k = 0, . . ., n (the index of the critical point). Over C, the substitution of a new complex variable ixj for xj changes xj2 to its negative, and so as complex germs all these are equivalent. Nevertheless, we do not attempt a systematic discussion of the relation between real and complex classification. Part II is concerned with the geometrical theory. Classical singularity theory is interested in preimages f −1 (y0 ), where f is a germ of map (Fn , 0) → (Fp , 0) with n > p. We do not cover this part of the theory; it is already the subject of several excellent books, in particular [Loo84, Dim92, GLS07]. There, the link between the semi-local theory and the local is the “Milnor–Tjurina relation”: the Milnor number of an isolated complete intersection singularity (the rank of the middle homology of a nearby non-singular fibre) is greater than or equal to its Tjurina number, its codimension for K -equivalence, with equality in the weighted homogeneous case (see [Gre80, LS85, Vos02]). For the theory of singularities of

1.2 Structure of the Book

5

mappings, one is interested in left-right equivalence, and the Milnor number bears no relation to the left-right codimension. Instead, it is in the topology of images and discriminants of stable perturbations that one finds invariants which do relate to the left-right codimension. In Chap. 8 we introduce image and discriminant Milnor numbers, due to Jim Damon and the first author, and explore their relation with left-right codimension. Part of this relation is still obstinately conjectural, though well-supported by examples. Images and discriminants of germs of maps Fn → Fp are in general highly singular, and this makes calculating their topology rather difficult. It turns out that a useful handle on this topology and, indeed, other useful invariants of singularities of maps come from the study of their multiple points. Here we take a rather naive definition: if f : X → Y , then D k (f ) is the closure in Xk of the set {(x1 , . . ., xk ) ∈ Xk : xi = xj if i = j, f (xi ) = f (xj ) for all i, j }. In Chap. 9 we show that for germs (Fn , 0) → (Fp , 0) with n ≤ p of corank 1 and with finite left-right codimension, the D k (f ) are isolated complete intersection singularities, of which the multiple point spaces of a stable perturbation ft of f are smoothings. The Milnor numbers of the D k (f ) are left-right invariants, but there is more: from certain invariants of the symmetric group action on D k (ft ) , permuting the copies of X (the “alternating homology”), one can calculate the homology of the image of ft by means of the “image computing spectral sequence” ICSS. The ICSS was introduced by Goryunov and the first author in [GM93] for rational homology, and then further developed by Goryunov in [Gor95] and by Houston in [Hou97] and other papers. In Chap. 10 we give a new construction of the ICSS, due to José Luis Cisneros and the first author, using the triangulability of stratified mappings proved by Hardt in [Har77] for light mappings and more generally by Verona in [Ver84]. With this approach, convergence of the ICSS to the homology of the image is easily proved. The fact that for a corank 1 map-germ the D k (f ) are isolated complete intersection singularities means that in this case the ICSS for a stable perturbation collapses at E 1 , making the homology of the image easy to read off from the alternating homology of the D k (ft ). We also discuss (though we do not prove) the remarkable results of Houston in [Hou97] which show that the same collapse occurs even in the case of germs of corank > 1, where the D k (f ) are neither complete intersection singularities nor necessarily isolated. Chapter 11 focuses on the case of germs of maps (Cn , 0) → (Cn+1 , 0) and studies the information which can be obtained from the matrix of a presentation of O Cn ,0 as O Cn+1 ,0 -module. In particular we are interested in the Fitting ideals, which, it turns out, give a natural analytic structure to the multiple point sets Mk (f ) = {y ∈ Y : |f −1 (y)| ≥ k, (counting multiplicity)}. We reprove results of Catanese, Mond–Pellikaan, de Jong–van Straten, KleimanLipman–Ulrich and Sharland.

6

1 Introduction

The book ends with a number of appendices on technical aspects of the theory which are needed in the main body of the text.

1.3 The Nearby Stable Object Much of Part II centres on the notion of the nearby stable object. This term will be made more precise in Chap. 8; for now, we make do with three examples. The first comes from classical singularity theory, concerned with preimages. The second comes from the singularity theory of images, an important part of the subject of this book. The third comes from the singularity theory of the discriminant, the set of critical values of a map. The first example is the nearby level set of a non-degenerate critical point x0 of a complex analytic function. The critical level set is unstable—by means of an arbitrarily small deformation (or choice of other level) it is smoothed, and in particular it is qualitatively changed. The nearby stable object is a nearby nonsingular level set, “localised” by being intersected with a small ball around the critical point. Being smooth, it does not change qualitatively under small changes in the function or level.  By the Morse lemma, f is equal to f (x0 ) + j xj2 in some local coordinate system centred on the point x0 . For simplicity let us assume f (0) = 0. The critical level set, f −1 (0), is contractible within a sufficiently small ball Bε (x0 ). For t = 0, f −1 (t) ∩ Bε (x0 ) has the homotopy-type of an n − 1-sphere. The extra homology acquired by f −1 (t) ∩ Bε (x0 ) as t leaves 0 is known as the vanishing homology of the singularity.1 The second example of nearby stable object and vanishing homology is provided by the third Reidemeister move of knot theory, where three immersed pieces of parameterised plane curve meet, two-by-two transversely, at a point. This configuration is evidently unstable: one can move any one of the three to form a triangle (see Fig. 1.1). Since now all intersections are transverse, the altered configuration is stable. It is the “nearby stable object” for this example, and its vanishing homology, generated by the 1-cycle highlighted in the drawing on the right, once again has rank 1. In this example, the inclusion of real in complex is a homotopy equivalence: the drawing on the right is a “good real picture” (cf [Mon96, MM96]). The left-right codimension here is equal to 1; that is why it is important in knot theory. Given two plane projections of the same knot, one can be deformed to the other in such a way that during the deformation, only three types of qualitative change occur. These are the three Reidemeister moves of classical knot theory

1 The term “emerging homology” might be more appropriate, but because algebraic geometers saw singularities as degenerations of the objects they really wanted to study, we are stuck with “vanishing homology”.

1.3 The Nearby Stable Object

Unstable

7

Stable

Fig. 1.1 Stable perturbation of the triple point

[Rei27a], each of which is passage through a codimension 1 singularity. Our example shows the third of these. They cannot be avoided in a 1-parameter family of projections, other more complicated singularities can be. The third example is familiar to every schoolchild. When the real cubic x 3 is deformed to x 3 − ux,√ the unstable critical point at 0 splits a non-degenerate√local maximum at c− := − u/3 and a non-degenerate local minimum at c+ := u/3, with critical values vmax and vmin . For u > 0, the discriminant consists of the two points vmax and vmin . Its relative 0th homology is generated by the class [vmax ] − [vmin ], which vanishes as u returns to 0. Example 1.1 (Looking at Bent Wires) A number of the themes of the book can be appreciated by considering plane projections of knots in 3-space. We return to this several times. The topic was first studied from this point of view by J.M. Soares David (published after a long delay) in [Dav83] and by C.T.C. Wall in [Wal77], and subsequently in [Mon95, Wal09, DNnB08, Dia15] among others. Take a thin copper wire, easy to bend but thick enough to support its own weight, and join the two ends after bending it to form a knot—which (making allowances for the fact that the wire is not infinitely thin) should be a C ∞ embedding of the circle in R3 . You should obtain something looking like Fig. 1.2.

Fig. 1.2 A knot in 3-space

8

1 Introduction

Regarding this picture as a subset of the plane, the view shown here is “a generic projection” of the knot to the plane: the only singular points on the image are nodes—transverse crossings of two branches. From certain points of view one can see other singularities than just nodes. If one spends a little time experimenting, looking at the knot from different points of view, then provided the knot is generic, one can convince oneself that there is only a rather short list of topologically distinct local pictures. The intuitive term “local picture” can be understood with the help of the fact that every analytic set X ⊂ RN is locally conical, in the sense that for every point x0 ∈ X there is an ε > 0 such that X ∩ Bε (x0 ) is homeomorphic to the cone on its boundary X ∩ ∂Bε (x0 ) (see Sect. B.3 below). By a local picture of a set X at x0 we mean such a neighbourhood of a point x0 in X. Of course, the planar pictures of our knot in 3-space may not be analytic subsets of the plane; but for a smooth generic knot, all of the germs of projection to the plane are finitely determined, and thus equivalent to the polynomial map-germs defined by finite segments of their Taylor series. So in some neighbourhood of each point, the image is analytic with respect to suitable local coordinates, and thus locally conical. In the case of plane projections of a knot, by slightly changing our viewpoint we can deform any unstable view so that it becomes stable. Each local picture then turns into a “nearby stable picture”: the portion of the deformed curve inside Bε (x0 ). To capture this notion more precisely, the deformation must be so small that the neighbourhood boundary X ∩ ∂Bε (x0 ) does not change during the deformation. Note that here the nearby stable picture associated to each unstable local picture is not unique: depending on the deformation we choose, we may end up with one of several different possibilities. This can be seen in the case of the first two Reidemeister moves, whose nearby stable pictures are shown below. To Do Experiment with a copper knot, as described, and make a list of topologically distinct local pictures in its planar projections. For each one, determine its nearby stable pictures by gently moving the viewpoint (or the knot). For each type X the following two numbers can be determined experimentally: 1. the codimension in R3 of the set View(X) of centres of projection (viewpoints) for which a singularity of type X appears, and 2. the maximum number of nodes, n, into which the singularity X splits when the centre of projection is moved. For example, one sees a first order cusp by looking at the curve along one of its tangent lines at a point of non-vanishing torsion; so when X is a cusp, View(X) is the tangent developable surface of the curve (minus the tangent lines at torsion zero points), and has codimension 1. To Do For each singularity in your list, determine the codimension of the viewing set and the maximum number of nodes into which it splits.

1.3 The Nearby Stable Object

9

Fig. 1.3 Sequence of projections of the trefoil knot

You should find two other singularities whose viewing set has codimension 1: the tacnode and the triple point. The cusp, tacnode and triple point are the Reidemeister moves mentioned above. Each one appears in the sequence of images of Fig. 1.3. In the neighbourhood of the cusp, tacnode and triple point in the sequence here, or in any other generic one-parameter family, the local picture changes in an entirely predictable way, shown in the diagram of Fig. 1.4. These changes can be observed with a copper knot, by moving one’s eye along a path which crosses any of the codimension 1 viewing sets transversely. For a “generic” knot, the deformations of the local pictures obtained by moving the viewpoint are, with one exception, universal, in the sense that, up to equivalence, they contain all possible deformations: they are versal deformations. The exception is the quadruple point, where four smooth branches of the projected curve cross at a point, pairwise transversely. Here there is a modulus in the smooth classification, namely the cross ratio of the four tangent lines. This cannot be altered merely by moving the viewpoint: the only movement retaining a view of the quadruple crossing is along the line of sight, and this does not change the cross ratio.

10

1 Introduction

Fig. 1.4 The three Reidemeister moves

1.4 Exercises and Open Questions We suppose that the book falls somewhere between a textbook and a monograph. Nevertheless every reader will no doubt want to practice their skills, and we have tried to provide exercises for every section. We hope that enough of them are easy and enough are hard. One of the pleasures of the subject is the unlimited number of examples for which one can make interesting calculations, which range from easy and gratifying to stubbornly difficult. At the end of some of the sections of Part II of the book we have listed some open questions. We would welcome questions and answers from anyone interested in tackling them.

1.5 Notation Proofs are ended with  and Remarks and Examples are ended with ♦.

Part I

Thom-Mather Theory: Right-Left Equivalence, Stability, Versal Unfoldings, Finite Determinacy

Chapter 2

Manifolds and Smooth Mappings

This chapter gives a brief account of the definitions and results we need from elementary differential topology, ranging from the notion of manifold to transversality. We assume the reader is already familiar with these concepts, so we do not give many details. The only novelty is that we consider real C ∞ and complex analytic manifolds together, highlighting the differences between the two cases.

2.1 Germs In order to study local properties of mappings, we introduce the notion of germ of a mapping f : X → Y between topological spaces at a subset S ⊂ X. Definition 2.1 Let X, Y be topological spaces, and let S ⊂ X. 1. Let f : U → Y and g : V → Y be maps, where U and V are open neighbourhoods of S in X. We say that f and g have the same germ at S, if there is a neighbourhood W ⊂ U ∩ V of S in X such that f and g coincide on W . This is evidently an equivalence relation, and a germ of mapping at S or map-germ at S is an equivalence class under this relation. 2. Two subsets X1 and X2 of X have the same germ at S if there is a neighbourhood U of S in X such that X1 ∩ U = X2 ∩ U . A germ of subset of X or set-germ at S is an equivalence class of subset under this relation. 3. We denote a germ at S of a mapping f : U → Y by f : (X, S) → Y , or f : (X, S) → (Y, T ) if f (S) ⊂ T ⊂ Y . Given a map-germ f : (X, S) → Y , each member f : U → Y of the corresponding equivalence class is called a representative. Analogously, the germ at S of a subset X1 of X is denoted by (X1 , S) and each member X1 of the corresponding equivalence class is called a representative of the set-germ.

© Springer Nature Switzerland AG 2020 D. Mond, J. J. Nuño-Ballesteros, Singularities of Mappings, Grundlehren der mathematischen Wissenschaften 357, https://doi.org/10.1007/978-3-030-34440-5_2

13

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2 Manifolds and Smooth Mappings

Terminology A germ at a one point set is sometimes called a mono-germ, to distinguish it from multi-germ, a germ at a finite point set with more than one point. When S is not a finite point set, it is common to use the expression germ along S instead of germ at S. In fact we will not be concerned with cases where S is infinite. We say that a map-germ f : (X, S) → Y is continuous if there exists a continuous representative f : U → Y . If a germ f : (X, S) → (Y, T ) is continuous and g : (Y, T ) → Z is another germ, then we can define the composition in the obvious way: take f : U → Y and g : V → Z representatives with f (U ) ⊂ V and define the composition g ◦ f : (X, S) → Z as the germ of g ◦ f : U → Z at S. The continuity of f is necessary to ensure the existence of such representatives and it is easy to prove that the definition does not depend on the choice of representatives. A map-germ φ : (X, S) → (Y, T ) is called a homeomorphism if there exists a representative φ : U → V which is a homeomorphism, where U and V are open neighbourhoods of S and T in X and Y , respectively. Equivalently, φ is invertible as a continuous map-germ, that is, there exists another continuous mapgerm φ −1 : (Y, T ) → (X, S) such that φ ◦ φ −1 = id(Y,T ) ,

φ −1 ◦ φ = id(X,S) .

Let f : (X, S) → (Y, T ) be a continuous map-germ and let (Y1 , T ) be a set-germ in Y . We take representatives f : U → Y and Y1 ⊂ Y , respectively. The inverse image f −1 (Y1 , T ) is defined as the set-germ (f −1 (Y1 ), S). Again it is not difficult to prove that the definition does not depend on the choice of representatives. Example 2.1 In general, the image of a continuous map-germ is not well defined, since it may depend on the choice of the representative. For instance, consider the continuous mapping f : [0, 2π) → R2 given by f (x) = (cos x, sin x). For each  with 0 <  < 2π, f and the restriction f |[0,) : [0, ) → R2 have the same germ at 0, but their images S 1 and f ([0, )) do not have the same germ at (1, 0). ♦ A mapping f : X → Y is called open onto its image at S ⊂ X if for every W neighbourhood of S in X, f (W ) is a neighbourhood of f (S) in f (X). We say that a map-germ f : (X, S) → Y is open onto its image if there is representative f : U → Y which is open onto its image at S. If f : (X, S) → Y is open onto its image, the image f (X, S) is defined as the set-germ (f (U ), f (S)) in Y , where f : U → Y is any representative which is open onto its image at S. The fact that the definition does not depend on the choice of the representative is shown in the next lemma. Lemma 2.1 Assume that f : U → Y and g : V → Y have the same germ at S ⊂ X and that each mapping is open onto its image at S. Then f (U ) and g(V ) have the same germ at f (S) in Y .

2.2 Manifolds and Their Tangent Spaces

15

Proof On the one hand, there exists an open neighbourhood W ⊂ U ∩ V of S such that f and g coincide on W . On the other hand, there exist A, B ⊂ Y , neighbourhoods of f (S), in Y such that f (W ) = A ∩ f (U ) and g(W ) = B ∩ g(V ). Thus, f (U ) ∩ A ∩ B = f (W ) ∩ A ∩ B = g(W ) ∩ A ∩ B = g(V ) ∩ A ∩ B, which implies that f (U ) and g(V ) have the same germ at f (S) in Y .

 

The following result provides a simple and useful criterion to guarantee that a continuous map-germ is open onto its image. Proposition 2.1 Assume that X and Y are locally compact, Hausdorff and first countable spaces. Then every continuous map-germ f : (X, S) → (Y, y) with S finite and such that (f −1 ({y}), S) = (S, S) is open onto its image. Proof We first take a continuous representative f : U  → Y such that f −1 (y) = S. Since X is locally compact and S is finite, there exists another open neighbourhood U ⊂ X of S whose closure U is compact and such that U ⊂ U  . We will show that the restriction f : U → Y is open onto its image at S. In fact, if this was false, there would be a neighbourhood W ⊂ U of S whose image f (W ) is not a neighbourhood of y in f (U ). Since Y is first countable, this means that there exists a sequence {yn } in f (U )  f (W ) which converges to y. For each n ≥ 1, there is an xn ∈ U  W such that f (xn ) = yn . By the compactness of U , we can take a subsequence {xnk } which converges to some x  ∈ U . Now f is continuous, so {ynk } converges to f (x  ) and hence f (x  ) = y, since Y is Hausdorff. This implies that x  ∈ S, but this is in contradiction with the fact that W is a neighbourhood of S.   The notion of germ is particularly natural in the (real or complex) analytic category, because of uniqueness of analytic continuation: if U1 and U2 are open sets in Fn with U1 ∩ U2 connected, and fi : Ui → Cp are analytic maps which coincide on some open V ⊂ U1 ∩ U2 , they coincide on all of U1 ∩ U2 . Thus, if the fi have the same germ at some point of U1 ∩ U2 , then they coincide on all of U1 ∩ U2 . The same is not true of C ∞ maps, of course.

2.2 Manifolds and Their Tangent Spaces We consider the affine space Fn , where F is either R or C. By convention, a smooth mapping will mean any mapping f : A → Fp , where A ⊂ Fn is an open subset, which is differentiable of class C ∞ in the case F = R or holomorphic (complex analytic) in the case F = C. Definition 2.2 Let X be a topological space and let n ≥ 1.

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2 Manifolds and Smooth Mappings

• A smooth atlas in X over Fn is a family A = {φi : Ui → Ai }i∈I of homeomorphisms called charts, where Ui ⊂ X and Ai ⊂ Fn are open subsets for all i ∈ I , such that:  1. X = i∈I Ui , 2. for each i, j ∈ I , the restriction φj ◦ φi−1 : φi (Ui ∩ Uj ) −→ φj (Ui ∩ Uj ) is a smooth map (between open subsets of Fn ). • Two smooth atlases A and A are called compatible if A ∪ A is again a smooth atlas. This defines an equivalence relation. Each equivalence class [A] is called a smooth structure in X over Fn . • A manifold of dimension n is a pair (X, [A]), where X is a Hausdorff and second countable topological space and [A] is a smooth structure in X over Rn . • A complex manifold of dimension n is a pair (X, [A]), where X is a Hausdorff and second countable topological space and [A] is a smooth structure in X over Cn . • A manifold of dimension 0 is any discrete and countable topological space. Observe that a complex manifold of dimension n is a C ∞ manifold of dimension 2n. We will use the word “manifold” to refer either to a C ∞ manifold or to a complex manifold, whenever there is no risk of confusion. It is also common to use the notation X for a manifold, instead of (X, [A]), as long as only one smooth structure is in play. Definition 2.3 Let f : X → Y be a continuous mapping between manifolds. • We say that f is smooth if for every x ∈ X, there exist charts φ : U → A in X and ψ : V → B in Y such that x ∈ U ⊂ f −1 (V ) and the mapping ψ ◦f ◦φ −1 : A → B is smooth. • We say that f is a diffeomorphism if it is bijective and f and f −1 are both smooth. • A germ f : (X, S) → Y is called smooth if there exists a smooth representative f : U → Y , where U is an open neighbourhood of S in X. Next, we introduce the notion of tangent space of a manifold X at a point x0 ∈ X. We introduce the following notation: CX,x0 is the set of smooth curve germs α : (F, 0) → (X, x0 ) and OX,x0 is the set of smooth function germs h : (X, x0 ) → F. We define an equivalence relation in CX,x0 as follows: α ∼ β ⇐⇒ (h ◦ α) (0) = (h ◦ β) (0), ∀h ∈ OX,x0 . Definition 2.4 The tangent space of X at x0 is defined as Tx0 X :=

CX,x0 . ∼

2.2 Manifolds and Their Tangent Spaces

17

The elements of Tx0 X are called tangent vectors of X at x0 . Given a tangent vector v = [α] ∈ Tx0 X and a function h ∈ OX,x0 , the derivative of h in the direction v is defined as v(h) = Dv (h) := (h ◦ α) (0). To endow Tx0 X with the structure of F-vector space, we make use of the action of tangent vectors as derivations on OX,x0 . Germs of smooth functions can be added and multiplied, so OX,x0 has the structure of a commutative F-algebra. In fact, it is a local algebra whose maximal ideal, mX,x0 , is the subset of germs h ∈ OX,x0 such that h(x0 ) = 0. Let φ : U → A be a chart in X with x0 ∈ U . After composing φ with a translation of Fn , we can assume that φ(x0 ) = 0. The associated germ φ : (X, x0 ) → (Fn , 0) is a germ of diffeomorphism which induces an isomorphism of F-algebras φ ∗ : OFn ,0 → OX,x0 given by h → h ◦ φ. The maximal ideal mFn ,0 of OFn ,0 is generated by the coordinate functions u1 , . . . , un of Fn (this is easy to prove in the complex analytic case, although is not so trivial in the C ∞ real case, where it is a consequence of the Hadamard’s lemma, see Appendix C for details). As a consequence, mX,x0 is generated in OX.x0 by xi := φ ∗ (ui ) = ui ◦ φ, for i = 1, . . . , n. Thus, we have: Lemma 2.2 The maximal ideal mX,x0 is generated by the coordinate functions x1 , . . . , xn of the chart φ : (X, x0 ) → (Fn , 0).  A derivation of OX,x0 is a linear map D : OX,x0 → F such that D(gh) = D(g)h(p) + g(p)D(h),

∀g, h ∈ OX,x0 .

The following proposition ensures that tangent vectors have a double life: they can be seen at the same time as tangent vectors to curves (as we have already defined), or as derivations of functions. Proposition 2.2 The mapping v → Dv gives a bijection between Tx0 X and Der(OX,x0 ), the set of all derivations of OX,x0 . Proof Let v = [α] ∈ Tx0 X. For all g, h ∈ OX,x0 and λ, μ ∈ F, we have Dv (λg + μh) = ((λg + μh) ◦ α) (0) = λ(g ◦ α) (0) + μ(h ◦ α) (0) = λDv (g) + μDv (h), and Dv (gh) = ((gh) ◦ α) (0) = (g ◦ α) (0)(h ◦ α)(0) + (g ◦ α)(0)(h ◦ α) (0) = Dv (g)h(p) + g(p)Dv (h).

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2 Manifolds and Smooth Mappings

Hence, Dv ∈ Der(OX,x0 ). Moreover, the mapping v → Dv is one-to-one by construction. It only remains to show that it is surjective. Take a chart φ : (X, x0 ) → (Fn , 0) and let x1 , . . . , xn be its coordinate functions. Given D ∈ Der(OX,x0 ), we set λi := D(xi ) ∈ F, for each i = 1, . . . , n. We define v = [α] ∈ Tx0 X, where α(t) = φ −1 (t (λ1 , . . ., λn )). We have Dv (xi ) = (xi ◦ α) (0) = (tλi ) (0) = λi = D(xi ),

i = 1, . . . , n.

By Lemma 2.2, m2X,x0 is generated by the functions xi xj , with 1 ≤ i, j ≤ n. Thus, every function h ∈ OX,x0 can be written as h = a0 +

n 

ai xi +

n 

xi xj gij ,

i,j =1

i=1

for some ai ∈ F and gij ∈ OX,x0 , with i, j = 1, . . . , n. Since D is a derivation, D(a0 ) = 0 and D(xi xj gij ) = 0, for all i, j and the same is true for Dv . Thus, D(h) =

n  i=1

ai D(xi ) =

n 

ai Dv (xi ) = Dv (h),

i=1

and hence, D = Dv .

 

Since Der(OX,x0 ) has a natural structure of F-vector space, we can use the above bijection to add tangent vectors or to multiply a tangent vector by an element of F. Definition 2.5 The tangent space Tx0 X has a unique structure of F-vector space such that the mapping v → Dv is an isomorphism of F-vector spaces between Tx0 X and Der(OX,x0 ). Note that if X is a smooth submanifold of some FN , as in most of the early examples one meets, then Tx0 X is naturally a vector subspace of the vector space Tx0 FN  FN . As we show below in Sect. 2.4, the vector space structure defined by Definition 2.5 coincides with this subspace structure. Let φ : U → A be a chart in X with x0 ∈ U and let u0 = φ(x0 ) ∈ A. Let {e1 , . . . , en } be the canonical basis of Fn . For each i = 1, . . . , n we define the i-th coordinate tangent vector as:  ∂  := [γi ] ∈ Tx0 X, ∂xi x0 where γi ∈ CX,x0 is the curve γi (t) = φ −1 (u0 + tei ).

2.2 Manifolds and Their Tangent Spaces

19

Proposition 2.3 1. For each h ∈ O X,x0 , the derivative of h in the direction of ∂/∂xi |x0 is ∂h ∂(h ◦ φ −1 ) (x0 ) := (u0 ), ∂xi ∂ui where u1 , . . . , un are the coordinates in Fn . 2. For every v ∈ Tx0 X we have v=

n  i=1

 ∂  v(xi ) , ∂xi x0

where xi is the i-th coordinate function of the chart φ. 3. The coordinate tangent vectors form a basis of Tx0 X, in particular, dim Tx0 X = n. Proof 1. By definition,    ∂  d  −1  h ◦ φ (h) = (h ◦ γ ) (0) = (u + te ) i 0 i   ∂xi x0 dt t =0 ∂(h ◦ φ −1 ) h ◦ φ −1 (u0 + tei ) − h ◦ φ −1 (u0 ) = (u0 ). t →0 t ∂ui    2. Let wi := ni=1 v(xi ) ∂x∂ i  . As in the proof of Proposition 2.2, in order to show = lim

x0

that v = w, it is enough to show that v(xj ) = w(xj ), for all j = 1, . . . , n. By item 1,  n  n   ∂xj ∂  (x w(xj ) = v(xi ) ) = v(xi ) (x0 ) j  ∂xi x0 ∂xi i=1

=

n  i=1

i=1

v(xi )

∂uj (u0 ) = ∂ui

n 

j

v(xi )δi = v(xj ).

i=1

3. By item 2, Tx0 X is generated by the vectors ∂/∂xi |x0 , i = 1, . . . , n. Suppose that n  i=1

 ∂  ai = 0, ∂xi x0

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2 Manifolds and Smooth Mappings

with a1 , . . . , an ∈ F. For each j = 1, . . . , n, 0=

 n  i=1

 n  ∂xj ∂  ai ) = ai (x0 ) = aj . (x j  ∂xi x0 ∂xi i=1

Thus, the vectors ∂/∂xi |x0 , i = 1, . . . , n are also linearly independent.

 

Remark 2.1 There is another equivalent definition of the tangent space Tx0 X in terms of the maximal ideal m := mX,x0 of OX,x0 . In fact, there is a canonical ∗  isomorphism between Tx0 X and m/m2 , the dual vector space of m/m2 , defined ∗  as follows: to each tangent vector v ∈ Tx0 X we associate an element vˆ ∈ m/m2 2 ) = v(h), for each h ∈ m. This alternative definition of the tangent given by v(h+m ˆ space is very useful in algebraic geometry, since it allows us to extend the concept of tangent space to varieties over any field. ♦ Remark 2.2 When X is a complex manifold of dimension n, the tangent space is usually called the holomorphic tangent space, and we can denote it by Tx0 X, if we want to distinguish between it and the (real) tangent space Tx0 X as a (real) vector space of dimension 2n. The relationship between the two spaces is obtained via the complexification of Tx0 X. This is the complex space Tx0 X ⊗R C, which admits the following decomposition: Tx0 X ⊗R C = Tx0 X ⊕ Tx0 X, where Tx0 X is the so-called anti-holomorphic tangent space. Given a chart φ : U → A with x0 ∈ U , we denote by z1 , . . . , zn the complex coordinates of φ. The associated real coordinates are x1 , y1 , . . . , xn , yn , where xk and yk are the real and imaginary parts of each zk , respectively. A basis over C of Tx0 X ⊗R C is then given by the coordinate tangent vectors

 ∂ ∂ ∂ ∂ , ,..., , . ∂x1 ∂y1 ∂xn ∂yn

For each k = 1, . . . , n, the tangent vectors ∂ 1 = ∂zk 2



∂ ∂ −i ∂xk ∂yk

,

∂ 1 = ∂zk 2



∂ ∂ +i ∂xk ∂yk

,

are called holomorphic and anti-holomorphic coordinate tangent vectors, respectively. It follows that

∂ ∂ ∂ ∂ , ,..., , ∂z1 ∂z1 ∂zn ∂zn



2.2 Manifolds and Their Tangent Spaces

21

is also a basis of Tx0 X ⊗R C and the holomorphic and anti-holomorphic tangent spaces in this new basis are precisely Tx0 X = SpC



 ∂ ∂ , ,..., ∂z1 ∂zn

Tx0 X = SpC

 ∂ ∂ . ,..., ∂z1 ∂zn ♦

Let f : A → Fp be a smooth mapping where A ⊂ Fn is an open subset. We recall that the differential of f at a point u0 ∈ A is the mapping du0 f : Fn → Fp defined as f (u0 + tv) − f (u0 ) . t →0 t

du0 f (v) = lim

(2.1)

This is a linear mapping whose matrix in the canonical bases of Fn and Fp is the Jacobian matrix

∂fi (u0 ) , ∂uj with i = 1, . . . , p and j = 1, . . . , n. The tangent space we have introduced allows to extend the concept of differential to smooth mappings between manifolds. Definition 2.6 Let f : X → Y be a smooth mapping between manifolds and let x0 ∈ X. The differential of f at x0 is the mapping dx0 f : Tx0 X → Tf (x0 ) Y given by dx0 f (v) = [f ◦ α], for every tangent vector v = [α] ∈ Tx0 X. The main properties of the differential are given in the next proposition. Proposition 2.4 With the above notation, we have: 1. dx0 f (v)(h) = v(h ◦ f ), for every h ∈ O Y,f (x0 ) . 2. dx0 f (v) is well defined, that is, it does not depend on the choice of α such that v = [α]. 3. The differential is linear. 4. The chain rule: given another smooth mapping g : Y → Z, then dx0 (g ◦ f ) = df (x0 ) g ◦ dx0 f. 5. If φ : U → A and ψ : V → B are charts on X, Y , respectively, such that x0 ∈ U ⊂ f −1 (V ), then the matrix of dx0 f in the bases of coordinate tangent vectors

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2 Manifolds and Smooth Mappings

of Tx0 X and Tf (x0 ) Y coincides with the Jacobian matrix of ψ ◦ f ◦ φ −1 at φ(x0 ) and is thus equal to:

∂fi (x0 ) , ∂xj

where x1 , . . . , xn and y1 , . . . , yp are the coordinate functions of the charts φ and ψ, respectively, and fi := yi ◦ f is the i-th coordinate function of f . Proof 1. By construction, dx0 f (v)(h) = [f ◦ α](h) = (h ◦ (f ◦ α)) (0) = ((h ◦ f ) ◦ α) (0) = v(h ◦ f ). 2. This is an immediate consequence of item 1 and Proposition 2.2. 3. Given v, w ∈ Tx0 X, λ, μ ∈ F and h ∈ OX,x0 , dx0 f (λv + μw)(h) = (λv + μw)(h ◦ f ) = λv(h ◦ f ) + μw(h ◦ f ) = λdx0 f (v)(h) + μdx0 f (w)(h), hence dx0 f (λv + μw) = λdx0 f (v) + μdx0 f (w). 4. We use item 1: for every h ∈ OZ,g(f (x0 )) , dx0 (g ◦ f )(v)(h) = v(h ◦ (g ◦ f )) = v((h ◦ g) ◦ f ) = dx0 (v)(h ◦ g) = df (x0 ) g(dx0 f (v))(h), so dx0 (g ◦ f )(v) = df (x 0 ) g(dx0 f (v)).  5. Let wi = dx0 f ∂x∂ i  , then by item 2 of Proposition 2.3, x0

 dx0 f

  p  ∂  ∂  wi (yj ) , = wi = ∂xi x0 ∂yj f (x0 ) j =1

hence the components of the matrix of dx0 f are the derivatives wi (yj ). Now apply item 1 again:  wi (yj ) = dx0 f

 ∂(yj ◦ f ) ∂fj ∂  (yj ) = (x0 ) = (x0 ). ∂xi p ∂xi ∂xi

2.2 Manifolds and Their Tangent Spaces

23

Moreover, by item 1 of Proposition 2.3, ∂(fj ◦ φ −1 ) ∂(uj ◦ (ψ ◦ f ◦ φ −1 )) ∂fj (x0 ) = (φ(x0 )) = (φ(x0 )), ∂xi ∂ui ∂ui which are the components of the Jacobian matrix of ψ ◦ f ◦ φ −1 at φ(x0 ).

 

Remark 2.3 Let f : X → Y be a holomorphic mapping between complex manifolds. As in Remark 2.2, we denote by dx 0 f : Tx0 X → Tf (x0 ) Y the holomorphic differential between the holomorphic tangent spaces and by dx0 f : Tx0 X → Tf (x0 ) Y the (real) differential between the (real) tangent spaces. The induced mapping between the complexifications is dx0 f ⊗ id : Tx0 X ⊗ C −→ Tf (x0 ) Y ⊗ C, and it follows that dx 0 f is the restriction of dx0 f ⊗ id to the holomorphic tangent spaces. In fact, one can show more, namely, that dx0 f ⊗ id is the direct sum of dx0 f  with the anti-holomorphic differential: dx0 f  : Tx0 X −→ Tf (x0 ) Y . If dx 0 f has matrix A = (aij ) with respect to the holomorphic coordinate bases of Tx0 X and Tf (x0 ) Y , then the matrix of dx 0 f with respect to the anti-holomorphic

bases of Tx0 X and Tf (x0 ) Y is the conjugate matrix A = (aij ) and the matrix of dx0 f ⊗ id is A⊕A=

aij 0 0 aij

.

One consequence of this fact is that dx 0 f has rank r over C if and only if dx0 f has rank 2r over R. ♦ From now on we treat the real and complex cases together wherever possible, and use the notation dx0 f to refer both to the real differential in the real case, and the holomorphic differential hitherto dx 0 f in the complex case.

Exercises for Sect. 2.2 1 Show that an open U subset of a manifold X has an induced structure of manifold of the same dimension. 2 Show that a product X × Y of manifolds X, Y has an induced structure of manifold. What is its dimension?

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2 Manifolds and Smooth Mappings

3 Show that a topological space Y which is homeomorphic to a manifold X has an induced structure of manifold of the same dimension. 4 Show that every chart φ : U → A of a manifold X is a diffeomorphism. 5 Conversely, show that every diffeomorphism φ : U → A, where U ⊂ X and A ⊂ Fn are open subsets, can be considered as a chart of X. 6 Let U be an open subset of a manifold X. Show that the inclusion U → X is smooth. 7 Let X ×Y be a product of manifolds X, Y . Show that the projections X ×Y → X and X × Y → Y are smooth. 8 Show that when X = Fn , for each x0 ∈ Fn there exists a canonical identification between Tx0 Fn and Fn given by [α] → α  (0).  ∗ 9 Prove that for every v ∈ Tx0 X, vˆ is a well-defined element of m/m2 (see Remark 2.1) and that the mapping v → vˆ gives an isomorphism between Tx0 X and  ∗ m/m2 . 10 Let U be an open subset of a manifold X. Show that there is a canonical isomorphism Tx0 U ≡ Tx0 X, for each x0 ∈ U . 11 Let X × Y be a product of manifolds X, Y . Show that there is a canonical isomorphism T(x0 ,y0 ) (X × Y ) ≡ Tx0 X × Ty0 Y , for each x0 ∈ X and y0 ∈ Y . 12 Let f : A → Fp be a smooth mapping where A ⊂ Fn is an open subset. Show that via the canonical identifications Tx0 A ≡ Fn and Tf (x0 ) Fp ≡ Fp , the definition of dx0 f in Definition 2.6 coincides with that of (2.1). 13 Cauchy–Riemann equations in complex coordinates. Let f : A → Cp be any mapping, where A ⊂ Cn is an open subset. Show that f is holomorphic if and only if f is C ∞ and ∂fi ≡ 0, ∀i = 1, . . . , p, ∀j = 1, . . . , n. ∂zj 14 Coordinate-free version of Cauchy–Riemann equations. Let f : X → Y be any mapping between complex manifolds. Show that f is holomorphic if and only if f is C ∞ and for all x0 ∈ X, (dx0 f ⊗ id)(Tx0 X) ⊂ Tf (x0 ) Y.

2.3 Inverse Mapping Theorem and Consequences Most of the results of this section are local, so it is more convenient to work with germs of mappings instead of mappings. A smooth germ f : (X, x0 ) → (Y, y0 ) is called a diffeomorphism if there exists a representative f : U → V which is a

2.3 Inverse Mapping Theorem and Consequences

25

diffeomorphism, where U and V are open neighbourhoods of x0 and y0 in X and Y , respectively. This is equivalent to say that f is invertible as a smooth germ, that is, there exists another smooth germ f −1 : (Y, y0 ) → (X, x0 ) such that f ◦ f −1 = id(Y,y0 ) ,

f −1 ◦ f = id(X,x0 ) .

By the chain rule, we have dx0 f ◦ dy0 f −1 = idTy0 Y ,

dy0 f −1 ◦ dx0 f = idTx0 X .

In particular, dx0 f : Tx0 X → Ty0 Y is an isomorphism whose inverse is dy0 f −1 . The inverse mapping theorem gives the converse of this fact. Theorem 2.1 (Inverse Mapping Theorem for Manifolds) A smooth germ f : (X, x0 ) → (Y, y0 ) is a diffeomorphism if and only if dx0 f : Tx0 X → Ty0 Y is an isomorphism. Proof We have already seen the “only if” part, so we need to only show “if”. Consider first the case F = R. Here the statement is a simple consequence of the classical inverse mapping theorem for C ∞ -mappings from Rn to Rn . In fact, we take charts φ : U → A in X and ψ : V → B in Y , with x0 ∈ U ⊂ f −1 (V ) and consider their germs φ : (X, x0 ) → (Rn , u0 ) and ψ : (Y, y0 ) → (Rn , v0 ), which are diffeomorphisms. By the chain rule, the differential of g := ψ ◦ f ◦ φ −1 : (Rn , u0 ) → (Rn , v0 ) is an isomorphism. By the classical inverse mapping theorem, g is a diffeomorphism and hence, so is f . For the case F = C we use the real version and Remark 2.3. By taking charts as above, we only need to prove that if g : (Cn , u0 ) → (Cn , v0 ) is a holomorphic germ whose holomorphic differential is a C-isomorphism, then g is a biholomorphism (i.e., bijective with holomorphic inverse). Let A = (aij ) be the Jacobian matrix of the holomorphic differential du 0 g in complex coordinates. It follows that the (real) differential du0 g is also an R-isomorphism whose complexification has matrix A⊕A=

aij 0 0 aij

.

By the classical inverse mapping theorem, g is a C ∞ -diffeomorphism. Hence, g −1 is C ∞ and the complexification of dv0 g −1 has matrix of the form A

−1

⊕A

−1

=

bij 0 0 bij

.

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2 Manifolds and Smooth Mappings

But this implies that ∂gi−1 /∂zj ≡ 0, for all i, j = 1, . . . , n. Hence, g −1 is holomorphic, by the Cauchy–Riemann equations in complex coordinates (see Exercise 2.2.13).   One of the consequences of the inverse mapping theorem is the constant rank theorem, which states that every smooth germ of constant rank between manifolds can be “linearised”, that is, we can choose coordinates in the source and target such that the mapping in these coordinates is the restriction of a linear mapping. The rank of a smooth mapping at a point is the rank of the differential at that point. We say that a smooth germ f : (X, x0 ) → (Y, y0 ) has constant rank r if there exists a representative f : U → Y such that f has rank r at every point x ∈ U . The corank of a mapping at a point x0 is the difference between the maximum possible and the rank at x0 . Thus if the dimension of the domain is less than or equal to the dimension of the target, the corank is the dimension of ker dx0 f ; if greater than or equal, the corank is the dimension of coker dx0 f . Lemma 2.3 Let f : (Fn , 0) → (Fp , 0) be a smooth mapping of rank r at 0. There exist a diffeomorphism φ : (Fn , 0) → (Fn , 0) and a linear isomorphism ψ : Fp → Fp such that ψ ◦ f ◦ φ −1 (u) = (u1 , . . . , ur , gr+1 (u), . . . , gp (u)), for some smooth functions gr+1 , . . . , gp : (Fn , 0) → (F, 0). Proof After taking linear isomorphisms in Fn and Fp we can assume that the Jacobian matrix of f at 0 is

Ir 0 . 0 0

Let φ : (Fn , 0) → (Fn , 0) be the smooth germ given by φ(u) = (f1 (u), . . . , fr (u), ur+1 , . . . , un ). The Jacobian matrix of φ at 0 is the identity, so φ is a diffeomorphism by the inverse mapping theorem 2.1. Define g := f ◦ φ −1 : (Fn , 0) → (Fp , 0), which is written as g(s) = (s1 . . . , sr , gr+1 (s), . . . , gp (s)), for some smooth functions gr+1 , . . . , gp : (Fn , 0) → (F, 0).

 

Coordinates in which f takes the form specified in the lemma are called “linearly adapted coordinates”. Theorem 2.2 (Constant Rank Theorem) Let f : (X, x0 ) → (Y, y0 ) be a smooth germ of constant rank r. There exist diffeomorphisms φ : (X, x0 ) → (Fn , 0) and ψ : (Y, y0 ) → (Fp , 0) such that ψ ◦ f ◦ φ −1 : (Fn , 0) → (Fp , 0) is the germ of the

2.3 Inverse Mapping Theorem and Consequences

27

linear mapping (u1 , . . . , un ) −→ (u1 , . . . , ur , 0, . . . , 0). Proof By taking charts in X and Y , it is enough to prove the statement for a smooth germ f : (Fn , 0) → (Fp , 0) of constant rank r. By Lemma 2.3, we can assume that f is written as f (u) = (u1 . . . , ur , gr+1 (u), . . . , gp (u)), for some smooth functions gr+1 , . . . , gp : (Fn , 0) → (F, 0). The Jacobian matrix of f at u is equal to 

Ir ∂gi ∂uj (u)

0 ∂gi ∂uk (u)

,

∂gi ≡ 0, where 1 ≤ j ≤ r and r + 1 ≤ k ≤ n. Since f has constant rank r, ∂u k for all i = r + 1, . . . , p and k = 1 . . . , n − r. This implies that the functions gr+1 , . . . , gp only depend on the first variables u1 , . . . , ur and we can write gi (u) = gi (u1 , . . . , ur ), for all u and for each i = r + 1, . . . , p. Now consider the smooth germ ψ : (Fp , 0) → (Fp , 0) given by

  ψ(v) = v1 , . . . , vr , vr+1 − gr+1 (v1 , . . . , vr ), . . . , vp − gp (v1 , . . . , vr ) . It is obvious that ψ is a diffeomorphism, since its inverse mapping is   ψ −1 (w) = w1 , . . . , wr , wr+1 + gr+1 (w1 , . . . , wr ), . . . , wp + gp (w1 , . . . , wr ) , which is also smooth. We have constructed ψ precisely so that ψ ◦ f (u) = (u1 , . . . , ur , 0, . . . , 0), which concludes the proof.

 

Definition 2.7 A smooth mapping f : X → Y is called an immersion (resp. submersion) at x0 ∈ X if dx0 f is a monomorphism (resp. epimorphism). We say that f is an immersion (resp. submersion) if it is an immersion (resp. submersion) at every point x0 ∈ X. A germ f : (X, x0 ) → (Y, y0 ) is called an immersion (resp. submersion) if dx0 f is a monomorphism (resp. epimorphism). The rank of a smooth map f : X → Y at a point x0 ∈ X is an upper semicontinuous function. This means that if the rank of f at x0 is r, then there exists an open neighbourhood U of x0 in X such that the rank of f at every x ∈ U is ≥ r. In particular, if f is an immersion or submersion at x0 , then f has constant rank in

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2 Manifolds and Smooth Mappings

a neighbourhood of x0 . The following corollary is a particular case of the constant rank theorem. Corollary 2.1 (Local Form of the Immersion/Submersion) Let f : (X, x0 ) → (Y, y0 ) be an immersion or submersion. There exist diffeomorphisms φ : (X, x0 ) → (Fn , 0) and ψ : (Y, y0 ) → (Fp , 0) such that ψ ◦ f ◦ φ −1 : (Fn , 0) → (Fp , 0) is the germ of the linear mapping  (u1 , . . . , un ) −→

(u1 , . . . , un , 0, . . . , 0),

if f is an immersion,

(u1 , . . . , up ),

if f is a submersion.

Another consequence is that every immersion admits a left inverse which is a submersion and every submersion admits a right inverse which is an immersion. Corollary 2.2 Let X and Y be manifolds. Then: 1. If f : (X, x0 ) → (Y, y0 ) is an immersion, there exists a submersion g : (Y, y0 ) → (X, x0 ) such that g ◦ f = id(X,x0 ) . 2. If f : (X, x0 ) → (Y, y0 ) is a submersion, there exists an immersion g : (Y, y0 ) → (X, x0 ) such that f ◦ g = id(Y,y0 ) . 

Exercises for Sect. 2.3 1 A smooth germ f : (X, x0 ) → (Y, y0 ) gives rise to a map f ∗ : OY,y0 → OX,x0 given by f ∗ (h) = h ◦ f , for each h ∈ OY,y0 . Show: (a) (b) (c) (d)

f ∗ is an F-algebra homomorphism. For any other smooth germ g : (Y, y0 ) → (Z, z0 ), we have (g ◦ f )∗ = f ∗ ◦ g ∗ . f is a diffeomorphism if and only if f ∗ is an isomorphism. f is an immersion if and only if f ∗ is an epimorphism.

2 Show that a submersion f : X → Y is an open map (that is, f (U ) is open in Y , for every open subset U ⊂ X). 3 Let f : X → Y be a submersion, where X is compact and Y is connected. Show that f is surjective. 4 A smooth mapping f : X → Y is called a local diffeomorphism if for each x0 ∈ X, the germ f : (X, x0 ) → (Y, f (x0 )) is a diffeomorphism. Show that if f is a local diffeomorphism and is one-to-one, then f (X) is open in Y and the restriction f : X → f (X) is a diffeomorphism. 5 Give an example of a mapping f : F2 → F2 which is a local diffeomorphism but is not a diffeomorphism onto its image. 6 Show that up to right-left equivalence, the germ on the right-hand side in the equality of Lemma 2.3 can be assumed to be an unfolding of a germ of rank 0.

2.4 Submanifolds

29

2.4 Submanifolds A submanifold of a manifold Y is a subset X ⊂ Y which is also a manifold whose smooth structure is compatible with that of Y . More precisely: Definition 2.8 An immersion f : X → Y is called an embedding if the restriction f : X → f (X) is a homeomorphism. A manifold X is a submanifold of the manifold Y if X ⊂ Y and the inclusion X → Y is an embedding. In the definition of submanifold, we always assume that X has the topology induced as a subspace of Y , hence if X is a manifold, then the inclusion mapping i : X → Y is an embedding if and only if it is an immersion. When X is a submanifold of Y , then at each point x0 ∈ X the inclusion mapping i : X → Y induces a canonical monomorphism dx0 i : Tx0 X → Tx0 Y . In fact, this monomorphism can be easily visualised: given a tangent vector v = [α] ∈ Tx0 X, where α ∈ CX,x0 , the associated tangent vector in Y is v˜ = [i ◦ α] ∈ Tx0 Y . Since α and i ◦ α parameterise the same curve, but considered either in X or Y , we will identify the two tangent vectors v ≡ v˜ and hence, we consider Tx0 X as a subspace of Tx0 Y . Lemma 2.4 Let f : X → Y and g : Y → Z be mappings between manifolds such that g is an embedding. If g ◦ f is smooth, then f is smooth. Proof The restriction g¯ : Y → g(Y ) is a homeomorphism, hence it has a continuous inverse mapping g¯ −1 : g(Y ) → Y . We put f = g¯ −1 ◦ (g ◦ f ), so f is continuous. For each point x0 ∈ X, the germ g : (Y, f (x0 )) → (Z, g(f (x0 ))) is an immersion. By Corollary 2.2, there exists a submersion h : (Z, g(f (x0 ))) → (Y, f (x0 )) such that h ◦ g = id(Y,f (x0 )) . The germ f : (X, x0 ) → (Y, f (x0 )) can be written as f = (h ◦ g) ◦ f = h ◦ (g ◦ f ), so f is smooth at x0 .   The following proposition is usually referred to as the uniqueness of the smooth structure of a submanifold. Proposition 2.5 Let X be any subset of a manifold Y . There exists at most one smooth structure on X such that X is a submanifold of Y . Proof Let [A] and [A ] be two smooth structures in X such that (X, [A]) and (X, [A]) are both submanifolds of Y . We have two commutative diagrams: i

(X, [ ]) i

id

(X, [

])

Y,

(X, [

i

])

Y,

i id

(X, [ ])

where i, i  are the inclusion mappings and id, id the corresponding identity mappings. Since i  is an embedding and i = i  ◦ id is smooth, id is smooth by Lemma 2.4. Analogously, id is also smooth, which implies [A] = [A ].  

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2 Manifolds and Smooth Mappings

As a corollary, the property that X is a submanifold of Y is local. A set-germ (X, x0 ) is called a submanifold of (Y, y0 ) if there exists a representative X which is a submanifold of Y . Corollary 2.3 Let X be any subset of a manifold Y and assume that (X, x0 ) is a submanifold of (Y, x0 ), for each x0 ∈ Y . Then X is a submanifold of Y . Proof Take an open covering {Ui }i∈I of X in Y such that X ∩ Ui is a submanifold of Y , for all i ∈ I . If Ai is an atlas in X ∩ Ui , then by Proposition 2.5, A = ∪i∈I Ai is an atlas in X which provides it a structure of submanifold of Y .   Definition 2.9 Let f : X → Y be a smooth mapping. We say that x0 ∈ X is a critical point if f is not a submersion at x0 and the set C of all critical points of f is called the critical set of f . A critical value is the image of a critical point and the set of all critical values  = f (C) is called the discriminant of f . A regular value of f is a point y0 ∈ Y which is not a critical value (this includes also the case that y0 is not in the image of f ). Theorem 2.3 (Regular Value Theorem) Let f : X → Y be a smooth mapping and let y0 ∈ f (X)  . Then Z = f −1 (y0 ) is a submanifold of X of dimension dim X − dim Y , whose tangent space at each point x0 ∈ Z is Tx0 Z = ker dx0 f. Proof Given x0 ∈ X, f is a submersion at x0 and by Corollary 2.1, there exist diffeomorphisms φ : (X, x0 ) → (Fn , 0) and ψ : (Y, y0 ) → (Fp , 0) such that ψ ◦ f ◦ φ −1 : (Fn , 0) → (Fp , 0) is the germ of the projection (u1 , . . . , un ) → (u1 , . . . , up ). In particular, φ(Z, x0 ) = ({0} × Fn−p , 0). Now {0} × Fn−p is obviously a submanifold of Fn of dimension n − p and φ is a diffeomorphism, so (Z, x0 ) is a submanifold of (X, x0 ) of dimension n − p. By Corollary 2.3, Z is a submanifold of X dimension n − p. To see the second part, consider the inclusion i : Z → X, then the composition f ◦ i is the constant mapping y0 . By the chain rule, dx0 f ◦ dx0 i = 0, for all x0 ∈ Z. Thus, Tx0 Z ⊂ ker dx0 f and since both spaces have the same dimension, we have the equality.   The converse of Theorem 2.3 also holds locally: Proposition 2.6 If Z is a submanifold of X, of codimension k, then for each x0 ∈ Z there is a neighbourhood U of x0 in X and a smooth map g : U → Fk with 0 as regular value, such that Z ∩ U = g −1 (0).

2.4 Submanifolds

31

Proof The inclusion Z → Y is an immersion. It follows from the local normal form of immersions that there is a neighbourhood U of x0 and a diffeomorphism (U, Z ∩ U )

φ

  / V , V ∩ ({0} × Fn−k ) where V is open in Fn . Let

y1 , . . ., yn be coordinates on V . Take, as g, the map (y1 , . . ., yk ) ◦ φ.

 

The component functions of the map g in the proposition are often referred to as regular equations for Z in the neighbourhood of x0 .

Exercises for Sect. 2.4 1 Suppose that Xk ⊂ Y n , and that both are manifolds, with possibly unrelated smooth structures. Show that (i) X is a submanifold of Y if the restriction to X of every smooth function on an open set in Y is smooth with respect to the smooth structure on X, and, conversely, every smooth function on X is locally the restriction of a smooth function on Y . (ii) X is a submanifold of Y if for each x ∈ X there is a neighbourhood U of x in Y and a chart ϕ : U → V ⊂ Rn such that ϕ|X∩U is a diffeomorphism of X ∩ U to Rk ∩ V . 2 Let f : X → Y be an embedding. Show that f (X) is a submanifold of Y such that the restriction f : X → f (X) is a diffeomorphism. 3 Let Y be a manifold of dimension p and X ⊂ Y a subset. A point x0 ∈ X is called a regular point of X of dimension n if (X, x0 ) is a submanifold of (Y, x0 ) of dimension n. If x0 is not regular (of any dimension), then x0 is called a singular point of X. Show that the following statements are equivalent: (a) (b) (c) (d)

x0 is a regular point of X of dimension n; there exists a chart φ : (Y, x0 ) → (Fp , 0) such that φ(X, x0 ) = (Fn × {0}, 0); there exists an immersion f : (Fn , 0) → (Y, y0 ) such that f (Fn , 0) = (X, x0 ); there exists a submersion g : (Y, x0 ) → (Fp−n , 0) such that g −1 (0) = (X, x0 ).

4 Let X be the subset of Y = F2 of points (x, y) such that x 2 = y 3 . Show that X has only a singular point at the origin. Hint: the hardest part is to prove that the origin is not regular. Assume that 0 is regular and take an immersion f : (F, 0) → (F2 , 0) such that f (Fm , 0) = (X, 0). Now take a Taylor expansion of f and substitute in the equation x 2 = y 3 to get a contradiction. 5 Find the singular points of the subset X of the manifold Y in the following cases: (a) Y = F2 and X is the subset of points (x, y) such that xy = 0;

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(b) Y = F3 and X is the subset of points (x, y, z) such that x 2 + z2 = y 3 (1 − y)3 . (c) Y = R2 and X is the boundary of the square [0, 1] × [0, 1]. 6 Let f : F3 → F be the function f (x, y, z) = x 2 + y 2 − z2 . (a) Find the discriminant  of f . (b) Show that if a ∈ , then f −1 (a) is not a submanifold of F3 . (c) Show that if a and b are regular values in the same connected component of F  , then the submanifolds f −1 (a) and f −1 (b) are diffeomorphic. (d) Is there any difference between the real C ∞ and the complex analytic cases in this example? (e) Consider F = R and make a picture of the transition between the submanifolds f −1 (a) and f −1 (b) when passing through a critical value.

2.5 Vector Fields and Flows In order to classify functions and mappings up to right- or right-left equivalence, a central aim in singularity theory, one needs a good supply of diffeomorphisms. In almost all cases, these are obtained by integrating vector fields. A vector field on a manifold X is a rule ξ which assigns to each point x ∈ X a tangent vector ξ(x) ∈ Tx X, in such a way that ξ(x) depends smoothly on x. To give a precise meaning to the term “smooth” here, we need a smooth structure on the set of all tangent vectors to X. Definition 2.10 Let X be a manifold. A vector bundle on X is a manifold E and a smooth mapping π : E → X such that: 1. For each x ∈ X, the fibre Ex := π −1 (x) is a vector space over F. 2. For each x0 ∈ X, there exists an open neighbourhood U ⊂ X of x0 and a diffeomorphism ϕ : U × Fk → π −1 (U ) such that a. π ◦ ϕ(x, v) = x, for all (x, v) ∈ U × Fk , b. the map ϕx : Fk → Ex given by ϕx (v) = ϕ(x, v) is a linear isomorphism, for all x ∈ U . The number k in item 2 is equal to the difference dim E − dim X and is called the rank of the vector bundle. A section of the vector bundle is a smooth mapping ξ : X → E such that π ◦ ξ = idX . This means that ξ(x) ∈ Ex for all x ∈ X. It is common to use the notation ξx := ξ(x).

2.5 Vector Fields and Flows

33

Given two vector bundles π : E → X and π  : F → Y , a morphism between them is a pair of smooth mappings (g, f ) where g : E → F and f : X → Y such that 1. f ◦ π = π  ◦ g, that is, we have a commutative diagram g

E

F

π

π f

X

Y.

2. For each p ∈ X, the restriction gx : Ex → Ff (x) is linear. Definition 2.11 Let X be a manifold of dimension n. We define the tangent bundle of X as follows. As a set, it is the disjoint union of all tangent spaces, that is, T X :=



Tx X.

x∈X

The projection π : T X → X is the mapping defined as v → x such that v ∈ Tx X. We will show that T X has a natural topology and a structure of smooth manifold induced by those of X. Let A = {φi : Ui → Ai }i∈I be an atlas in X. For each i ∈ I we define the subset U˜ i := π −1 (Ui ) ⊂ T X and the mapping φ˜ i : U˜ i → Ai × Fn given by φ˜ i (v) = (π(v), v(x1 ), . . . , v(xn )), where x1 , . . . , xn are the coordinate functions of φi on Ui . The mapping φ˜ i is bijective and its inverse is φ˜i−1 (u, w) =

x  i=1

 ∂  wi . ∂xi φ −1 (u) i

We consider in T X the topology whose basis is the family of subsets of the form φ˜ i−1 (A), where i ∈ I and A ⊂ Ai × Fn is any open subset. It follows that each U˜ i is open in T X and that each mapping φ˜ i is a homeomorphism. Proposition 2.7 The family A˜ = {φ˜ i : U˜ i → Ai × Fn }i∈I defines a smooth structure in T X such that the projection π : T X → X is a vector bundle of rank n. Proof For each i, j ∈ I , the mapping φ˜j ◦ φ˜ i−1 (u, w) = (φj ◦ φi−1 (u), du (φj ◦ φi−1 )(w)),

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is smooth and thus, A˜ defines a smooth structure in T X. Moreover, the projection π : T X → X is smooth since for each i ∈ I , we have φi ◦ π ◦ φ˜ i−1 (u, w) = u. The fact that π : T X → X is a vector bundle of rank n follows directly from the definitions.   Given a smooth mapping f : X → Y , the differential of f is the mapping df : T X → T Y such that df (v) = dx f (v), for each v ∈ Tx X. By construction, we have a commutative diagram TX

df

TY π

π

X

f

Y

Proposition 2.8 The pair (df, f ) is a morphism between the tangent bundles of X and Y . Proof We only need to prove that df : T X → T Y is smooth. Given any v ∈ T X, let x = π(v) ∈ X. There exist charts φ : U → A in X and ψ : V → B in Y with x ∈ U ⊂ f −1 (V ) such that f¯ = ψ ◦ f ◦ φ −1 : A → B is smooth. Consider the induced charts φ˜ : U˜ → A × Fn in T X and ψ˜ : V˜ → B × Fp in T Y . We have v ∈ U˜ ⊂ (df )−1 (V˜ ) and moreover, ψ˜ ◦ df ◦ φ˜ −1 (u, w) = (f¯(u), du f¯(w)),  

which is smooth.

Definition 2.12 A vector field on X is a section ξ : X → T X of the tangent bundle. The set of vector fields on X is denoted by θX and has a structure of O X -module. Given any chart φ : U → A in X, for each i = 1, . . . , n, the ith-coordinate vector  ∂ ∂  field is the vector field ∂xi ∈ θU given by the mapping x → ∂xi  . Since the x coordinate tangent vectors at a point form a basis of the tangent space, it follows that θU is a free O U -module of rank n generated by the coordinate vector fields. In particular, every vector field ξ ∈ θX can be written locally in a unique way as ξ |U =

n  i=1

ξi

∂ , ∂xi

2.5 Vector Fields and Flows

35

for some functions ξ1 , . . . , ξn ∈ OU , which are called components of ξ in the chart φ : U → A. Given a subset S ⊂ X, we can also consider the set of germs of vector fields on X at S. This set is denoted by θX,S and has a structure of OX,S module. Again, by taking charts, it follows that θX,S is a free OX,S -module of rank n generated by the germs of the coordinate vector fields ∂x∂ i ∈ θX,S , with i = 1, . . . , n. Definition 2.13 Given a smooth mapping f : X → Y , a vector field along f is a smooth mapping ξ : X → T Y such that π ◦ ξ = f , where π : T Y → Y is the canonical projection. The set of vector fields along f is denoted by θ (f ) and again has the structure of OX -module. For instance, if we consider ξ ∈ θX and η ∈ θY , then both df ◦ ξ and η ◦ f are vector fields along f . In general not all vector fields along f are of this type. The quotient of the set of all vector fields along f by the set obtained in this way plays a major rôle in the subject, as we will see in Chap. 3. Definition 2.14 Given a map f : X → Y , we say that vector fields ξ ∈ θX and η ∈ θY are f-related if df ◦ ξ = η ◦ f . Given any chart ψ : V → B in Y with coordinate functions y1 , . . . , yn , the coordinate vector fields along f are the vector fields ∂y∂ i ◦ f ∈ θ (f |U ), where U = f −1 (V ). As in the case of vector fields on a manifold, θ (f |U ) is a free O U -module of rank p generated by the coordinate vector fields. Any vector field ξ ∈ θ (f ) can be written locally in a unique way as ξ |U =

p 

ξi

i=1

∂ ◦f ∂yi

,

for some functions ξ1 , . . . , ξp ∈ OU . When we consider a smooth map-germ f : (X, S) → Y instead of a mapping, then we have the set of germs of vector fields along f which is also denoted by θ (f ). It is a free OX,S -module of rank p generated by the germs of the coordinate vector fields. Let X be a manifold and γ : D → X a smooth curve, where D ⊂ F is an open subset. The tangent vector field of γ is defined as γ˙ := dγ ◦

d ∈ θ (γ ). dt

Given a vector field ξ ∈ θX , we say that γ is an integral curve of ξ if ξ ◦ γ = γ˙ . If t0 ∈ D and γ (t0 ) = x0 , then we say that the integral curve γ passes through x0 at t = t0 . All the above definitions can be adapted easily to germs instead of mappings or vector fields. The following theorem is better stated in terms of germs, due to its local nature.

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2 Manifolds and Smooth Mappings

Theorem 2.4 Let ξ ∈ θX,S be a germ of vector field. There exists a unique smooth map-germ φ˜ : (X × F, S × {0}) → (X, S) with the following property: ˜ • for each x in a neighbourhood of S, the curve γx given by γx (t) = φ(x, t) is an integral curve of X passing through x at t = 0. Proof Take a chart ψ : (X, S) → (Fn , S  ), then we can write ξ=

n  i=1

ξi

∂ , ∂xi

for some functions ξ1 , . . . , ξn ∈ OX,S . We put Ai = ξi ◦ ψ −1 ∈ OFn ,S  , for each i = 1, . . . , n. On the other hand, any curve γ is expressed in the chart by its coordinate functions γ1 , . . . , γn . The condition that γ is an integral curve of ξ passing through x at t = 0 is equivalent to its being a solution of the following system of ordinary differential equations: γ˙i (t) = Ai (γ1 (t), . . . , γn (t)), i = 1, . . . , n, with initial conditions γi (0) = xi , i = 1, . . . , n (here x1 , . . . , xm are the coordinates of x in the chart). The existence and uniqueness of solutions of system of ordinary differential equations with initial conditions imply the existence and uniqueness of integral curves γx passing through a point x at t = 0. Moreover, the fact that the solutions depend smoothly on the initial condition implies that the mapping ˜ φ(x, t) = γx (t) is also smooth. (These results on systems of differential equations are well known for the real C ∞ case; a holomorphic version of these results can be found for instance in [Car95].)   Definition 2.15 The map-germ φ˜ : (X × F, S × {0}) → (X, S) obtained in Theorem 2.4 is called the local flow of ξ ∈ θX,S . Let us denote by  : (X × F, S × {0}) → (X × F, S × {0}) the map-germ ˜ (x, t) = (φ(x, t), t).

(2.2)

This map-germ is determined uniquely by the two following properties: 1. (x, 0) = (x, 0), for all x, 2.  ◦ ∂t∂ = ξ ◦ . It follows from item 1 that the differential of  at each (x, 0) is the identity, hence  is a diffeomorphism by the Inverse Mapping Theorem 2.1. Moreover,  is an

2.5 Vector Fields and Flows

37

unfolding of the identity in (X, S), that is, it makes commutative the following diagram: id

(X, S)

(X, S)

i

i

(X × F, S × {0})

(X × F, S × {0}) π

π

(F, 0)

(2.3)

where i(x) = (x, 0) and π(x, t) = t. Definition 2.16 The diffeomorphism  : (X × F, S × {0}) → (X × F, S × {0}) obtained from the local flow as in (2.2) is called the local unfolding of ξ ∈ θX,S . Fix a representative  : U × D → X × F which is a diffeomorphism onto its image, where U ⊂ X and D ⊂ F are open neighbourhoods of S and 0, respectively. ˜ For each t ∈ D, we denote by φt : U → X the mapping φt (x) = φ(x, t). Lemma 2.5 For all t ∈ D, φt (U ) ⊂ X is open and φt : U → φt (U ) is a diffeomorphism. Proof Let it : X → X × F be the mapping it (y) = (y, t). Since (U × D) is open in X × F, it follows that φt (U ) = it−1 ((U × D)) is open in X. Moreover, since the restriction  : U × D → (U × D) is a diffeomorphism, it has an inverse −1 : (U × D) → U × D. Necessarily, −1 will be of the form −1 (y, t) = ˜ ˜ (ψ(y, t), t) and it follows that ψt : φt (U ) → U given by ψt (y) = ψ(y, t) is the inverse of φt .   The 1-parameter family of diffeomorphisms φt : U → φt (U ), with t ∈ D, is also sometimes referred to as the local flow of the vector field ξ . The following property of flows is known as the Thom–Levine lemma. In [Lee13] it is called the property of naturality of flows. Let f : X → Y be a smooth mapping and consider ξ ∈ θX and η ∈ θY . For each x ∈ X, we take local flows of ξ, η at x, f (x), respectively. By shrinking the neighbourhoods if necessary, we can assume that the local flows are, respectively, of the form φt : U → φt (U ) and ψt : V → ψt (V ), with |t| <  and such that f (U ) ⊂ V . Lemma 2.6 Assume ξ, η are f -related. Then the following diagram is commutative for all t, with |t| < : f

U φt

φt (U )

V ψt

f

ψt (V )

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2 Manifolds and Smooth Mappings

Proof Given x ∈ U and t ∈ D, γx (t) = φt (x) is the integral curve of ξ passing through x at t = 0. That is, γx (0) = 0 and ξ(γx (t)) = γ˙x (t). We claim that (f ◦ γx )(t) is the integral curve of η passing through f (x) at t = 0. In fact, f (γx (0)) = f (x) and also η((f ◦ γx )(t)) = (η ◦ f )(γx (t)) = (df ◦ ξ )(γx (t)) = df (ξ(γx (t))) = df (γ˙x (t)) = (f ◦˙ γx )(t). But this implies that ψt (f (x)) = (f ◦˙ γx )(t) = f (γx (t)) = f (φt (x)).

 

When f is a diffeomorphism, then for each ξ ∈ θX there exists a unique η ∈ θN which is f -related to ξ and which is given by η = df ◦ ξ ◦ f −1 . This vector field is called the push-forward of ξ by f and is denoted by f∗ ξ . Unfortunately, the construction of local flow and local unfolding of a vector field is not enough for our purposes. We need these notions for a more general class, namely the time-dependent vector fields. We explain how the constructions of the local flow and the local unfolding can be adapted easily for this type of vector fields just by adding an extra parameter. By definition, a time-dependent vector field on X is a smooth mapping ξ˜ : X × D → T X, where D ⊂ F is open, such that for each t ∈ D, the mapping ξt : X → T X given by ξt (p) = ξ˜ (p, t) is a vector field on X. Given a time-dependent vector field ξ˜ , we can associate another vector field ξ ∈ θX×D just by setting ξ := ξ˜ +

∂ , ∂t

where we identify T (X × D) with T X ⊕ T D, and ∂/∂t is the coordinate vector field on D. Conversely, if ξ ∈ θX×D is such that dt (ξ ) = ξ(t) = 1, then we can associate a unique time-dependent vector field ξ˜ just by subtracting ∂/∂t to ξ . Hence, the set of time-dependent vector fields on X can be identified canonically with the subset of vector fields ξ ∈ θX×D such that ξ(t) = 1. Now, given a point x ∈ X, consider (x, 0) ∈ X × D. The local flow of ξ at (x, 0) is a map ψ˜ : U × B (0) × Bδ (0) → X × D, where U is an open neighbourhood of x in X and , δ > 0. Of course, we can assume without loss of generality that ˜  = δ and since X(t) = 1, we have ψ(x, t, s) = (ψ˜ 1 (x, t, s), s), where ψ˜ 1 is the ˜ We define φ˜ : U × B (0) → X as first component of ψ. ˜ φ(x, t) = ψ˜ 1 (x, t, t), and we call it the local flow of the time-dependent vector field X˜ at p. Now, the construction of the 1-parameter family of diffeomorphisms φt : U → φt (U ) and of the local unfolding (x, t) = (φt (x), t) follows the same steps as in the case of a vector field.

2.5 Vector Fields and Flows

39

Exercises for Sect. 2.5 1 Let X be a smooth manifold. We have defined a vector field v (a map assigning a tangent vector v(x) ∈ Tx X to each point x ∈ X) to be smooth if it is smooth as a section of the tangent bundle. Show that this is equivalent to the following characterisation: v is smooth at x if for all f ∈ OX,x , the germ v · f is smooth. 2 Let π : E → X be a vector bundle on a manifold X. The dual vector bundle is defined as π  : E ∗ → X, where  E ∗ := Ex∗ , x∈X

where Ex∗ is the dual vector space of Ex and π  is defined in the obvious way. Show that π  : E ∗ → X is indeed a vector bundle on X. 3 The cotangent bundle of X is defined as the dual of the tangent bundle, that is, T ∗ X := (T X)∗ . Let f : X → F be a smooth function. For each x ∈ X, we denote by dx f ∈ Tx∗ X the linear form given by dx f (v) = v(f ), for all v ∈ Tx X. Show that the mapping df : X → T ∗ X given by x → dx f is a section of the cotangent bundle (this is usually called the differential 1-form of f ). 4 Let f : (X, x0 ) → (Y, y0 ) be an immersion. Show that for each ξ ∈ θX,x0 there exists η ∈ θY,y0 such that ξ and η are f -related. 5 Let f : (X, x0 ) → (Y, y0 ) be a submersion. Show that for each η ∈ θY,y0 there exists ξ ∈ θX,x0 such that ξ and η are f -related. 6 Find the local flow of the following vector fields on F2 : ∂ ∂ (a) x ∂x + y ∂y , ∂ ∂ + x ∂y . (b) −y ∂x

Draw also a picture of these vector fields and of their local flows in the case F = R. 7 Let ξ ∈ θX have local flow φt : U → φt (U ), t ∈ D. Show that: (a) φs ◦ φt = φs+t in the subset of U where the composition φs ◦ φt is defined; (b) the inverse of φt is φ−t . 8 Let ξ ∈ θX have local flow φt : U → φt (U ), t ∈ D. Let x ∈ U be a point such that X(x) = 0. Show that φt (x) = x, for all t ∈ D. 9 Let ξ ∈ θX×D be a vector field such that ξ(t) = 1. Show that the local unfolding of the associated time-dependent vector field at any point x ∈ X is characterised uniquely by the following two conditions: (a)  is an unfolding of the identity (see (2.3)), (b) ξ = ∗ (∂/∂t).

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10 Let  : (X × F, S × {0}) → (X × F, S × {0}) be a diffeomorphism which is an unfolding of the identity (see (2.3)). Write (x, t) = (φt (x), t) and assume that φs ◦φt = φs+t , whenever the composition is defined. Show that there exists ξ ∈ θX,S whose local unfolding is .

2.6 Transversality The Regular Value Theorem 2.3 says that if y ∈ Y is a regular value of a smooth mapping f : X → Y , then the inverse image f −1 (y) is a submanifold of X. One can ask when the inverse image f −1 (Z) of a submanifold Z ⊂ Y is also a submanifold of X. It turns out that the answer is provided by the notion of transversality. Definition 2.17 Let f : X → Y be a smooth mapping between manifolds and Z ⊂ Y a submanifold. Given a point x ∈ X such that f (x) ∈ Z, we say that f is transverse to Z at x if dx f (Tx X) + Tf (x) Z = Tf (x)Y.

(2.4)

We say that f is transverse to Z it is transverse at every point x ∈ X such that f (x) ∈ Z. Note that in particular, if f −1 (Z) = ∅, then f is transverse to Z automatically. We use the symbol f −  Z to indicate that f is transverse to Z. The following properties are immediate. 1. If Z ⊂ Y is an open submanifold, then f −  Z for every smooth mapping f : X → Y. 2. If f : X → Y is a submersion, then f −  Z for every submanifold Z ⊂ Y . 3. For a single point y ∈ Y , f : X → Y is transverse to {y} if and only if y is a regular value of f . 4. When dim X < codim Z then f −  Z if and only if f (X) ∩ Z = ∅. The next theorem extends the Regular Value Theorem 2.3 to transverse preimages. Theorem 2.5 Let f : X → Y be a smooth mapping between manifolds and Z ⊂ Y a submanifold such that f −  Z. Then, provided it is not empty, f −1 (Z) is a submanifold of X with codim f −1 (Z) = codim Z and moreover, for each x ∈ f −1 (X), Tx f −1 (Z) = (dx f )−1 (Tf (x) Z). Proof By Corollary 2.3, we only have to prove that (f −1 (Z), x) is a submanifold of (X, x) for each x ∈ f −1 (Z). Let y = f (x) and let g1 , . . ., gk be regular equations

2.6 Transversality

41

for Z in some neighbourhood U of y. From the hypothesis of transversality we have T0 Fk = dy g(Ty Y ) = dy g(dx f (Tx X) + Ty Z). Since Ty Z = ker dy g, the right-hand side reduces to dy g(dx f (Tx X)), showing that g ◦ f is a submersion at x. Thus 0 is a regular value of g ◦ f , and so (g ◦ f )−1 (0) is a smooth submanifold of f −1 (U ), of codimension k, by the Regular Value Theorem 2.3. But (g ◦ f )−1 (0) = f −1 (Z) ∩ f −1 (U ). The statement on tangent spaces holds also by the regular value theorem: Tx (f −1 (Z)) = ker dx (g ◦ f ) = (dx f )−1 (ker dy g) = (dx f )−1 (Ty Z).   Definition 2.18 Let X, Z ⊂ Y be submanifolds and let x ∈ X ∩ Z. We say that X and Z are transverse at x if the inclusion i : X → Y is transverse to Z at x. We say that X and Z are transverse if they are transverse at every point of X ∩ Z. Since the differential of the inclusion is the inclusion of tangent spaces, the transversality of X and Z at x means that Tx X + Tx Z = Tx Y. In particular, the condition is the same if we interchange the order of the submanifolds X, Z. We will represent the transversality between X and Z by the symbol X−  Z. The following corollary is an immediate consequence of Theorem 2.5. Corollary 2.4 Let X, Z ⊂ Y be transverse submanifolds. Then X ∩ Z is a submanifold, codim(X ∩ Z) = codim X + codim Z and for each x ∈ X ∩ Z, we have Tx (X ∩ Z) = Tx X ∩ Tx Z. Transversality is a crucial notion. A transversal intersection of submanifolds is “stable”—move either one slightly, and the transversality persists, and the intersection does not change significantly. Transversality is also a “dense” property: a non-transverse intersection can be perturbed an arbitrarily small amount to become transverse (see Fig. 2.1). These simple facts are at the root of the notions of stability and stable perturbation that are central in singularity theory. They are made more precise in the elementary transversality theorem, Exercise A.3 in Appendix A, and in Exercise B.1.1 in Appendix B. We recall that N ⊂ Rk is a null subset if for each  > 0 there exists a countable ∞ family of cuboids {Ci }∞ such that N ⊂ i=1 Ci and moreover i=1 ∞  i=1

vol(Ci ) < .

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2 Manifolds and Smooth Mappings

Non-transverse

Transverse

Transverse

Fig. 2.1 Transverse perturbations of a non-transverse intersection

By a cuboid we mean any product of closed intervals C = [a1 , b1 ] × · · · × [ak , bk ] with ai < bi for i = 1, . . . , k. Its volume is vol(C) = (b1 − a1 ) . . . (bk − ak ). In the complex case, we say that N ⊂ Ck is a null subset if it is a null subset when considered as a subset of R2k . Definition 2.19 We say that N ⊂ X is a null subset of a manifold X if for each x ∈ N, there exists a chart φ : U → A on X, with x ∈ U , such that φ(U ∩ N) is a null subset of Fn . Theorem 2.6 (Sard’s Theorem) f : X → Y is a null subset of Y .

The discriminant  of a smooth mapping

A proof for the C ∞ real version of Sard’s theorem can be found, for instance, in [GG73, Theorem 1.12]. The complex holomorphic case follows from the real one, since the discriminant of a holomorphic mapping coincides with the discriminant when considered as a C ∞ real mapping. It is common to say that a certain property holds “for almost every” point in a manifold if the property holds outside a null subset. Thus, Sard’s theorem can be also stated by saying that almost every y ∈ Y is a regular value of f . The following result, known as the basic transversality lemma, is an easy consequence of Sard’s theorem; it is a key ingredient in the proof of the Thom Transversality Theorem A.1, and, thus, in the proof of all genericity theorems , such as the density of stable mappings in the nice dimensions discussed in Sect. 5.2. Given a mapping F : X × S → Y , for each s ∈ S we denote by fs : X → Y the mapping fs (p) = F (p, s). Theorem 2.7 (Basic Transversality Lemma) Let F : X × S → Y be a smooth mapping and Z ⊂ Y a submanifold. If F −  Z, then fs −  Z for almost every s ∈ S. Proof By Theorem 2.5, W = F −1 (Z) is a submanifold of X × S. Let π : W → S be the restriction of the natural projection from X × S into S. We will show that  Z whenever s is a regular value of π. Thus, the result follows from Sard’s fs − Theorem 2.6.

2.6 Transversality

43

Suppose first that s is a regular value of π and let x ∈ X such that fs (x) = y ∈ Z. Since F −  Z, we have d(x,s)F (T(x,s)(X × S)) + Ty Z = Ty Y. That is, given any a ∈ Ty Y there exists b ∈ T(x,s)(X × S) such that d(x,s)F (b) − a ∈ Ty Z. We want to find a vector v ∈ Tx X such that dx (fs )(v) − a ∈ Ty Z. Since T(x,s)(X × S) = Tx X × Ts S, we can write b = (w, e) with w ∈ Tx X and e ∈ Ts S. Moreover, π is the restriction of the projection onto the second component, hence its differential d(x,s)π : T(x,s)W → Ts S is also the restriction of the projection of Tx X × Ts S into Ts S. Since s is a regular value of π, d(x,s)π is an epimorphism and hence, there exists u ∈ Tx X such that (u, e) ∈ T(x,s)W . On the other hand, since W = F −1 (Z), its tangent space is T(x,s)W = (d(x,s)F )−1 (Ty Z), so d(x,s)F (u, e) ∈ Ty P . Let v = w − u ∈ Tx X, then: dx (fs )(v) − a = d(x,s)F (v, 0) − a = d(x,s)F [(w, e) − (u, e)] − a = d(x,s)F (b) − a − d(x,s)F (u, e) ∈ Ty Z.  

Exercises for Sect. 2.6 1 Suppose that f : X → Y is a smooth map of manifolds, Z ⊂ Y is a submanifold, and x0 ∈ X such that f (x0 ) ∈ Z. Suppose also that Z has regular defining equation g : U → Fc , where U is a neighbourhood of f (x0 ). Show that f−  Z at x0

⇐⇒

g ◦ f is a submersion at x0 .

This was already used in the proof of Theorem 2.5. The condition on the right is often easier to check than the definition of transversality. 2 Let X be a manifold of dimension n and let N ⊂ X be a null subset. Show that for every x ∈ N and for every chart φ : U → A with x ∈ U , φ(N ∩ U ) is a null subset of Fn .

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2 Manifolds and Smooth Mappings

3 Let f : X → Fp be a smooth mapping and Z ⊂ Fp a submanifold. Show that for almost every a ∈ Fp , the mapping f + a : X → Fp is transverse to Z. Hint: consider the family F : X × Rp → Rp defined by F (x, a) = f (x) + a. 4 Let f : X → F be a smooth function. A critical point x0 ∈ X of f is called non-degenerate if given a chart φ : (X, x0 ) → (Fn , u0 ), the Hessian determinant of f ◦ φ −1 is not zero at u0 , that is, det

∂ 2 (f ◦ φ −1 ) (u0 ) =

0. ∂ui ∂uj

Show that the definition of non-degenerate critical point does not depend on the choice of the chart φ. 5 Let x0 be a critical point of f : X → F. Show that x0 is non-degenerate if and only if the differential 1-form df : X → T ∗ X is transverse to the zero section Z ⊂ T ∗ X at x0 (see Exercise 2.5.3). 6 A Morse function is a smooth function f : X → F whose critical points are all non-degenerate and have pairwise distinct critical values. Suppose that X ⊂ FN . Show (i) that for almost every a ∈ FN , the function fa : X → F given by fa (x) = f (x) + a1 x1 + · · · + aN xN , has only non-degenerate critical points, and (ii) for almost all a ∈ Fn , fa is a Morse function.

2.7 Local Conical Structure A further justification for the use of the notion of germ in singularity theory comes from the fact that analytic spaces are “locally conical”. That is, for every point x0 in a closed analytic subset X of some open set in Cn (or, indeed, in a closed semialgebraic subset of an open set of Rn ), the intersection of X with a sufficiently small ball Bε (x0 ) centred at x0 is homeomorphic to the cone on its boundary, X∩∂Bε (x0 )—see Theorem B.7 in Sect. B.3. The locally conical structure is particularly important in the definition of the vanishing homology, so we discuss it in some detail in Sect. B.3.

Chapter 3

Left-Right Equivalence and Stability

As in Chap. 2, by a smooth map between manifolds we mean either a C ∞ -map between C ∞ -manifolds when F = R or a holomorphic map between complex manifolds when F = C. In the complex case a smooth map with smooth inverse is usually called “bianalytic”; we prefer the term “diffeomorphism” for both the real and the complex case. Definition 3.1 Let fi : (Xi , Si ) → (Yi , yi ), i = 1, 2 be germs of smooth maps between manifolds. They are 1. right-equivalent, if there exists a germ of diffeomorphism ϕ : (X1 , S1 ) → (X2 , S2 ) such that f2 = f1 ◦ ϕ −1 ; 2. left-equivalent, if there exists a germ of diffeomorphism ψ : (Y1 , y1 ) → (Y2 , y2 ) such that f2 = ψ ◦ f1 ; 3. left-right-equivalent, if there exist germs of diffeomorphism ϕ : (X1 , S1 ) → (X2 , S2 ) and ψ : (Y1 , y1 ) → (Y2 , y2 ) such that f2 = ψ ◦ f1 ◦ ϕ −1 . By taking charts in X and Y , any germ from X to Y is right-left equivalent to a germ f : (Fn , S) → (Fp , 0), so for our purposes it is enough to consider this class of germs. The advantage when the source and target are fixed is that the equivalences can be seen as group actions. Let A = Diff(Fn , S) × Diff(Fp , 0) be the group of pairs of diffeomorphisms. We have an action of A on the set of germs f : (Fn , S) → (Fp , 0), given by (ϕ, ψ) · f = ψ ◦ f ◦ ϕ −1 . Analogously, we can consider the groups R = Diff(Fn , S) and L = Diff(Fp , 0) and the corresponding actions. If G = R, L or A , we say that f1 , f2 are G equivalent if they are in the same G -orbit. In this situation, we will use the terms R, L or A -equivalences instead of right, left or right-left equivalences, respectively. For a very good survey of these groups and their actions, see [Wal81]. © Springer Nature Switzerland AG 2020 D. Mond, J. J. Nuño-Ballesteros, Singularities of Mappings, Grundlehren der mathematischen Wissenschaften 357, https://doi.org/10.1007/978-3-030-34440-5_3

45

46

3 Left-Right Equivalence and Stability

A big part of singularity theory has always been concerned with the problem of classification. Generally one classifies germs of smooth maps (Fn , S) → (Fp , 0) up to A -equivalence, and up to R-equivalence if p = 1. A key ingredient in classification is the notion of finite determinacy. For a smooth germ f : (Fn , S) → (Fp , 0) where S = {x1 , . . . , xr }, and for each k ≥ 0, we denote by j k f the k-jet of f at S, that is, j k f = (j k f (x1 ), . . . , j k f (xr )), where j k f (xi ) is the k-jet of f at xi (see Appendix A for the definition of j k f (xi )). Definition 3.2 Let f : (Fn , S) → (Fp , 0) be a smooth germ, and let G be one of the groups listed above, or a subgroup. We say f is k-determined for G -equivalence if whenever the k-jet at S of another germ g coincides with that of f , then f, g are G -equivalent, and finitely determined if it is k-determined for some k ∈ N. In [Mat68b], John Mather showed that for all of the groups listed above, in the complex analytic case, finite determinacy is equivalent to isolated instability. We will prove this for R and A . The key is to understand how to construct diffeomorphisms. In all of singularity theory (except where the inverse and implicit function theorems apply), the diffeomorphisms we use arise as the local flow of vector fields, as described in Sect. 2.5.

3.1 Classification of Functions by Right Equivalence The Thom–Levine Lemma 2.6 shows how an infinitesimal condition gives rise to a family of diffeomorphisms. The equality df ◦ ξ = η ◦ f of Definition 2.14 is linear in ξ and η, and these vector fields can often be constructed by the methods of commutative algebra. This is the entry-point of commutative algebra, which, through it, has a huge input into singularity theory. As an example of what is involved, let us prove the simplest of the determinacy theorems of John Mather [Mat68b], also proved independently by Jean-Claude Tougeron in [Tou68]. We will return to the theme of finite determinacy, this time for A -equivalence, in Chap. 6. Let On := OFn ,0 be the local ring of germs of smooth functions f : (Fn , 0) → F, with maximal ideal mn := mFn ,0 . In the real C ∞ case On is often denoted by En , but in order not to have to distinguish between real and complex except when necessary, we prefer to use On in both cases. Remark 3.2 and Proposition 3.8 below will justify treating the two cases together in this way. If f ∈ O n , then the ideal in O n generated by the first order partial derivatives ∂f/∂x1 , . . ., ∂f/∂xn is called the Jacobian ideal and denoted Jf . Theorem 3.1 Let f ∈ On . (i) Suppose that f is k-determined for R-equivalence. Then mn Jf ⊃ mk+1 .

3.1 Classification of Functions by Right Equivalence

47

(ii) Conversely, suppose that mn Jf ⊃ mk .

(3.1)

Then f is k-determined for R-equivalence. Notice that different powers of mn appear in (i) and in (ii). It is not true that mn Jf ⊃ mk+1 n

⇒

f is k-determined for R.

Example 3.1 The Morse Lemma, which gave rise to Morse theory in topology (see e.g. [Mil63]), states that if a C ∞ function f : M n → R has a non-degenerate critical point at x0 (i.e. the Hessian determinant of f at x0 is non-zero) then in suitable coordinates around x0 , f takes the form f (x1 , . . ., xn ) = f (x0 ) −

k  i=0

xi2 +

n 

xi2 .

(3.2)

i=k+1

This follows easily from Theorem 3.1. For as Exercise 3.1.2 shows, non-degeneracy is equivalent to having Jf = mn , so that mn Jf = m2n and, by the theorem, f is 2-determined for right equivalence, and is thus right-equivalent to f (x0 ) + non-degenerate quadratic form. The form given in (3.2) then follows from the classification of quadratic forms. ♦ Proof of Theorem 3.1 (i) Let h ∈ mk+1 n . Then since f is k-determined for R, for all t there exists ϕt ∈ Diff(Fn , 0) such that f + th = f ◦ ϕt . If we could assume the existence of a smoothly parameterised family of diffeomorphisms ϕt with ϕ0 = id such that f ◦ ϕt = f + th then we could reason as follows:

d(f ◦ ϕt )  ∂f d(f + th) ∂ϕt,i = = . ◦ ϕt h= dt dt ∂xi ∂t

(3.3)

i

Note that since ϕt (0) = 0 for all t it follows that ∂ϕt,i /∂t ∈ mn . When t = 0, since ϕ0 = id, this gives h=

 ∂f ∂ϕt,i ∈ mn Jf ∂xi ∂t

(3.4)

i

so that mk+1 ⊂ mn Jf as required. n However, our hypothesis does not allow us immediately to assert that the diffeomorphisms ϕt fit together to give a smooth family. So instead we look in

48

3 Left-Right Equivalence and Stability

jet space J k+1 (n, 1) = mn / mk+2 n . As f is k-determined, the set k+1 (n, 1) L := {j k+1 (f + h) : h ∈ mk+1 n }⊂J

lies entirely in the R (k+1) -orbit of f , where R (k+1) is the finite-dimensional quotient of Diff(Fn , 0) acting on jet space. Now R (k+1) can be identified with the set {j k+1 ϕ(0) : ϕ ∈ Diff(Fn , 0)} and has a natural structure of algebraic group: the composite of two polynomial mappings depends polynomially on their coefficients, and in R (k+1) one composes and then truncates at degree k + 1. This group acts algebraically on J k+1 (n, 1). Thus, as the set L lies in the orbit of j k+1 f (0), writing z = j k+1 f (0), and R (k+1)z for the R (k+1)-orbit of z, one has mk+1 n mk+2 n

= Tz L ⊂ Tz (R (k+1)z) =

mn Jf + mk+2 n mk+2 n

,

(3.5)

and thus mk+1 ⊂ mn Jf + mk+2 . n n The conclusion we want follows by Nakayama’s Lemma C.1, which we apply taking (O n , mn ) as (R, m), mkn as M and mn Jf as N. The second equality in (3.5) is important and not completely obvious. It can be obtained along the lines of the argument leading up to (3.4), but using the crucial fact that if the Lie group G acts on the manifold M and for x ∈ M we denote by αx the orbit map g ∈ G → gx, then for each x ∈ M with smooth orbit Gx, Tx Gx = de αx (Te G). Now

de αx (Te G) =

   d (γ (t) · x) : γ : (F, 0) → (G, e) is a curve germ ; dt t =0

every curve in (R (k+1), id) is of the form j k+1 ϕt for a 1-parameter family of diffeomorphisms ϕt , so now it really is true that Tz R (k+1) z is equal to

   d k+1 n  j (f ◦ ϕt ) : ϕt is a 1-parameter family in Diff(F , 0) with ϕ0 = id , dt t=0

3.1 Classification of Functions by Right Equivalence

49

which is equal to 

  d j k+1 : ϕt is a 1-parameter family in Diff(Fn , 0) with ϕ0 = id . (f ◦ ϕt ) dt t=0

This shows that (3.4) holds even without the assumption that there is a smoothly parameterised family of diffeomorphisms ϕt such that f + th = f ◦ ϕt . (ii) Suppose that g has the same degree k Taylor polynomial as f . Then g − f ∈ mk+1 n . Let F (x, t) = f (x) + t (g(x) − f (x)), and write ft (x) = F (t, x). The idea of the proof is to show that for each value t0 of t, there is a neighbourhood U (t0 ) of t0 in F such that ft and ft0 are R-equivalent for all t ∈ U (t0 ). Thus, the set of t such that ft is R-equivalent to f is open, and so is its complement. Since F is connected, f = f0 and g = f1 are R-equivalent. We construct the set Ut0 first for t0 = 0. As F is a function of the n + 1 variables x1 , . . ., xn , t, we consider the germ F ∈ O n+1 . Notice that ∂F /∂t = g − f ∈ Mnk+1 , where by Mn we mean the ideal in O n+1 generated by (x1 , . . ., xn ). Because mn Jf ⊃ mkn ,

∂f ∂f . , . . ., ∂x1 ∂xn

(3.6)

∂F ∂F . , . . ., ∂x1 ∂xn

(3.7)

∂F ∂F ∂F = ξ1 + · · · + ξn ∂t ∂x1 ∂xn

(3.8)

∂F ∈ Mn ∂t



We would like to show ∂F ∈ Mn ∂t



For if we have

for some functions ξi ∈ Mn , then defining a germ of vector field ξ on (Fn+1 , 0) by ξ=

 ∂ ∂ − ξi , ∂t ∂xi i

(3.8) becomes dF (ξ ) = 0. This is exactly the condition of Definition 2.14 with η = 0. Let (x, t) = (ϕt (x), t) be the integral flow of the vector field ξ . Since ξi ∈ Mn for i = 1, . . ., n, we have (0, t) = (0, t) for all t. The integral flow of the zero vector field is the identity

50

3 Left-Right Equivalence and Stability

(0,t)

(0,0)

Fig. 3.1 The arrows show the vector field ξ of the proof. At all points of the t-axis, the vector field is tangent to the axis, so any trajectory beginning at a point on the axis remains on the axis. Thus ϕt (0) = 0

map, and therefore by the Thom–Levine lemma we have ˜ = F. F ◦

(3.9)

Since the component of ξ in the t-direction is 1, it follows that ϕt maps Fn × {0} to Fn × {t}. Restricting both sides of (3.9) to Fn × {0} we therefore get ft ◦ ϕt = f, with ϕt (0) = 0 (see Fig. 3.1). Now take representatives of the germs involved in (3.9). Since the germs on the left- and right-hand sides are equal, it follows that the representatives coincide on some neighbourhood of (0, 0) in Fn × F. We deduce that there exists η > 0 such that for |t| < η, the germ of ft is R-equivalent to the germ of f . Now we set about deducing (3.7) from (3.6). Since ∂F /∂t = g − f ∈ Mnk+1 , to show (3.7), it will be enough to show Mnk+1

⊂ Mn

∂F ∂F , . . ., . ∂x1 ∂xn

(3.10)

Because ∂F ∂(g − f ) ∂f = −t ∂xi ∂xi ∂xi

(3.11)

3.1 Classification of Functions by Right Equivalence

51

it follows that ∂F ∂f ≡ mod mn+1 Mnk ∂xi ∂xi and therefore Mnk+1 ⊂ Mn

∂f ∂f , . . ., ∂x1 ∂xn

⊂ Mn

∂F ∂F , . . ., ∂x1 ∂xn

+ mn+1 Mnk+1 .

(3.12)

Again, Nakayama’s Lemma C.1  comes to our  aid. We apply it taking (O n+1 , mn+1 ) ∂F ∂F , . . ., as (R, m), Mnk as M and Mn ∂x ∂xn as N to conclude that 1 Mnk+1

⊂

∂F ∂F , . . ., ∂x1 ∂xn

.

(3.13)

This completes the proof that the deformation f + t (g − f ) is trivial for t in some neighbourhood of 0. The remainder of the proof involves showing that the same procedure can be employed for every value of t: we want to show that for any t0 the deformation f + t (g − f ) is trivial in a neighbourhood of t0 . This deformation can be written in the form (f + t0 (g − f )) + (t − t0 )(g − f ), and taking as new parameter s = t −t0 , the problem reduces to what we have already discussed, except that instead of our original f we now have a new function, ft0 := f + t0 (g − f ). In order that our earlier argument should apply, we have to show that ft0 also satisfies the hypothesis of this argument: that mkn ⊂ m Jft0 .

(3.14)

Once again this is done by a simple argument involving Nakayama’s Lemma: we have ∂f ∂ft0 ≡ mod mkn ∂xi ∂xi and so mkn ⊂ mn Jf ⊂ mn Jft0 + mn mkn , from which (3.14) follows by Nakayama’s Lemma. The space

  ∂ft  : f0 = f ∂t t =0

 

52

3 Left-Right Equivalence and Stability

of derivatives of one-parameter deformations of f is evidently equal to O n , since each g ∈ O n gives a one-parameter  deformation ft = f + tg. If we insist that t ft (0) = 0 for all t then clearly ∂f ∂t t =0 ∈ mn . In the first part of the proof of Theorem 3.1 we saw that if ft = f ◦ ϕt is a deformation of f obtained by composing it with a one-parameter family of diffeomorphisms ϕt with ϕ0 = id, then   n  dft  dϕi,t  ∂f = .   dt t =0 dt t =0 ∂xi i=1

If ϕt (0) = 0 for all t, then we can think of ϕt as a path in the group R = Diff(Fn , 0), and of f ◦ϕt as a path in the R-orbit of f , passing through f itself when t = 0. Thus d(f ◦ϕt )  dt t =0 is (heuristically only—the R-orbit of f in mn is not really a manifold) a tangent vector to that orbit at f . n t The derivative dϕ dt t =0 defines a vector field germ on (F , 0); to each point x it associates the tangent vector at x to the trajectory ϕt (x). The condition that  dϕi,t  ∈ mn for i = 1, . . ., n, and ϕt (0) = 0 for all t means that in particular dt  t =0   t thus df ∈ mn Jf . Conversely, if ξ1 , . . ., ξn ∈ mn then the germ at 0 of dt  t =0

ϕt (x) = x + t (ξ1 , . . ., ξn )   t = i ξi ∂f/∂xi . Thus we have shown is a diffeomorphism for t near 0, with df dt  t =0

Proposition 3.1   dft  : f0 = f = O n dt t =0  : f0 = f, ft (0) = 0 for all t = mn

 dft  dt t =0  

d(f ◦ ϕt )  : ϕ is a path in R, ϕ = id = mn Jf . t 0  dt t =0

This provides the heuristic for the following definition. Definition 3.3 Given f ∈ On we define the extended R-tangent space T Re f and the R-tangent space T Rf , respectively, by T Re f = Jf ,

T Rf = mn Jf ,

3.1 Classification of Functions by Right Equivalence

53

and we define quotients TR1 e f =

On , Jf

TR1 f =

mn , mn Jf

and the Re -codimension and Re -codimension of f , respectively, as codimRe f = dimF TR1 e f,

codimR f = dimF TR1 f.

Proposition 3.2 Let f ∈ On . The following statements are equivalent: 1. 2. 3. 4.

codimR f < ∞, codimRe f < ∞, mkn ⊂ Jf , for some k ∈ N, f is finitely determined for R-equivalence.

Proof We first claim that if I ⊂ On is any ideal, then dimF On /I < ∞ if and only if mkn ⊂ I , for some k ∈ N. In fact, if mkn ⊂ I , then dimF On /I ≤ dimF On /mkn < ∞, since On /mkn is generated over F by all the monomials of degree < k. For the converse, we consider the sequence of ideals I + mn ⊃ I + m2n ⊃ · · · ⊃ I. If dimF On /I < ∞ the sequence must stop at some point, so I + mkn = I + mk+1 n , k ⊂ I by Nakayama’s for some k ≥ 1. In particular, mkn ⊂ I + mk+1 and hence m n n Lemma C.1. Now we see the equivalences: (2) ⇐⇒ (3) This follows directly from the claim when I = Jf . (1) ⇐⇒ (3) If codimR f < ∞, then dimF On /mn Jf = dimF mn /mn Jf + 1 < ∞ and hence, mkn ⊂ mn Jf ⊂ Jf for some k. Conversely, if mkn ⊂ Jf , then mk+1 ⊂ mn Jf and thus, dimF mn /mn Jf = dimF On /mn Jf − 1 < ∞. n (4) ⇐⇒ (3) If f is finitely determined, then mk+1 ⊂ mn Jf ⊂ Jf for some n k, by Theorem 3.1. Moreover, if mkn ⊂ Jf , then mk+1 ⊂ mn Jf and hence, f is n (k + 1)-determined, again by Theorem 3.1.   The extended codimension codimRe f is usually called the Milnor number of f and it is denoted by μ(f ). We say that f is R-finite if μ(f ) < ∞. The R-finiteness has now the following geometrical interpretation: Corollary 3.1 Let f ∈ On . If 0 < μ(f ) < ∞ then f has an isolated singularity at 0. When F = C we also have the converse, that is, if f has an isolated singularity at 0 then 0 < μ(f ) < ∞. Proof If 0 < μ(f ) < ∞ then mkn ⊂ Jf ⊂ mn for some k, by Proposition 3.2. The inclusion Jf ⊂ mn implies that 0 is a singular point of f . The other inclusion

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3 Left-Right Equivalence and Stability

mkn ⊂ Jf gives xik

n 

=

aij

j =1

∂f , ∂xj

∂f for some aij ∈ On . If ∂x = 0 for all j , then xik = 0 for all i and hence, x = 0. j Assume now that F = C. If f has an isolated singularity at 0, then V (Jf ) = {0}, where V (Jf ) is the zero locusof the Jacobian ideal.  By the Rückert Nullstellensatz (see Theorem D.4), we have Jf = mn , where Jf is the radical of Jf . But this implies mkn ⊂ Jf ⊂ mn for some k and hence, 0 < μ(f ) < ∞.  

In the real case, it is not true in general that if f : (Rn , 0) → R has isolated singularity, then it is R-finite, even if f is real analytic. When f is real analytic, we can consider its complexification fC : (Cn , 0) → C. Proposition 3.3 Let f : (Rn , 0) → R be real analytic. Then μ(f ) < ∞ if and only if μ(fC ) < ∞. Moreover, if μ(f ) < ∞ then TR1 e fC ∼ = TR1 e f ⊗R C and in particular μ(f ) = μ(fC ). Proof In order to distinguish between the real and complex cases, we use here the notations (En , mn ) and (On , mn ) for the local rings of C ∞ -germs (Rn , 0) → R and complex analytic germs (Cn , 0) → C, respectively. If μ(f ) < ∞, then mkn ⊂ Jf for some k. This implies that for any multi-index α, with |α| = k, xα =

n 

ai

i=1

∂f , ∂xi

for some ai ∈ En . Replacing each ai by its k-jet, we get a polynomial version of these equalities modulo mk+1 n . These equalities are also valid if we take the corresponding complexifications, giving the inclusion mn k ⊂ JfC + mn k+1 . By Nakayama’s Lemma C.1, mn k ⊂ JfC and hence, μ(fC ) < ∞. Conversely, if μ(fC ) < ∞, then mn k ⊂ JfC . As above, for each α, with |α| = k, xα =

n  i=1

ai

∂fC , ∂xi

for some aij ∈ On . Since the partial derivatives have real coefficients, the same equalities hold with ai∗ , the function obtained from ai by complex conjugating the coefficients in its Taylor expansion, in place of ai . It follows that 1 ∂fC (ai + ai∗ ) , x = 2 ∂xi n

α

i=1

3.1 Classification of Functions by Right Equivalence

55

where now the functions ai + ai∗ have real coefficients. We deduce that mkn ⊂ Jf and thus, μ(f ) < ∞. For the second part, take k such that mkn ⊂ Jf and mn k ⊂ JfC . The polynomial complexification gives an isomorphism En On ⊗R C ∼ = k, mkn mn which maps (Jf /mkn ) ⊗R C onto JfC /mn k . Therefore, μ(f ) = μ(fC ).

 

Example 3.2 Let f : (R2 , 0) → R be the function f (x1 , x2 ) = (x12 + x22 )2 . The singular locus of f is given by the equation x12 +x22 = 0. This gives only the origin in R2 , so f has isolated singularity. However, the same equation in C2 gives the union of two lines passing through the origin and thus, fC has not isolated singularity. By Corollary 3.1 and Proposition 3.3, f is not R-finite. ♦ The heuristic justification of the extended tangent space is less immediately clear than that of T Rf ; it can be obtained by the same argument as T Rf if we remove the requirement that ϕt (0) = 0 for all t. This loosening of requirements is exactly what is needed “in the field”, where there is no distinguished base-point. For both versions, something precise is true, nevertheless. We have already seen, in the first part of the proof of Theorem 3.1, that modulo mk+1 n , mn Jf is the tangent space at j k f to the R (k) -orbit of f in J k (n, 1), and moreover, if f is k-determined for R-equivalence then by Theorem 3.1(i), the codimension of this orbit in J k (n, 1) is equal to codimR f . For the second, we need a preparatory lemma. Lemma 3.1 Suppose f ∈ O n has an isolated singularity at 0, then (i) dimF Jf / mn Jf = n, and (ii) codimR f = codimRe f + n − 1. Proof (i) As every function c ∈ O n can be written in the form c = c(0) + c, ˜ where c˜ ∈ mn , it follows that Jf ∂f ∂f = SpF { , . . ., }, mn Jf ∂x1 ∂xn where ∂f/∂xi means the class of ∂f/∂xi in the quotient. If the dimension of the quotient is less than n, the ∂f/∂xi are not linearly independent, so that there is some non-trivial relation  i

ci

∂f = 0, ∂xi

56

3 Left-Right Equivalence and Stability

or in other words 

ci

i

 ∂f ∂f = αi ∂xi ∂xi i

for some functions α1 , . . ., αn ∈ mn , where c1 , . . ., cn ∈ F are not all zero. Subtracting the right-hand side from the left, we get  ∂f (ci − αi ) = 0. ∂xi

(3.15)

i

The vector field ϑ :=

 ∂ (ci − αi ) ∂xi i

has a smooth 1-dimensional orbit though 0, since not all of its coefficients vanish at 0. Since df (ϑ) = 0, f is invariant under the integral flow of ϑ. This means that at all points of the orbit through 0, f has a critical point, since it has a critical point at 0. This contradicts the assumption that f has isolated singularity at 0. Thus, the ∂f/∂xi must be linearly independent in Jf / mn Jf , which therefore has dimension n. (ii) The exactness of 0→

Jf mn mn → → →0 mn Jf mn Jf Jf

and 0 →

mn On On → → →0 Jf Jf mn

now shows that dimF

mn mn = n + dimF mn Jf Jf

and

dimF

On mn = dimF + 1, Jf Jf  

and the conclusion follows.

Remark 3.1 A theorem from commutative algebra proves a slightly stronger result in the complex analytic case. If the zero locus of the n partials ∂f/∂x1 , . . ., ∂f/∂xn has codimension n, then these partials form a regular sequence (see Definition C.5). A relation(or syzygy) between the n partials in O n is an n-tuple (b1 , . . ., bn ) such that i bi ∂f/∂xi = 0. The fact that the ∂f/∂xi form a regular sequence is equivalent to the property that the O n -module of relations between them is generated by the so-called trivial relations (0, . . ., 0, −

∂f ∂f , 0, . . ., 0, , 0, . . ., 0) ∂xj ∂xi

3.1 Classification of Functions by Right Equivalence

57

(where −∂f/∂xj is in the i’th place and ∂f/∂xi is in the j ’th place). In particular, all the bj must lie in the ideal Jf , and thus in mn , so that a relation like (3.15), where not all of the coefficients ci − αi are in mn , is impossible. Now we justify the label “codimR f ” for dimF O n /Jf . Suppose that F : X × U → F, (x, u) → fu (x) is a family of functions, where X and U are open in Fn and Fd , respectively. Let Re(k) f ⊂ J k (X, F) be the set of k-jets of germs g : (X, x0 ) → (F, 0) which are right-equivalent to f in the sense that there is a germ of diffeomorphism (k) ϕ : (X, x0 ) → (Fn , 0) such that g = f ◦ ϕ. Similarly, let Re f be the set of jets with the same property except that now g = f ◦ ϕ + c for some constant c. Each fibre of the projection J k (X, F) → X × F is diffeomorphic to J k (n, 1) and contains (k) a copy of R (k)j k f , and Re f is the disjoint union of these copies. Thus codimension of Re

(k)

f in J k (X, F) = codimension of R (k)f in J k (n, 1)

codimension of Re(k) f in J k (X, F) = codimension of R (k)f in J k (n, 1) + 1. (3.16) If f is k-determined for R-equivalence then these codimensions are, respectively, codimR f and codimR f + 1, by Theorem 3.1(i). Let j k F /U : X × U → J k (X, F) be the relative jet extension map, defined by j k F /U (x, u) = j k fu (x). Let π : X × U → U be the projection. Proposition 3.4 Suppose that f is k-determined for R-equivalence. Then (k) (k) (i) if j k F /U −  R f and (j k F /U )−1 (R f ) = ∅, then the codimension in U e

e

of π((j k F /U )−1 (Re f )) is equal to codimRe f , and (k) (k) (ii) if j k F /U −  Re f and (j k F /U )−1 (Re f ) = ∅, the codimension in U of (k) π((j k F /U )−1 (Re f ) is equal to codimRe f − 1. (k)

Thus, in a “generic family” fu , the set of parameter values u such that fu has a singularity right-equivalent to f has codimension codimRe (f ), and the set of parameter values such that fu has a singularity right-equivalent to f + c, for some constant c, has codimension codimRe f − 1. Proof As f has an isolated singularity, so does any germ equivalent to f . Thus (k) in particular the projections to U of (j k F /U )−1 (Re(k)f ) and (j k F /U )−1 (Re f ) are both finite-to-one. It follows that projection to U does not reduce the dimension of either set, and thus reduces the codimension of each by dim X = n. By the assumption of transversality, and by (3.16), their codimensions in X × U are codimR f + 1 and codimR f , respectively. The conclusion follows by Lemma 3.1.  

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3 Left-Right Equivalence and Stability

Exercises for Sect. 3.1 1 Show (i) If f (x1 , . . ., xn ) = x12 + · · · + xn2 then Jf is the maximal ideal mn and dimF O n /Jf = 1. (ii) If f (x1 , . . . , xn ) = x12 + · · · + xn2 + higher order terms, then Jf = mn (use Nakayama’s Lemma C.1). (iii) If dimF O n /Jf = 1 then Jf = mn . (iv) Jf = mn if and only if the Hessian determinant det(∂ 2 f/∂xi ∂xj )1≤i,j ≤n does not vanish at 0. To prove this, use Nakayama’s Lemma and the fact that modulo m2n , ⎞ ⎛ 2 ∂ f/∂x12 ∂f/∂x1 ⎜ .. ⎟ ⎜ .. ⎝ . ⎠=⎝ . ⎛

∂f/∂xn

⎞⎛ ⎞ · · · ∂ 2 f/∂xn ∂x1 x1 ⎟ ⎜ .. ⎟ .. .. ⎠⎝ . ⎠. . .

∂ 2 f/∂xn ∂x1 · · ·

∂ 2 f/∂xn2

xn

2 Show (i) If f (x1 , x2 ) = x12 + x2k+1 then Jf = (x1 , x2k ) and dimF O n /Jf = k. (ii) If f (x1 , x2 ) = x12 x2 + x2k−1 then Jf = (x1 x2 , x12 + (k − 1)x2k−2 ) and dimF O n /Jf = k. (iii) If f (x1 , x2 ) = x12 x2 then Jf = (x1 x2 , x12 ) and dimF O n /Jf = ∞. 3 Find the lowest value of k for which (3.1) holds for (i) f1 (x1 , x2 ) = x12 + x2k+1 (ii) f2 (x1 , x2 ) = x12 x2 + x2k−1 (iii) f3 (x1 , x2 ) = x1k + x2 . 4 Show by an example that it is not true that if mn Jf ⊃ mk+1 then f is kn determined for R. You do not have to look far. 5 At what point does the proof of part (ii) of Theorem 3.1 fail if we assume only the weaker hypothesis that mn Jf ⊃ mk+1 n ?

3.2 Left-Right Equivalence and Stability In this book we are interested in A -equivalence more than R-equivalence. But Theorem 3.1 is a good indication of what is true and how, in principle, one goes about proving it. In this section we define a tangent space for A -equivalence, which will play a rôle in the theory similar to that of T Rf = mn Jf for R-equivalence. In Chap. 6 we then develop criteria for finite determinacy in terms of it. For A the

3.2 Left-Right Equivalence and Stability

59

process is more complicated than for R, since one has simultaneously to produce families of diffeomorphisms of source and target. But the overall strategy is the same. Mather and Thom, in their work in the 1960s on smooth maps, thought in global terms: a C ∞ map of manifolds f : N → P is stable if its orbit under the natural action of Diff(N) × Diff(P ) is open in C ∞ (N, P ), with respect to a suitable topology. Here, we are interested in local geometry, and so we give a local version of this definition: a map-germ f : (Fn , S) → (Fp , 0) is stable if every deformation is trivial: roughly speaking, if ft is a deformation of f then there should exist deformations of the identity maps of (Fn , S) and (Fp , 0), ϕt and ψt , such that ft = ψt ◦ f ◦ ϕt−1 .

(3.17)

A substantial part of Mather’s six papers on the stability of C ∞ mappings [Mat68a, Mat68b, Mat69a, Mat69b, Mat70, Mat71] is devoted to showing that if all the germs of a proper mapping f are stable in this local sense then f is stable in the global sense. We will discuss global stability later in Chap. 5, but do not prove the statement of Mather just referred to. Definition 3.4 (1) A d-parameter unfolding of f is a map-germ F : (Fn × Fd , S × {0}) → (Fp × Fd , 0) of the form F (x, u) = (f˜(x, u), u) such that f˜(x, 0) = f (x). If we denote the map x → f˜(x, u) by fu , then the above condition becomes simply f0 = f . Retaining the parameters u in the second component of the map makes the following definition easier to write down: (2) Two unfoldings F, G of f are equivalent if there exist germs of diffeomorphisms  : (Fn × Fd , S × {0}) → (Fn × Fd , S × {0}) and  : (Fp × Fd , 0) → (Fp × Fd , 0), which are themselves unfoldings of the identity in Fn and Fp , respectively, such that G =  ◦ F ◦ −1 .

(3.18)

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3 Left-Right Equivalence and Stability

(3) The unfolding F is trivial if it is equivalent to f × id (the “constant” unfolding (x, u) → (f (x), u)). (4) The map-germ f : (Fn , S) → (Fp , 0) is stable if every unfolding of f is trivial. By writing (x, u) = (ϕu (x), x) and (y, u) = (ψu (y), u), from (3.18) we recover the heuristic definition (3.17) in the particular case d = 1. We do not insist that the mappings ϕu and ψu preserve the base-points of the germ f . After all, if the interesting behaviour merely changes its location, we should not regard the unfolding as non-trivial. Example 3.3 (i) Consider the map-germ f : (F, 0) → (F, 0), f (x) = x 2 , and its unfolding F (x, u) = (x 2 + ux, u). This is trivialised by the families of diffeomorphisms (x, u) = (x + u/2, u), (y, u) = (y − u2 /4, u). Both  and  are just families of translations. (ii) The unfolding F (x, u) = (x 3 + ux, u) of f (x) = x 3 is not trivial. Two arguments are available to prove this. The first is that if we write fu (x) = x 3 + ux, then for u = 0, fu has two distinct critical points, whereas f has only one. If F were equivalent to f × 1 then this could not happen. We clarify this: if  ◦ F = (f × 1) ◦ , where  and  are diffeomorphisms unfolding the identity, then  restricts to a diffeomorphism from the critical set F of F to the critical set f ×1 of f × 1, and moreover since  is an unfolding, the diagram |

f ×1

F πu

πu

F commutes, where πu is projection to the parameter space. The projection on the left is two-to-one on the complement of 0, whereas the projection on the right is one-to-one. The second argument involves determining precisely what is the space

  dft  : F (x, t) = (f (x), t) is a trivial unfolding of f . t dt t =0

(3.19)

and showing that the derivative we obtain from the unfolding F above, x, does not belong to it. We cannot yet make this calculation, but turn to it shortly. ♦ The space (3.19) can also be described as

  dft  −1 : ft = ψt ◦ f ◦ ϕt , ϕ0 = id, ψ0 = id . dt t =0

(3.20)

3.2 Left-Right Equivalence and Stability

61

The families of diffeomorphisms ψt and ϕt here do not necessarily preserve the base-points of the germ f . The germs  and  are unfoldings of the identity, so that ϕ0 (S) = S and ψ0 (0) = 0 but are not required to satisfy ϕt (S) = S or ψt (0) = 0 for t = 0. Determining an algebraic formula for the space (3.19) will also provide us with a simple and computable criterion for stability. We consider (3.20) as a subspace of the space of all infinitesimal deformations of f , which we define as

I D(f ) :=

  dft  : F (x, t) = (f (x), t) is any unfolding of f . t dt t =0

(3.21)

As we will see later, I D(f ) has a richer algebraic structure, but for the moment, we only consider in I D(f ) the structure of vector space over F. A special case of infinitesimal deformation is when we require the unfolding F to preserve the base-points of the germ f , that is, when ft (S) = {0}, for all t. We will denote by I D0 (f ) the subspace of I D(f ) of infinitesimal deformations which preserve the base-points. In the following definition, F is either R or C. When F = R then f and the deformations ft are assumed to be C ∞ , and when F = C then f and F = C then f and the ft are assumed to be complex analytic. Definition 3.5 Given a map-germ f : (Fn , S) → (Fp , 0), we define    d (ψt ◦ f ◦ ϕt−1 ) : ϕ0 = id, ψ0 = id , dt t=0 

  d (ψt ◦ f ◦ ϕt−1 ) : ϕ0 = id, ψ0 = id, ϕt (S) = S, ψt (0) = 0, ∀t , TAf = dt t=0

T Ae f =

and the quotients TA1 e f =

I D(f ) , T Ae f

TA1 f =

I D0 (f ) . TAf

The space T Ae f plays a central rôle in the subject, and is called the extended tangent space of f. Its subspace T A f is the (heuristic) tangent space to the A -orbit of f . For just as in the discussion preceding Definition 3.3, if both ϕt and ψt fix the base-points, then the families ϕt and ψt can be regarded as curves in the groups R and L , so that ψt ◦ f ◦ ϕt is a curve in the A -orbit of f . We use the term“TA1 e f ” here because, as we shall see in Chap. 5 and in particular in Theorem 5.1 , it is the tangent space at 0 to the base space of a miniversal (i.e. semi-universal) unfolding of f ; the term “T 1 ” is the standard notation for the corresponding tangent space, in the deformation theory of analytic spaces. The upper index 1 is slightly misleading because it suggests that we might later meet a T 2 or a T 3 . The deformation theory of map-germs that we study here, developed

62

3 Left-Right Equivalence and Stability

by Thom, Mather, Martinet and others, is, by construction, unobstructed—we place no requirement that any of the spaces involved should be flat over the base of the deformation—and there is no need for these higher cotangent functors. Definition 3.6 (Continued) We define the Ae -codimension and the A codimension of f , respectively, as codimAe (f ) = dimF TA1 e f,

codimA (f ) = dimF TA1 f.

Definition 3.7 A germ f : (Fn , S) → (Fp , 0) is A -finite if dimF TA1 e f < ∞. Proposition 3.5 If f : (Fn , S) → (Fp , 0) is stable, then TA1 e f = 0. Proof Let F (x, t) = (ft (x), t) be a 1-parameter unfolding of f . By hypothesis it is trivial, so F =  ◦(f ×1)◦ for some  and , unfoldings of the identity. Writing (x, t) = (ϕt (x), t) and (y, t) = (ψt (y), t), we have ft = ψt ◦ f ◦ ϕt .   If TA1 e f = 0 we say that f is infinitesimally stable. Mather [Mat69a] proved the converse of Proposition 3.5, giving Theorem 3.2 Infinitesimal stability is equivalent to stability: f is stable if and only if TA1 e f = 0. We will prove this later in the chapter. For an unstable germ f , the Ae -codimension of f is an important invariant, measuring the failure of stability of f . In this chapter we develop techniques for calculating T Ae f , and apply them in some examples. Lemma 3.2 If ϕt and ψt are parameterised families of diffeomorphisms, we have    −1   d  dϕ dψt   t −1  ψt ◦ f ◦ ϕt  = df ◦ + ◦ f.  dt dt  dt t =0 t =0 t =0

Proof Although this statement uses nothing but the chain rule, there is some ˜  and ψ, ˜ and G and subtlety in its application, so we give the proof. Define  and φ, ˜ g, ˜ by the equalities (x, t) = (ϕ(x, ˜ t), t) = (ϕt (x), t), (y, t) = (ψ(y, t), t) = (ψt (y), t), and G(x, t) = (g(x, ˜ t), t) = (f (ϕt−1 (x)), t). Pick representatives of , f and . Let (x, 0) be a point in the domain of  ◦ (f × 1) ◦ −1 . Then  ∂ d ψt ◦ f ◦ ϕt−1  (x) = (ψ˜ ◦ G)(x, 0). dt ∂t t =0 By the chain rule this is equal to p  ∂ ψ˜ i=1

∂yi

(G(x, 0))

∂ ψ˜ ∂ g˜ i (x, 0) + (G(x, 0)). ∂t ∂t

3.2 Left-Right Equivalence and Stability

63

˜ Because ψ(y, 0) = y, and because g(x, ˜ 0) = f (x), this sum is just  dψt  ∂ g˜ (x, 0) + (f (x)). ∂t dt t =0 Again by the chain rule, and because ϕ0 (x) = x, the first summand here is  n  ∂f d(ϕt−1 )i  (x)  ∂xi dt  i=1

and this is equal to dx f

 dϕt−1  dt 

(x), t =0

.

 

t =0

As discussed before Proposition 3.1, both (dϕt /dt) |t =0 and (dψt /dt)|t =0 determine germs of vector fields, on (Fn , S) and (Fp , 0), respectively: (dϕt (x)/dt)|t =0 is the tangent vector at x to the trajectory ϕt (x). The set of all germs of vector fields on (Fn , S) is denoted by θn := θFn ,S . It is a module over On := OFn ,S , the ring of smooth functions (Fn , S) → F. The subset of those (dϕt (x)/dt)|t =0 which preserve the base-points is equal to the submodule mn θn , where mn := mFn ,S is the ideal of On of functions vanishing on S. In the same way, the infinitesimal deformations of f should be thought of as vector fields along f (cf. Definition 2.13); (dft /dt) |t =0 is the tangent vector at f (x) to the trajectory x → ft (x). By associating to (dft /dt)|t =0 the map  dft (x)  ˆ f : x → f (x), ∈ T Fp , dt t =0 we obtain a commutative diagram: T Fn Fn

df fˆ f

T Fp Fp

(3.22)

in which the vertical maps are the bundle projections. Conversely, any vector field ξ along f is a smooth map ξ : (Fn , S) → T Fp of the form ξ(x) = (f (x), ξ˜ (x)), for some ξ˜ : (Fn , S) → Fp . It follows that ξ˜ is the infinitesimal deformation (dft /dt)|t =0 associated to the deformation ft = f + t ξ˜ . Thus, we have a canonical identification I D(f ) ≡ θ (f ), the space of vector fields along f . The space θ (f ) is also a module over On and the infinitesimal deformations which preserve the base-points correspond to the submodule mn θ (f ). From now on, we will refer to vector fields along f rather than infinitesimal deformations of f , and abandon the notation I D(f ) in favour of θ (f ).

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3 Left-Right Equivalence and Stability

As an O n -module, θn is freely generated by the germs of the coordinate vector fields ∂/∂x1 , . . ., ∂/∂xn . Thus an element ξ of θn can be written in various ways: as a sum ξ(x) =

n 

ξj (x)∂/∂xj ,

j =1

where each ξj is a germ of function at S, or, suppressing mention of the coordinate system, simply as an n-tuple, ξ = (ξ1 (x), . . ., ξn (x)) (sometimes as a column rather than a row). In calculations below we will usually write the elements of θ (f ) and of θn as columns. We discuss the modifications to the notation needed in dealing with multi-germs in Sect. 3.4 below. Important Piece of Notation We denote by tf : θn → θ (f ) the map ξ → df ◦ ξ and by ωf : θp → θ (f ) the map η → η ◦ f . The notation “tf ” is slightly fussy. We use it instead of df here because we think of df as the bundle map between tangent bundles induced by f , as in the diagram (3.22), whereas tf is the map “left composition with df ” from θn to θ (f ). Some authors use “df ” for both. Now using this notation, from Lemma 3.2 we deduce Corollary 3.2 For any map-germ f : (Fn , S) → (Fp , 0), we have T Ae f = tf (θn ) + ωf (θp ),

T A f = tf (mn θn ) + ωf (mp θp )

and also TA1 e f =

θ (f ) , T Ae f

TA1 f =

mn θ (f ) . TAf

Remark 3.2 (Real or Complex?) The singularity theory of real maps is usually concerned with smooth maps and map-germs, whereas complex singularity theory is concerned with complex analytic germs. Our notation, and indeed our definitions, do not distinguish between the two cases, but at first sight they are rather different. Suppose that f : (Fn , 0) → (Fp , 0) is a smooth (real) germ or a complex analytic

3.2 Left-Right Equivalence and Stability

65

germ. In the real smooth case, θ (f ) is a free module of rank p over the ring En of germs of smooth functions, whereas in the complex analytic case, θ (f ) is a free module of rank p over the ring O n of germs of complex analytic functions. There is no simple relation between O n and En . In particular, En contains non-zero germs whose Taylor series about 0 is identically zero, which have no counterpart in O n . Equally, for every formal power series, convergent or not, there is a smooth germ with this as its Taylor series [Brö75, 4.9]. Thus there is no reason to expect a straightforward relation between the R or A tangent spaces and T 1 ’s of a smooth germ (Rn , 0) → (Rp , 0) defined by real power series, and the corresponding tangent spaces of its complexification fC : (Cn , 0) → (Cp , 0) defined by the same power series, now with complex variables. Nevertheless, for A -finite germs the relation is transparent: Proposition 3.8 below shows that a real analytic germ is A -finite in the C ∞ category if and only if its complexification is A -finite in the complex analytic category, and in this case, TA1 e fC = TA1 e f ⊗R C—so a basis for the T 1 of a real analytic germ is also a basis for the T 1 of its complexification. In particular, when we discuss A -finite germs, it is not necessary to distinguish in our notation between the real smooth and the complex analytic cases. ♦ Remark 3.3 Since tf : θn → θ (f ) is O n -linear, θ (f )/tf (θn ) is naturally an O n module. By means of the ring homomorphism f ∗ : O p → O n (right composition with f ), every O n -module becomes an O p -module, and every O n -linear map becomes O p -linear. Thus θ (f ) and θ (f )/tf (θn ) become O p -modules. As θ (f ) is a finitely generated O n -module, it is finitely generated over O p when O n is. By the Preparation Theorem D.2, this holds if and only if O n /f ∗ (mp ) O n is a finitedimensional F-vector space. In the complex case, by the Nullstellensatz, this holds if and only if the germ of f −1 (0) consists just of S. This cannot happen when n > p. However, even when n > p then provided f is A -finite, θ (f )/tf (θn ) is finitely generated over O p (Exercise 7 below). The property of being finitely generated is important in many of the algebraic arguments in singularity theory; for example, it is required in Nakayama’s Lemma (Proposition C.1 in the Appendix on Commutative Algebra). ♦ Proposition 3.6 Suppose f : (Fn , S) → (Fp , 0), with S = {s1 , . . . , sr } ⊂ C(f ). Then: (i) codimAe f < ∞ if and only if codimA f < ∞. (ii) If 0 < codimAe f < ∞ then codimA f = codimAe f + p − r(n − p). Proof Denote the coordinates near the i’th base-point si by x1(i) , . . . , xn(i) , for i = 1, . . ., r. The quotient T Ae f/T A f is generated as vector space over F by rn + p elements:  tf

∂ (i)

∂xk



, i = 1, . . ., r, k = 1, . . . , n,

and

ωf

∂ ∂yj

, j = 1, . . ., p.

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3 Left-Right Equivalence and Stability

If they are not linearly independent, then there are constants aik and bj , not all zero, such that ⎛ ⎞    ∂ ∂ ⎠ ∈ T A f, aik (i) + ωf ⎝ bj tf ∂yj ∂x ik

j

k

and hence vector fields ξ0 ∈ mn θn and η0 ∈ mp θp such that

tf

 

aik

ik



∂ ∂xk(i)

⎛ ⎞  ∂ ⎠ = tf (ξ0 ) + ωf (η0 ). bj + ωf ⎝ ∂yj j

Rearranging, we get  tf

 ik

aik



∂ ∂xk(i)

− ξ0

⎛ = ωf ⎝η0 −

 j

⎞ ∂ ⎠. bj ∂yj

(3.23)

  Write ξ := ik aik ∂/∂xk(i) − ξ0 and η := η0 − j bj ∂/∂yj . Integrating the vector fields ξ and η we get flows ϕt in the source and ηt in the target such that f ◦ ϕt = ψt ◦ f (here we are using the Thom–Levine Lemma, 2.6). It follows that for each value of t, the germ f : (Fn , ϕt (S)) → (Fp , ψt (0)) is left-right equivalent to the germ of f at S. Because not all of the aik and bj are equal to 0, either ξ is not zero at some point of S, or η is not zero at 0. If the former, say ξ(si ) = 0, so that the trajectory ϕt (si ) is a smooth curve germ, which lies in the germ of the critical set of f at si . The germ of f at si is A -finite (Exercise 3.4.3.4 below) and therefore finite-to-one on its critical set. It follows that we cannot have f (ϕt (si )) ≡ 0 for all t. Hence the trajectory of η through 0 is a curve and not just a point (and thus in fact η(0) = 0). But this is impossible: because 0 < codimAe f < ∞, the germ at 0 ∈ Fp of the set of points {y ∈ (Fp , 0) : the germ off at (Fn , (f −1 (y) ∩ C(f )) is not stable } consists only of 0, by the Mather–Gaffney criterion for A -finiteness, Theorem 4.5. This contradiction shows that all of the aik and bj must be zero, so that dimF T Ae f/T A f = rn + p. As in the proof of Lemma 3.1(ii), the conclusion follows from the exactness of the sequences 0→

T Ae f θ (f ) θ (f ) → → →0 TAf TAf T Ae f

3.2 Left-Right Equivalence and Stability

67

and 0→

θ (f ) θ (f ) mn θ (f ) → → →0 TAf TAf mn θ (f )

and the fact that dimF θ (f )/mn θ (f ) = rp.

 

Remark 3.4 When f is a mono-germ then r = 1, and codimA f = codimAe f +n. The number p − r(p − n) which gives the difference between the A and Ae codimensions in the multi-germ case will appear later in Chap. 9, where it is denoted by dr (n, p). It is the dimension of the set of r-multiple points in Xr of an immersion with normal crossings f : X → Y with dim X = n and dim Y = p. ♦ The proof of the Proposition uses the Mather–Gaffney criterion for A -finiteness, which we discuss in detail in Sect. 4.5. For now, we highlight a consequence, and introduce an important piece of terminology. Proposition 3.7 Let f : (Fn , S) → (Fp , 0) be a smooth germ and suppose that S ⊂ C(f ) (where C(f ) is the critical set of f ). If 0 < codimAe f < ∞ then the germ at 0 of the set of points 

y ∈ Fp : the germ of f at f −1 (y) ∩ C(f ) is left-right equiv. to the germ of f at S

consists only of 0.





The set-germ defined in the proposition is called the isosingular locus of f . The following Lemma is proved in Chap. 5, but may be proved using a sequence of exercises at the end of this section. Lemma 3.3 In both the real C ∞ and complex analytic cases, a germ f : (Fn , S) → (Fp , 0) is A -finite if and only if T Ae f ⊃ mkn θ (f ) for some finite k. Proof “If” is trivial in both cases, since θ (f )/ mkn θ (f ) is generated by the finite collection of monomials x α ∂/∂yi , i = 1, . . ., p, |α| < k. The opposite implication is proved in Exercises 5–8 below.   In Chap. 6 we will prove the general version of Mather’s Finite Determinacy Theorem 6.2. For G = R and p = 1, we already proved this theorem in Sect. 3.1. For G = A , we get the following equivalent statements: 1. f is A -finite, 2. f is finite determined for A -equivalence, 3. mkn θ (f ) ⊂ T Ae f , for some k ∈ N. We now use the equivalence of (i) and (iii) in order to prove the version for A equivalence of Proposition 3.3.

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3 Left-Right Equivalence and Stability

Proposition 3.8 Suppose that f : (Rn , S) → (Rp , 0) is a real analytic germ and let fC be its complexification. Then f is A -finite ⇐⇒ fC is A -finite, and when they are A -finite,   TA1 e fC ∼ = TA1 e f ⊗R C

  TA1 fC ∼ = TA1 f ⊗R C.

and

(3.24)

In particular codimAe f = codimAe fC

and

codimA f = codimA fC .

Proof In the proof, all of the objects associated with f are in the C ∞ category, while those associated with fC are complex analytic. Suppose that f is A -finite. Then by Exercises 8 and 9 below, tf (θn ) + f ∗ mp θ (f ) ⊃ mn θ (f )

for some finite 

(3.25)

and tf (θn ) + ωf (θp ) ⊃ mkn θ (f )

for some finite k.

(3.26)

We claim that the same statements hold with fC in place of f . For the first, it is enough, by Nakayama’s Lemma, to show that tfC (θn ) + fC∗ mp θ (fC ) +  m+1 n θ (fC ) ⊃ mn θ (fC ). This is easy: we know that for each α with |α| =  and for i = 1, . . ., p, there exist smooth ξ ∈ θn and βj ∈ θ (f ) such that xα

 ∂ = tf (ξ ) + fj βj . ∂yi

(3.27)

Replacing ξ and the βj by their Taylor polynomials of degree , we deduce that a polynomial version of (3.25) holds modulo m+1 n θ (f ). This can equally be regarded as an equality for fC . Then Nakayama’s Lemma proves the required inclusion. For the second, we can use the same argument, for it is enough, by Lemma 6.3, to show that k tfC (θn ) + ωfC (θp ) + mk+ n θ (fC ) ⊃ mn θ (fC ).

(3.28)

We use Lemma 6.3 with C = θ (fC )/tfC (θn ), A = T Ae fC /tfC (θn ), I = mkn and M = mm . Thus, smooth vector fields ξ and η giving exact solutions to xα

∂ = tf (ξ ) + ωf (η) ∂yi

3.2 Left-Right Equivalence and Stability

69

can be replaced by polynomial vector fields giving solutions modulo mk+ n θ (f ), which can be viewed as solutions modulo mk+ θ (f ). C n The same arguments work in reverse. It is necessary to add one extra step: if ξ and η are complex analytic vector fields such that tfC (ξ ) + ωfC (η) = x α ∂/∂yi , then we can assume that ξ and η have real analytic coefficients. For if ξ ∗ and η∗ are obtained from ξ and η by complex conjugating the coefficients in their Taylor series, then also tfC (ξ ∗ ) + ωfC (η∗ ) = x α ∂/∂yi , since the coefficient on the right-hand side is real. Thus, we can replace ξ and η by 12 (ξ + ξ ∗ ) and 12 (η + η∗ ). So (3.25) and (3.26) hold for f if and only if they hold with fC in place of f . In particular, f is A -finite if and only if fC is. The last part of the statement is now straightforward. We have θ (fC ) k+ mn θ (fC )

=

θ (f ) mk+ n θ (f )

⊗R C

(3.29)

and tfC (θn ) + ωfC (θp ) + mk+ n θ (fC ) mk+ n θ (fC )

=

tf (θn ) + ωf (θp ) + mk+ n θ (f ) mk+ n θ (f )

⊗R C, (3.30)

so taking quotients, TA1 e fC = TA1 e f ⊗R C. The proof for TA1 f is analogous.

 

The finite determinacy theorem (6.2 below) says that A -finiteness is equivalent to finite A -determinacy. Thus, by Proposition 3.8, a real analytic germ f is finitely determined for A -equivalence if and only if its complexification fC is. It is not evident that the determinacy degrees are the same, but in fact this is the case—we briefly discuss a theorem of [BdPW87] which shows this in 6.11 below. The next proposition is the analogue for A of Proposition 3.4, and highlights the importance of Ae –codimension. To state it we need a definition. Definition 3.8 Given a germ f : (Fn , S) → (Fp , 0) and manifolds N and P of dimensions n and p, for each k ∈ N we define Ae(k) f to be the set of k-jets in k  r J (N, P ) of germs g : (N, S ) → (P , y0 ) which are left-right equivalent to f . As mentioned above, the finite determinacy theorem says that if f is A -finite then it is finitely determined for A -equivalence. Suppose now that f is k-determined. An ×p On argument like the one in the proof of Theorem 3.1(i) shows that T A f ⊇ mk+1 n k (k) (see Theorem 6.2(ii) below). It follows that the codimension in r J (n, p) of A f is equal to codimA f (see Corollary 6.1). Each fibre of the natural projection k (r) × P r is isomorphic to J k (n, p) and is contained in the r J (N, P ) → N r submanifold W of multi-jets with the same target in P , which has codimension (r − 1)p. Moreover, because each fibre meets Ae(k) f in an orbit isomorphic (k) to A (k) f , it follows that the codimension of Ae f in r J k (N, P ) is equal to codimA f + (r − 1)p.

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3 Left-Right Equivalence and Stability

Proposition 3.9 Suppose that 0 < codimAe f < ∞. Let F : N × U → P be a family of mappings, and suppose that r j k (F /U ) : N (r) × U → r J k (N, P ) is transverse to Ae(k) f . Then if f is k-determined for A , the codimension in U of the projection of (r j k (F /U ))−1 (Ae(k) f ) is equal to codimAe f . That is, for a generic family F (x, u) = (fu (x), u), the set of parameter values in U for which fu has a singularity in P left-right equivalent to f , has codimension equal to codimAe f . Proof By the preceding remarks and by transversality, the codimension of the submanifold (r j k (F /U ))−1 (Ae(k) f ) in N (r) × U is equal to codimA f + (r − 1)p. Because the isosingular locus of f at 0 consists of the single point 0, the isosingular locus of fu at any point y ∈ P where the germ of fu is left-right equivalent to the germ of f consists of the single point y. It follows that the projection N (r) ×U → U is locally finite on (r j k (F /U ))−1 (Ae(k) f ), so that projection does not reduce its dimension. Hence codim π(r j k (F /U ))−1 (Ae(k) f ) = codim(r j k (F /U ))−1 (Ae(k) f ) − rn = codimA f − rn + (r − 1)p = codimAe f, by Proposition 3.6.

 

Example 3.4 (Looking at Bent Wires II) In the Introduction we discussed the different local views one can see by looking at a knot in R3 from different positions with one eye closed, and encouraged the reader to experiment, and make a list of “essentially distinct” classes of local views. For a generic knot, one observes three classes of local views whose viewing set (the set of points in R3 from which one can see a local view of the given class) has codimension 1, corresponding to the three Reidemeister moves in knot theory. With the help of Proposition 3.9 we can recover this observation as a consequence of the theory we are developing, by finding parameterisations of the three local views shown in the second diagram in Example 1.1, and showing that their Ae -codimension is indeed equal to 1. The three local views at the centre of the Reidemeister moves are R1: First order cusp: a plane curve germ f : (R, 0) → (R2 , 0) having a nonimmersive point at 0, but doing so in the least degenerate way possible. Clearly j 2 f must take the form x → (ax 2 , bx 2) and the assumption of minimal degeneracy means that at least one of a, b is different from 0. Suppose a = 0. Then a scale change, and a target coordinate change subtracting a multiple of y1 from y2 , brings j 2 f to the form x → (x 2 , 0). Thus f takes the form f (x) = (x 2 (1 + α(x)), x 3 (β(x))) where α ∈ m1 . The coordinate change in the domain x¯ = x(1 + α)1/2 ), together with the assumption of minimal degeneracy and scale change on the y2 coordinate, brings f to the form f (x) = (x 2 , x 3 + h.o.t.). By subtracting monomials in y1 and y2 from y2 , one can increase the order of the higher order terms as far as one likes, so it is

3.2 Left-Right Equivalence and Stability

71

at least plausible that we can suppose f (x) = (x 2 , x 3 ). Later we will see that this germ has Ae -codimension 1 and is 3-determined for A -equivalence. Interruption to Discuss Notation The remaining Reidemeister moves involve multigerms, germs f : (Fn , S) → (Fp , 0) with |S| > 1. Notation can become a problem here. Suppose S = {s1 , . . ., sr }. For each point si ∈ S, we will denote the germ of f at si by f (i) , and describe f (i) using local coordinates centred on si (so that si itself becomes (0, . . ., 0)). Thus, to represent f , we use |S| distinct local coordinate systems in the source. It would be possible to denote the coordinates around si (i) (i) by x1 , . . ., xn , but the profusion of upper indices quickly becomes tiresome, especially when we want to raise xj(i) to the power k. So we will denote each set of coordinates in the source simply by x1 , . . ., xn . This may seem like a recipe for confusion, but seems to work. Our notation uses “place value” (see below) to distinguish between the different coordinate systems. R2: a tacnode, two immersive plane branches having a first order tangency at 0. A target coordinate change turns one of the two branches into the y1 -axis, and we take y1 ◦ f1 as new coordinate in the domain of the first branch. With respect to the new coordinates the second branch has equation y2 = g(y1 ), with g(0) = g  (0) = 0, and the minimality of the tangency implies g  (0) = 0. Because the two branches are tangent we can take y1 ◦ f2 as coordinate on the domain of the second branch. After a linear coordinate change, f2 has the form x → (x, x 2 (1 + α)). Replacing y1 by y¯1 = y1 (1 + α)1/2 brings f to the form

x→  (x, 0) . x→  (x, x 2 )

R3: a triple point, three immersive plane branches meeting two-by-two transversely at 0. Here a change of coordinates in the target turns two of the branches into the coordinate axes. The third, being transverse to each of the other two, has equation y2 = g(y1 ) with respect to the new coordinates, for some function g with g(0) = 0, g  (0) = 0. The target coordinate change y¯2 = y2 − g(y1 ), followed by a suitable scale change, brings f to the form ⎧  (x, 0) ⎨x → x→  (0, x) . ⎩ x→  (x, x) ♦

3.2.1 Right Equivalence and Left Equivalence In this book we are mainly interested in right-left equivalence of map-germs, but for some specific applications we may ask ourselves questions about right or left

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3 Left-Right Equivalence and Stability

equivalences. In fact, in Sect. 3.1 we already considered right equivalence, but only for functions. At this point of the book, it will not be difficult for the reader to predict how the definitions and properties of this section can be adapted for the groups R and L . Given a map-germ f : (Fn , S) → (Fp , 0), the extended tangent spaces and the tangent spaces for R and L are defined as follows: T Re f = tf (θn ), T Le f = ωf (θp ),

T Rf = tf (mn θn ), T L f = ωf (mp θp ).

Observe that T Ae f = T Re f + T Le f and T A f = T Rf + T L f , which is compatible with the fact that A = R × L . The corresponding spaces T 1 are TG1e f =

θ (f ) , T Ge f

TG1 f =

mn θ (f ) , TGf

as well as the codimensions codimGe (f ) = dimF TG1e f,

codimG (f ) = dimF TG1 f,

where G = R or L . We will say that f is G -finite if codimGe (f ) < ∞. Details of how to adapt the properties of this section for R and L are left for the reader.

Exercises for Sect. 3.2 1 Check that in part (i) of Example 3.3, F =  ◦ (f × id) ◦ −1 . 2 Check that the diagram of part (ii) of Example 3.3 does indeed commute. 3 Show that every unfolding of the identity is a germ of diffeomorphism. 4 Germs of submersions and immersions are infinitesimally stable and therefore stable. This is an easy calculation using the normal forms of Theorems 1.1 and 1.2. For submersions, show that tf : θn → θ (f ) is surjective, and for immersions, show that ωf : θp → θ (f ) is surjective. The following exercises are straightforward but useful consequences of the definitions. They are used only very occasionally in this chapter, but do give an introduction to the kind of algebra which plays an increasing role as the book goes on.

3.2 Left-Right Equivalence and Stability

73

5 (i) If M is a finitely generated O n -module, then mkn M is also finitely generated. (ii) If f is A -finite then for every k,  < ∞, dimF

θ (f ) < ∞. tf (mkn θn ) + ωf (mp θp )

Hint: dimF θn / mkn θn < ∞, dimF θp / mp θp < ∞ (iii) If f is A -finite then for every k,  < ∞, dimF

tf (mkn θn )

θ (f ) < ∞. + f ∗ mp θ (f )

Hint: compare f ∗ mp θ (f ) with ωf (mp θp ). 6 (i) If M is a finitely generated O n -module, and dimF M = k < ∞, then mk M = 0. Hint: Consider the chain M ⊃ m M ⊃ m2n M ⊃ · · · ⊃ mk+1 M. Not all the inclusions can be strict. Use Nakayama’s Lemma. (ii) If N ⊂ M are modules over O n , with M finitely generated, and dimF M/N = k < ∞, then N ⊃ mk M. 7 (i) If f : (Fn , 0) → (Fp , 0) is A -finite, then θ (f )/tf (θn ) is a finitely generated module over O p . Hint: if the classes of g1 , . . ., gd generate TA1 e f then   θ (f ) ∂ ∂ = O p · ωf ( ), . . ., ωf ( ) + SpF !g1 , . . ., gd ". tf (θn ) ∂y1 ∂yp (ii) If dimF TA1 e f = k < ∞ then T Ae f ⊃ f ∗ mkp θ (f ). Hint: consider TA1 e f as O p -module and use Exercise 6(ii) (with O p in place of O n ). 8 (i) If dimF TA1 e f = k < ∞ then tf (θn ) + f ∗ mkp θ (f ) ⊃ mn θ (f ) for some finite . Hint: Use Exercises 5, 6 and 7. (ii) If dimF TA1 e f = k < ∞ then T Ae f ⊃ mn θ (f ) for some  < ∞. Note:  may be considerably greater than k.

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3 Left-Right Equivalence and Stability

9 Let f : (Fn , S) → (Fp , 0) be A -finite. Show that (i) tf (θn ) + f ∗ mp θ (f ) ⊃ mkn θ (f ) for some finite k. (ii) Suppose that tf (θn ) + f ∗ mp θ (f ) ⊃ mkn θ (f ). Let Jf be the ideal in O n generated by the maximal minors of the Jacobian matrix of f . Show that Jf + kp f ∗ mp O n ⊃ mn . kp (iii) Suppose that Jf + f ∗ mp O n ⊃ mn . Show that f is finite-to-one on its critical set. Hint: V (Jf + f ∗ mp O n ) = f −1 (0) ∩ C(f ). Now use the Nullstellensatz, Theorem D.4. (iv) Let f : (Fn , 0) → (Fp , 0) be A -finite. Show that f is finite-to-one on its critical set.

3.3 First Calculations We determine an F-basis for TA1 e f = θ (f )/T Ae f for a series of simple examples. In each case, we may assume that θ (f )/T Ae f is generated by monomials—terms of the form x1α1 · · ·xnαn ∂/∂yj . Although it is not true that θ (f ) itself is generated by monomials (an infinite power series is not in the linear span of monomials), in each example it will be clear “by inspection” that f is stable outside 0, and from this it follows by the Mather–Gaffney criterion (Theorem 4.5 below) that T Ae f ⊇ mkn θ (f ) for some finite k. From this it follows that the quotient θ (f )/T Ae f has a basis consisting of monomials, of degree ≤ k − 1. Later in the chapter we develop techniques which remove the need for this kind of geometrical insight. Our calculations here are naive, and hard to implement in a computer algebra system. In later chapters we develop more effective techniques for calculation. In particular in Chap. 8 we give an algorithm, Algorithm 8.1, for the Ae -codimension of a polynomial map-germ (Fn , 0) → (Fn+1 , 0) with rational coefficients, which one can implement in the algebraic geometry software packages Macaulay 2 and Singular. Example 3.5 (1) Consider the germ of the first Reidemeister move, f : (R, 0) → (R2 , 0), defined by f (x) = (x 2 , x 3 ). Note that this is an injective immersion outside 0. We write elements of θ2 and θ (f ) as column vectors. Every monomial x k except x itself can be written as a product of powers of x 2 and x 3 and hence as a composite a ◦ f . It follows that ωf (θ2 ) + SpR

 x 0 , = θ (f ), 0 x

3.3 First Calculations

75

where we use SpR to denote the R-vector space spanned by the vectors which follow. Since ∂ 2x tf , = 3x 2 ∂x it follows that

 0 = θ (f ). tf (θ1 ) + ωf (θ2 ) + SpR x 0 does not lie in T Ae f , since x the order of the second component of

any element in T Ae f is at least 2. Hence 0 . It is also easy to see that the unfolding TA1 e f has as basis the class of x F (x, t) = (x 2 , x 3 + tx, t) = : (ft (x), t) is not trivial. The reader should sketch the image curves for ft for different values of t and compare the sequence, as t varies, with the sequence of views of a space curve as one’s eye crosses the tangent developable surface. (2) The map-germ Finally, it is easy to see that the missing term

f (x1 , x2 ) = (x1 , x22 , x1 x2 ) parameterises the Whitney umbrella (also known as pinch-point). We will show that it is stable by showing that TA1 e f = 0. From now on we will refer to this germ as a Whitney umbrella. We use coordinates (x1 , x2 ) on the source and (y1 , y2 , y3 ) on the target. Z y x

f0

0

X

Y

Elements of θF2 ,0 , θF3 ,0 and θ (f ) will be written as column vectors. We divide O F2 ,0 into even and odd parts with respect to the x2 variable, and denote them by O e and O o . Every element of O e can be written in the form a(x1, x22 ), and every element of O o in the form x2 a(1 , x22 ). This is obvious in the analytic case. It is also true for C ∞ germs (see Exercise 3.3.3 below). Then (we hope

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3 Left-Right Equivalence and Stability

the notation is self-explanatory) ⎛

⎞ Oe ⊕ Oo θ (f ) = ⎝ O e ⊕ O o ⎠ Oe ⊕ Oo and since ⎛

⎞ ⎛ ⎞ a(y1, y2 ) a(x1, x22 ) ωf ⎝ b(y1, y2 ) ⎠ = ⎝ b(x1, x22 ) ⎠ c(y1 , y2 ) c(x1 , x22 )

(3.31)

we see that the even part of θ (f ) is indeed contained in T Ae f , and we need look only for the odd part. Since tf

a(x1, x22 )

∂ ∂x

⎞ ⎞ ⎛

1 0 a(x1, x22 ) 2) a(x , x 1 2 ⎠, = ⎝ 0 2x2 ⎠ =⎝ 0 0 2 x2 x1 x2 a(x1 , x2 ) (3.32) ⎛

T Ae f contains all of the odd part of the third row. Since ⎞ ⎞ ⎛ ⎛

1 0 0 ∂ 0 tf a(x1 , x22 ) = ⎝ 0 2x2 ⎠ = ⎝ 2x2 a(x1, x22 ) ⎠ , a(x1, x22 ) ∂x2 x2 x1 x1 a(x1, x22 ) (3.33) T Ae f contains the odd part of the second row. Since ⎞ ⎛ ⎞ ⎛

1 0 x2 a(x1, x22 ) 2 ∂ x2 a(x1 , x2 ) ⎠, tf x2 a(x1, x22 ) = ⎝ 0 2x2 ⎠ =⎝ 0 0 ∂x1 x2 x1 x22 a(x1 , x22 ) (3.34) T Ae f contains all of the odd part of the first row. So T Ae f = θ (f ), TA1 e f = 0 and f is infinitesimally stable. (3) The map-germ f (x1 , x2 ) = (x1 , x22 , x23 + x12 x2 ) is not stable. It has a nonimmersive point at (0, 0), which splits into two in the unfolding F (u, x1 , x2 ) = (u, x1 , x22 , x23 + x12 x2 + yx2 ) = : (u, fu (x1 , x2 )), for u = 0. In determining TA1 e f , the calculation of (3.31), (3.33) and (3.34) still apply, with insignificant modifications. The only change from (3.33) is that (3.32) now

3.3 First Calculations

77

shows that   T Ae f ⊃ x O o ∂/∂y3

(3.35)

and we need an extra calculation ⎛ ⎞



1 0 ∂ 0 2 ⎝ ⎠ = 0 tf x2 a(x1, x2 ) 2x2 x2 a(x1, x22 ) ∂y 2x1 x2 x12 + 3x22 ⎞ ⎛ 0 ⎠. = ⎝ 2x22 a(x1, x22 ) x12 x2 a(x1, x22 ) + 3x23 a(x1, x22 ) In view of (3.35) and what we know about the even terms, this completes the proof that ⎛

⎞ Oe + Oo ⎠. TAe f = ⎝ O e + O o e o o 2 O +x1 O +x2 O 1

It follows that TA1 e f is generated, as a vector space over F, by x2 ∂/∂y3 .

(3.36)

♦

Notation We will often use the following notation to display the results of our calculations of tangent spaces. Let α ∈ Nn be a multi-index, and x α = x1α1 · · ·xnαn the corresponding monomial. For a collection α (1) , . . ., α (r) of such multi-indices, we denote by (1)

(r)

O n {x α , . . ., x α } the set of convergent power series in n complex variables in which the coefficients (1) (r) of x α , . . ., x α are all equal to 0. Thus for example O n {1} is just mn . The expression ⎛

⎞ (1) (r) O n {x α , . . ., x α } ⎜ (1) (s) ⎟ ⎜O n {x β , . . ., x β }⎟ , ⎝ ⎠ .. . where there are p rows, will denote the set of p-tuples of elements of O n in which (1) (r) the first component has coefficients of x α , . . ., x α all equal to zero, the second (1) (s) has coefficients of x β , . . ., x β all equal to zero, etc. Then for f : (Fn , 0) →

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3 Left-Right Equivalence and Stability

(Fp , 0), T Ae f can be written in the form ⎛

⎞ (1) (r) O n {x α , . . ., x α } ⎜ (1) (s) ⎟ ⎜O n {x β , . . ., x β }⎟ + Sp {v1 , . . ., vk , . . .}, F ⎝ ⎠ .. .

(3.37)

where each vi is a column-vector in (O n )p . The first summand here simply lists the monomials x α ∂ui which do not lie in T Ae f , and the second summand lists the linear combinations of these monomials which do lie in T Ae f . For example, if the expression (3.37) is irredundant then the Ae -codimension of f is the number of α (i) plus the number of β (j ) plus · · · (we are listing the missing monomials row by row here) minus the number of vk . Example 3.6 1. If f (x1 , x2 ) = (x1 , x23 + x1 x2 ) then O 2 /f ∗ m2 O 2 is generated over F by 1, x2 and x22 . It follows from the preparation theorem that O 2 (source) is generated over O 2 (target) by these germs of functions, and that θ (f ) = O 2 ⊕ O 2 is generated over O 2 (target) by 2 1 0 x 0 x 0 , 2 . (3.38) , 2 , 2 , , 0 0 0 x2 1 x2 Since T Ae f is a module over O 2 (target), to prove that θ (f ) = T Ae f it is enough to find each of these six generators in T Ae f . This is the objective of Exercise 3.3.7 below. 2. A similar calculation shows that if f (x1 , x2 ) = (x1 , x23 + x12 x2 ) then the same six elements generate θ (f ) over O 2 (target). In Exercise 3.3.8 you are asked to 0 are in T Ae f . This shows that TA1 e f is generated over show that all but x2 0 O 2 (target) by . However, the calculations in Exercise 3.3.8 show that m2 x2 0 0 is contained in T Ae f , so that now generates TA1 e f (target) times x2 x2 ♦ over O 2 / m2 (target), i.e. over F.

Exercises for Sect. 3.3 1 Using the methods of Example 3.5(2) and (3), calculate the Ae -codimension, and an F-basis for TA1 e f , when (i) f (x) = (x 2 , x 5 ) (ii) f (x) = (x 2 , x 2k+1 )

3.3 First Calculations

(iii) (iv) (v) (vi)

79

f (x) = (x 3 , x 4 ) f (x1 , x2 ) = (x1 , x22 , x23 + x1k+1 x) f (x1 , x2 ) = (x1 , x22 , x12 x2 + x25 ) f (x1 , x2 ) = (x1 , x22 , x12 x2 + x22k+1).

2 Generalise the method of Example 3.5(2) and (3) to show that if f : (Fn , 0) → (Fn+1 , 0) has the form f (x, y) = (x, y 2 , xn p(x, y 2 )) (where x ∈ Fn−1 ) with respect to target coordinates (X1 , . . ., Xn−1 , Y, Z), then  n−1 T Ae f = i=1

∂ On ∂Xi





∂ ∂p ∂ e e 2 ∂p ⊕ O +y O y ◦ h, ◦ h, p ◦ h . ⊕ On ∂Y ∂y ∂x ∂Z

3 Use the preparation theorem (Corollary D.1) to prove that if f : (Fn , 0) → (Fn , 0) is the “fold map” f (x1 , . . ., xn ) = (x1 , . . ., xn−1 , xn2 ) then every germ h ∈ O n can be written uniquely in the form h = h1 ◦ f + xn h2 ◦ f for some smooth germs h1 , h2 . Hint: find an F-basis for O 2 /f ∗ m2 O 2 . 4 Let f (x1 , . . ., xn ) = x12 + · · · + xn2 ; show f is stable. From the Morse Lemma and Exercise 3.3 it follows that the germ of any Morse function is stable. 5 Show that if f : (Fn , 0) → (F, 0) is a stable germ which is not a submersion then f has a non-degenerate critical point at 0. Hint: since f ∈ m2n , stability implies that Jf = mn . 2 + · · · + xn2 ) is stable. 6 Show that the fold map f (x1 , . . ., xn ) = (x1 , . . ., xk , xk+1

7 The Whitney cusp map is the germ f (x1 , x2 ) = (x1 , x1 x2 + x23 ). Show that the elements of θ (f ) listed in (3.38) in fact lie in T Ae f . In Example 3.6, we used the preparation theorem to deduce from this that this germ is therefore stable. 8 Let f (x1 , x2 ) = (x1 , x23 ± x12 x2 ). (i) Show that all of the elements of θ (f ) listed in (3.38), except for the fifth (which we call g5 ), lie in T Ae f .

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3 Left-Right Equivalence and Stability

(ii) Show that x1 g5 and (x23 + x1 x2 )g5 both lie in T Ae f . In Example 3.6 we used the Preparation Theorem to conclude that Ae -codimension of f is 1. 9 The calculation of a basis for TA1 e f in Example 3.5(1) above suggests that the unfolding F (u, x) = (u, x 2 , x 3 + ux) should be interesting. Make drawings of the images of fu for u < 0, u = 0 and u > 0, and show that as u passes through 0, the family fu parameterises the first Reidemeister move of knot theory. 10 Make an analogous sequence of drawings of the images of the maps in the family fu (x1 , x2 ) = (x1 , x22 , x23 + x12 x2 + ux2 ) suggested by Example 3.5(3). We will return to this example in Chap. 8. 11 Show (i) that if f : (F, 0) → (F2 , 0) is a non-immersive germ such that the two vectors f  (0) and f  (0) are linearly independent then f is left-right-equivalent to the cusp t → (t 2 , t 3 ). (ii) if f : (F, 0) → (F2 , 0) is a non-immersive germ such that the two vectors f  (0) and f  (0) are not linearly independent then codimAe f > 1. 12 Let f be the germ of Example 3.5(2). (i) Check that j 1 f : (R2 , 0) → L(R2 , R3 ) meets, and is transverse to, the submanifold  1 of linear maps of corank 1. Suggestion: rather than directly applying the definition of transversality, Definition 2.17, use the characterisation of transversality in Exercise 2.6.1. (ii) In fact transversality to  1 characterises the Whitney umbrella: any mapgerm g : (F2 , 0) → (F3 , 0) with this property is A -equivalent to f . To prove this, begin by choosing coordinates in which g takes the form g(u, v) = (u, g2 (u, v), g3 (u, v)) (cf Lemma 2.3). It is then not hard to find explicit coordinate changes which reduce g to the form g(u, v) = f (u, v) + higher order terms. (iii) Using the fact that f is 2- determined for A -equivalence, (see Chap. 6 below), deduce that any stable non-immersive germ (F2 , 0) → (F3 , 0) is A -equivalent to f . Hint: use Proposition A.1. 13 If f : (F2 , 0) → (F3 , 0) is not an immersion then the ideal f ∗ m3 generated in O 2 by the three component functions of f is strictly contained in m2 . It follows that dimF O 2 /f ∗ m3 ≥ 2. Show that every germ for which this dimension is exactly 2 (as in all the examples above) is A -equivalent to one of the form f (u, x) = (u, x 2 , xp(u, x 2 )). Details can be found in [Mon85]. 14 Suppose that f : (Fn , 0) → (Fp , 0), of the form f (x) = (x1 , . . ., xp−1 , fp (x)) has finite Ae -codimension. Then the map-germ F : (Fn+1 , 0) → (Fp , 0) defined by F (x, y) = (x1 , . . ., xp−1 , fp (x) + y 2 ) has TA1 e F  TA1 e f .

3.4 Multi-Germs

81

15 Given a germ f : (Fn , 0) → (Fp , 0) and diffeomorphisms ϕ : (Fn , 0) → (Fn , 0) and ψ : (Fp , 0) → (Fp , 0), find (i) a natural isomorphism of O p -modules K : θ (f ) → θ (f ◦ ϕ), for which K(T Ae f ) = T Ae (f ◦ ϕ) (ii) a natural isomorphism of O p -modules L : θ (f ) → θ (ψ ◦ f ), for which L(T Ae f ) = T Ae (ψ ◦ f ). For (i), the diagram T Fn

dϕ

T Fn

df

T Fp

Fn

ϕ

Fn

f

Fp

in which elements of θ (f ) and θ (f ◦ ϕ) are shown as dashed arrows, can help to guide the definition of K. A similar diagram will help with (ii). Conclude that if f and g are G -equivalent, for G = R, L or A , then θ (f )/T Ae f and θ (g)/T Ge g are isomorphic as O p -modules. 16 Use the approach of Example 3.6 to find a F-basis for TA1 e f when f (x, y) = (x, y 3 + x k y).

3.4 Multi-Germs We now turn to multi-germs, beginning with the Reidemeister moves R2 and R3. The first requirement is an adequate notation.

3.4.1 Notation For each positive integer r we define (Fnr , 0r ) =

r  (Fn , 0). i=1

We identify a multi-germ f : (Fnr , 0r ) → (Fp , 0) with an r-tuple of map-germs f (i) : (Fn , 0) → (Fp , 0), i = 1, . . ., r. When necessary, we will refer to the j ’th base-point as 0j . We will not distinguish in our notation between the local coordinates around 0i and 0j for i = j ; each set of coordinates will be denoted x1 , . . ., xn . The F-algebra of germs of functions on (Fnr , 0r ) (analytic or C ∞ ,

82

3 Left-Right Equivalence and Stability

depending on the case) will be denoted O n,r . Clearly r

O n,r =

O Fn ,0i ; i=1

we write its elements as row vectors. It is a semi-local ring, with maximal ideals mn,i = {g ∈ O n,r : gi (0i ) = 0}, i = 1, . . ., r and Jacobson radical (the intersection of its maximal ideals) mJ . Let f : (Fnr , 0r ) → (Fp , 0) be a multi-germ. Evidently r

θ (f ) =

θ (f (i) ). i=1

In keeping with our notation up to now, we write elements of θ (f (i) ) as column vectors, and therefore elements of θ (f ) as p × r matrices, in which the j ’th column represents an element of θ (f (j ) ), and has entries in O Fn ,0i . The space of germs of vector fields on (Fnr , 0r ) is denoted θn,r , and once again is a direct sum. The homomorphism tf : θn,r → θ (f ) is the direct sum of the homomorphisms tf (i) : θFn ,0i → θ (f (i) ). The homomorphism ωf : θp → θ (f ) has the form ⎛

η1 ◦ f (1) ⎜ .. (1) (r) ωf (η) = (η ◦ f , . . ., η ◦ f ) = ⎝ .

⎞ · · · η1 ◦ f (r) ⎟ .. .. ⎠. . .

ηp ◦ f (1) · · · ηp ◦ f (r) Warning In this notation, the action of O n,r on θ (f ) takes a slightly off-putting form in which the r-tuple (a1 , . . ., ar ) ∈ O Cn ,0r acts on a p × r matrix representing an element of θ (f ) by multiplying the j ’th column in the matrix by aj for j = 1, . . ., r. Example 3.7 (i) According to Example 3.4, the bi-germ of Reidemeister move R2 is equivalent to the bi-germ

x→  (x, 0) x→  (x, h(x))

for h(x) = x 2 . Let us calculate T Ae f . We have



∂ a(x) 0 tf a(x) , 0 = 0 0 ∂x

(3.39)

3.4 Multi-Germs

83

so

O1 0 T Ae f ⊃ ; 0 0

(3.40)

now since



∂ a(x) a(x) = ωf a(y1) 0 0 ∂y1 it follows from (3.40) that

0 O1 T Ae f ⊇ 0 0

(3.41)

also. Since



∂ 0 b(x) tf 0, b(x) = , 0 h (x)b(x) ∂x

(3.42)

it follows from (3.41) that

0 0 T Ae f ⊃ , 0 Jh

(3.43)

where Jh is the Jacobian ideal of h. Contributions to the bottom left-hand entry in T Ae f come only from ωf : ωf





∂ 0 0 η2 = . η2 (x, 0) η2 (x, h(x)) ∂y2

(3.44)

We have η2 (x, 0) = 0 if and only if η2 (y1 , y2 ) is divisible by y2 , in which case η2 (x, h(x)) ∈ (h) ⊂ Jh (recall that for h ∈ O 1 , h ∈ Jh always). Thus T Ae f

! 0 0 0 0 = . 0 O1 0 Jh

A map :

O1 θ (f ) → Jh T Ae f

(3.45)

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3 Left-Right Equivalence and Stability

may now be defined as follows. For a ∈ O 1 denote the class of a modulo Jh by a. ¯ Then 00 (a) ¯ = + T Ae f. 0a It is well defined and injective by (3.45). It is surjective, by (3.40), (3.41) and (3.44). Note that  is O 2 -linear, where O 1 is an O 2 -module via f (2) . We have proved Proposition 3.10 TA1 e f is isomorphic as O 2 -module to O 1 /Jh .



Recall that for R2, h(x) = x 2 , and dimC O 1 /Jh = 1. Thus we have shown that R2 has Ae -codimension 1, and that the node (the transverse crossing of two immersed branches), where h(x) = x, is infinitesimally stable. Since the germ h in (3.39) can be perturbed to have ν = order h non-degenerate zeros, the germ f of this example can be perturbed to a bi-germ with ν nodes.

So the number of nodes is one more than the codimension. In fact the image of a ν-nodal perturbation is homotopy-equivalent to a wedge of ν − 1 circles, and so we conclude Corollary 3.3 The bi-germ f of Example 3.7 can be perturbed to a germ whose image has the homotopy-type of a wedge of circles, with the number of circles in the image equal to the Ae -codimension of f . We will return to this theme many times! The relation between the Ae -codimension of a map-germ and the geometry and topology of a stable perturbation is one of the most interesting aspects of the subject. It is clearest in the complex analytic case, but we can see it in the real C ∞ case if we are observant—in particular in the three Reidemeister moves. Example 3.7 (continued) (ii) The third Reidemeister move, R3, is left-right equivalent to ⎧ ⎨ x → (x, 0) x → (0, x) ⎩ x → (x, x).

3.4 Multi-Germs

85

Let a ∈ O 1 . We have tf (a∂x , 0, 0) =



a00 000

tf (0, a∂x , 0) =

000 0a0

so T Ae f ⊃



O1 0 0 . 0 O1 0

(3.46)

Let a, b ∈ O 2 be functions of y1 and y2 , respectively. Then

  a a(0) a ωf a∂y1 = 0 0 0

  ωf b∂y2 =



0 00 b(0) b b

so taking into account (3.46),

0 0 m1 . T Ae f ⊃ 0 0 m1

(3.47)

Similarly, ωf



∂ 0 0 0 = a a a(0) a ∂y2

ωf



∂ b(0) b b = b 0 00 ∂y1

so taking into account (3.46) and (3.47), T Ae f ⊃

0 m1 0 m1 0 0

and thus

T Ae f ⊃ mJ θ (f ).

(3.48)

Let L be the 6-dimensional complement to mJ θ (f ) in θ (f ), L = SpF





 100 000 , . . ., . 000 001

The intersection of T Ae f with L is generated by the five elements tf (∂/∂x, 0, 0), tf (0, ∂/∂x, 0), tf (0, 0, ∂/∂x), ωf (∂y1 ), ωf (∂y2 ). These are linearly independent. So dim TA1 e f = 1.

♦

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3 Left-Right Equivalence and Stability

Exercises for Sect. 3.4 1 We say that the multi-germ g : (Fnr , 0r ) → (Fp , 0) is contained in (is strictly contained in) the multi-germ g : (Fns , 0s ) → (Fp , 0), and write g ⊆ f (g ⊂ f ) if the mono-germs making up g form a subset (form a proper subset) of the monogerms making up f . (i) Show that if g ⊆ f then codimAe g ≤ codimAe f . (ii) Show that if n < p, g ⊂ f , and g is not stable then codimAe g < codimAe f . 2 Show that if f : (Fr , 0r ) → (F2 , 0) has r > 3 then codimAe f > 1. Suggestion: consider dimF

θ (f ) . mJ θ (f ) + T Ae f

3 Show that R1, R2 and R3 are the only map-germs (Fr , 0r ) → (F2 , 0) of Ae codimension 1. Most of the proof can be assembled from earlier exercises. 4 Show that if f : (Fn2 , 02 ) → (F, 0) is a bi-germ with both branches nonsubmersive then f cannot be stable. What is the smallest possible Ae -codimension for such a germ? 5 Classifying bi-germs of immersions: we measure the contact between the two branches of a bi-germ (F2 , 02 ) → (F2 , 0)

g:

x → g1 (x) x → g2 (x)

as follows: pick an equation h1 for the image of g1 , and define the order of contact ν(g1 , g2 ) to be the order (lowest non-zero derivative at 0) of g2∗ (h1 ). (i) Find ν for the bi-germs



x → (x, 0) x → (x 2 , x 3 ) x→  (x, r(x)) (a) (b) (c) x → (x, r(x)) x → (x, r(x)) x→  (x 2 , x 3 ) (ii) Show (a) The definition of ν(g1 , g2 ) is independent of choice of h1 ; (b) ν(g2 , g1 ) = ν(g1 , g2 ); (c) If g1 and g2 are both immersions then g is A -equivalent to the germ

g

(ν)

:

x→  (x, 0) . x→  (x, x ν )

3.4 Multi-Germs

87

6 Suppose that f : (Fn2 , 02 ) → (Fn+1 , 0) is a bi-germ of immersions whose two branches meet transversely. Show that f is A -equivalent to

 (x1 , . . ., xn−1 , xn , 0) (x1 , . . ., xn ) → .  (x1 , . . ., xn−1 , 0, xn ) (x1 , . . ., xn ) →

Hint: let g1 and g2 be equations for the images of the two branches. Show that the transversality of the two branches is equivalent to the linear independence of d0 g1 and d0 g2 . So transversality implies that g1 and g2 can be taken as the last two members of a system of coordinates on Fn+1 . In this system, the image of f has equation xn xn+1 = 0. Now compose the new coordinates on Fn+1 with f (i) to get new coordinates on (Fn , 0i ). 7 Generalise the approach of the last question to the cases (a) where f : (Fnr , 0r ) → Fn+1 , 0), r ≥ 3, is a multi-germ of immersion with the images of the r branches meeting in general position, and (b) where f : (Fnr , 0r ) → (Fn+k , 0), k ≥ 2, is a multi-germ of immersion with the images of the r branches meeting in general position. 8 Show, by a direct calculation, that a multi-germ of immersions whose images meet in general position is infinitesimally stable. That this is true follows also from the much more general Theorem 3.3, below. 9 Let f : (F23 , 03 ) → (F4 , 0) be the tri-germ ⎧ ⎨ (x1 , x2 ) → (x1 , x2 , 0, 0) (x , x ) → (x1 , x2 , x1 , x2 ) . ⎩ 1 2 (x1 , x2 ) → (0, 0, x1 , x2 ) (a) Show that every tri-germ in these dimensions, consisting of three immersions meeting two-by-two transversely, is left-right equivalent to f . (b) Deduce that the Ae -codimension of f is 2. 10 Let f : (Fn2 , 02 ) → (Fn+1 , 0) be a bi-germ of immersions whose images meet tangentially at 0. (a) Show that in suitable coordinate f can be written in the form

where h ∈ m2 .

x→  (x, 0)) , x→  (x, h(x))

(3.49)

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3 Left-Right Equivalence and Stability

(b) Show that T Ae f

! 0 0 0 0 = 0 On 0 (h) + Jh

(3.50)

and deduce that TA1 e f 

On . (h) + Jh

Hint: the calculations of Example 3.7 go through unchanged until the line corresponding to (3.45). We call the function h the separation function of the bi-germ f . 11 Here we assume familiarity with the Milnor fibre of an isolated hypersurface singularity, and introduce the notion of stable perturbation without a formal definition. It is sufficient to consider a perturbation of the bi-germ in which the two immersions become transverse. (a) Show that when F = C, the image Xt of a stable perturbation of a bi-germ of the form (3.49) has reduced homology satisfying "q (Xt ) = H



Zμ(h) if q = n . 0 otherwise

(3.51)

(b) Show further that this image is homotopy-equivalent to a wedge of μ(h) (n−1)spheres.

3.5 Infinitesimal Stability Implies Stability Next, we prove some lemmas which are needed in order to prove Theorem 3.2. The first is a Thom–Levine type result similar to Lemma 2.6, namely, an infinitesimal condition for the triviality of a 1-parameter unfolding. Given any k, we consider germs of vector fields ζ on Fk × F such that ζ (t) = 1, that is, of the form ζ =

k  i=1

ζi (x, t)

∂ ∂ + . ∂xi ∂t

(3.52)

Lemma 3.4 Let f : (Fn , S) → (Fp , 0) be a smooth germ and F a 1-parameter unfolding of f . Then F is trivial  if and onlyif there exist germs of vector fields ξ on (Fn × F, S × {0}) and η on Fp × F, (0, 0) such that ξ(t) = 1, η(t) = 1 and dF ◦ ξ = η ◦ F.

3.5 Infinitesimal Stability Implies Stability

89

Proof Suppose first that F is trivial. Then there exist diffeomorphisms ,  which are unfoldings of the identity in Fn , Fp , respectively, such that F =  ◦ G ◦ −1 , where G = f × id. We define ξ, η as the vector fields given by ξ = d ◦

∂ ◦ −1 , ∂t

η = d ◦

∂ ◦  −1 . ∂t

(3.53)

Since ,  are unfoldings of the identity, we have ξ(t) = 1, η(t) = 1. We use the fact that dG ◦

∂ ∂ = ◦ G, ∂t ∂t

and the chain rule: ∂ ∂ ∂ ◦ −1 = d(F ◦ ) ◦ ◦ −1 = d( ◦ G) ◦ ◦ −1 ∂t ∂t ∂t ∂ ∂ = d ◦ dG ◦ ◦ −1 = d ◦ ◦ G ◦ −1 = η ◦  ◦ G ◦ −1 = η ◦ F. ∂t ∂t

dF ◦ ξ = dF ◦ d ◦

Conversely, assume there exist vector fields ξ, η such that ξ(t) = 1, η(t) = 1 and dF ◦ ξ = η ◦ F . By taking the integral flows of ξ, η we define ,  as the unique diffeomorphisms which satisfy Eqs. (3.53). Since ξ(t) = 1 and η(t) = 1, it follows that ,  are unfoldings of the identity on Fn , Fp , respectively. Now let G =  −1 ◦ F ◦ . Again, by the chain rule we have dG ◦

∂ ∂ ∂ = d( −1 ◦ F ◦ ) ◦ = d −1 ◦ dF ◦ d ◦ = d −1 ◦ dF ◦ ξ ◦  ∂t ∂t ∂t ∂ = d −1 ◦ η ◦ F ◦  = d −1 ◦ η ◦  ◦ G = ◦ G. ∂t

If the unfolding G is written as G(x, t) = (g(x, ˜ t), t), then the above condition means that ∂ g/∂t ˜ = 0. Thus, G is the constant unfolding.   If ζ is a germ of vector field in Fk × F as in (3.52), then we write ζ˜ = ζ −

 ∂ ∂ = ζi (x, t) . ∂t ∂xi k

i=1

We can view ζ˜ as a vector field along the projection πk : Fk × F → Fk onto the first factor—that is, as an element of θ (πk ). We can also view ζ˜ as a timedependent vector field on Fk : given a representative of ζ in an open neighbourhood U × D ⊂ Fk × F, for each t ∈ D, ζt is the vector field on U given by (ζt )x = ζ˜(x,t ). Let F : (Fn × F, S × {0}) → (Fp × F, 0) be a 1-parameter unfolding given by F (x, t) = (f˜(x, t), t). The infinitesimal condition dF ◦ ξ = η ◦ F can be written

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3 Left-Right Equivalence and Stability

now in matrix notation as 

˜

dft ∂∂tf 0 1



ξt 1

=

ηt ◦ ft 1

which turns out to be equivalent to ∂ f˜ + dft ◦ ξt = ηt ◦ ft . ∂t

(3.54)

In the next definition we introduce relative versions of the maps tf and ωf and of the module TA1 e f for unfoldings. We define them for unfoldings with several parameters, since we will need them later. Definition 3.9 We denote by θn+d/d the set of germs of relative vector fields: vector fields on Fn × Fd “in the Fn direction”, i.e. of the form n 

ξi (x, u)∂/∂xi .

i=1

We will use the same notation for germs at S × {0}. Likewise, θ (F /d) will denote the space of relative infinitesimal deformations of F : infinitesimal deformations in the Fp direction, i.e. vector fields along F (i.e.in θ (F )) of the form 

ηj (x, u)∂/∂yj .

j

We will sometimes denote the domain and target of a map-germ f : (Fn , S) → (Fp , 0) by X and Y , and the parameter space of its unfolding F by U ; in this case θn+d/d becomes θX×U/U , θp+d/d becomes θY ×U/U and θ (F /d) becomes θ (F /U ). Let F be any d-parameter unfolding of f given by F (x, u) = (f˜(x, u), u) = (fu (x), u). Then tF and ωF restrict to morphisms trel F : θn+d/d → θ (F /d) and ωrel F : θp+d/d → θ (F /d). We set TA1 e F /d =

θ (F /d) . trel F (θn+d/d ) + ωrel F (θp+d/d )

This is an O p+d -module via F , in the same way that TA1 e f is an O p -module via f . It is called the relative T 1 of the unfolding F .

3.5 Infinitesimal Stability Implies Stability

91

In effect, TA1 e F /d is just a version of TA1 e f with parameters u1 , . . . , ud , and moreover TA1 e F /d {u1 , . . . , ud } · TA1 e F /d

= TA1 e f.

(3.55)

Lemma 3.5 Let F : (Fn × Fd , S × {0}) → (Fp × Fd , (0, 0)) be an unfolding of the A -finite germ f : (Fn , S) → (Fp , 0), let G1 , . . . , Gk ∈ θ (F /d), and for i = 1, . . ., k let gi (x) = Gi (x, 0). The following statements are equivalent: (a) the classes of G1 , . . . , Gk generate TA1 e F /d over Od , (b) the classes of g1 , . . . , gk generate TA1 e f over F. Proof The implication (a) ⇒ (b) is immediate from (3.55). To see the converse, we define M :=

θ (F /d) , trel F (θn+d/d )

M0 :=

M ∼ θ (f ) . = {u1 , . . . , ud } · M tf (θn )

We apply the multi-germ version of the preparation theorem (Corollary D.2) simultaneously to M and M0 . These are finitely generated, over O n+d and O n , p p respectively, (recall that θ (F /d) ∼ = O n+d and θ (f ) ∼ = O n ). Assuming condition (b), θ (f ) = tf (θn ) + ωf (θp ) + SpF {g1 , . . . , gk } = tf (θn ) + O p ·{∂/∂y1 , . . . , ∂/∂yp } + SpF {g1 , . . . , gk }, which shows that M0 is finitely generated over O p . Hence M0 /(f ∗ mFp ,0 M0 ) is finitely generated over F. However, M M0 ∼ , = ∗ F ∗ mp+d M f mp M0 since F is an unfolding of f , and it now follows by the preparation theorem that M is finitely generated over O p+d . Next, we use the preparation theorem again, but this time with the module TA1 e F /U and the germ of the projection π : (Fp × Fd , 0) → (Fd , 0). Since TA1 e F /d is a quotient of M, it also is finitely generated over O p+d . Moreover, TA1 e F /d π ∗ md · TA1 e F /d

=

TA1 e F /d {u1 , . . . , ud } · TA1 e F /d

∼ = TA1 e f,

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3 Left-Right Equivalence and Stability

which is generated over F by the classes of g1 , . . . , gk . Since gi is the image of the class of Gi under the isomorphism of the previous line, it follows by the preparation theorem that TA1 e F /d is generated over O d by the classes of G1 , . . . , Gk .   Proof (of Theorem 3.2) Assume f is stable. For each ζ ∈ θ (f ), we consider the 1-parameter unfolding F (x, t) = (f (x, t), t) given by f (x, t) = f + tζ . Because f is stable, F is trivial and by Lemma 3.4 there exist vector fields ξ, η such that ξ(t) = 1, η(t) = 1 and dF ◦ ξ = η ◦ F . By (3.54) this means that ∂f (x, t) + dx f (x, t) ◦ ξ = η ◦ F, ∂t and evaluating at t = 0,

 ∂ft  ζ = = −df ◦ ξ0 + η0 ◦ f ∈ T Ae f. ∂t t =0

Conversely, assume that TA1 e f = 0. We first prove that any 1-parameter unfolding F (x, t) = (ft (x), t) is trivial. In fact, we know by Lemma 3.5 that TA1 e F /d = 0. Hence, there exist vector fields ξ˜ and η˜ such that ∂ft = dft ◦ ξt + ηt ◦ ft . ∂t Again by displayed equation (3.54) we have dF ◦ ξ = η ◦ F , where ξ = −ξ˜ + ∂/∂t and η = η˜ + ∂/∂t, so F is trivial by Lemma 3.4. We show now that any r-parameter unfolding F (x, u) = (fu (x), u) is trivial, by induction on r. We have already proved the case r = 1. Assume now the result is true for r − 1. Consider the (r − 1)-parameter unfolding F1 obtained from F by taking ur = 0. By the induction hypothesis, F1 is trivial and hence, equivalent to f × id. But this implies that F1 is also A -equivalent to f × id as a map-germ and hence TA1 e F1 = 0. Since F is a 1-parameter unfolding of F1 we deduce that F is a trivial unfolding of F1 and hence, a trivial unfolding of f .  

Exercises for Sect. 3.5 1 Carefully justify Eq. (3.55).

3.6 Stability of Multi-Germs Suppose S = {s1 , . . . , sr }. A natural question is how the stability of f : (Fn , S) → (Fp , 0) is related to the stability of each branch fi := f |(Fn ,xi ) : (Fn , si ) → (Fp , 0), i = 1, . . . , r. To answer this question we need to introduce a new concept.

3.6 Stability of Multi-Germs

93

Definition 3.10 For each multi-germ f : (Fn , S) → (Fp , 0), we define   τ (f ) = ev (ωf )−1 ((f ∗ mFp ,0 )θ (f ) + tf (θFn ,S )) , where ev : θFp ,0 → T0 Fp is evaluation at 0, given by ev(η) = η0 . Clearly, τ (f ) is an F-vector subspace of T0 Fp . In the case that f is stable, τ (f ) has a very nice geometrical interpretation as follows. Let us fix a representative f : U → V of the germ, where U, V are open neighbourhoods of S and 0 in Fn and Fp , respectively, and let C be the critical set of f . Denote by Iso(f ) the subset of points y ∈ V such that the multi-germ of f at f −1 (y) ∩ C is A -equivalent to the multi-germ of f at S. This subset is called the isosingular locus of f . Then Iso(f ) is in fact a submanifold of V , whose tangent space at the origin is precisely τ (f ). We will prove this in Sect. 7.3, but for now it may help to understand the arguments. Example 3.8 In the case of the fold germ (Fn , 0) → (Fk+1 , 0) given by 2 (x1 , . . ., xn ) → (x1 , . . ., xk , xk+1 + · · · + xn2 ),

the isosingular locus Iso(f ) is evidently the hyperplane {(y1, . . ., yk+1 ) : yk+1 = 0}. We compare this with the space τ (f ). We have   τ (f ) = ev (ωf )−1 ((f ∗ mFk+1 ,0 )θ (f ) + tf (θFn ,S ))   2 + · · · + xn2 )θ (f ) = ev (ωf )−1 (x1 , . . ., xk , xk+1 + O n {∂y1 , . . ., ∂yk , xk+1 ∂yk+1 , . . ., xn ∂yk+1 }))   = ev (ωf )−1 (O n ⊕· · · ⊕ O n ⊕ mn ) = ev (O k+1 ⊕· · · ⊕ O k+1 ⊕ mk+1 ) = Fk × 0, This is indeed the tangent space at 0 to Iso(f ).

♦

We introduce now an important mapping due to Mather, which can be seen as an analogue of the Kodaira–Spencer map in the context of complex manifolds. Note that ωf (mFp ,0 θFp ,0 ) ⊂ (f ∗ mFp ,0 )θ (f ), so ωf induces the homomorphism ωf :

θFp ,0 θ (f ) . −→ mFp ,0 θFp ,0 tf (θFn ,S ) + (f ∗ mFp ,0 )θ (f )

94

3 Left-Right Equivalence and Stability

Moreover, by means of the evaluation map, we can identify θFp ,0 /mFp ,0 θFp ,0 with T0 Fp , and thus consider ωf as a map T0 Fp −→

θ (f ) , tf (θFn ,S ) + (f ∗ mFp ,0 )θ (f )

whose kernel is precisely τ (f ). The following lemma now characterises stability in terms of ωf . Lemma 3.6 A multi-germ f : (Fn , S) → (Fp , 0) is stable if and only if ωf is an epimorphism. Proof In order to simplify the notation, we set: R = O Fp ,0 , m = mFp ,0 , S = O Fn ,S , N = θFp ,0 , M = θFn ,S , L = θ (f ). If f is stable then TA1 e f = 0, that is, ωf (N) + tf (M) = L. For any ζ ∈ L, there exist ξ ∈ M and η ∈ N such that ζ = tf (ξ ) + ωf (η). This gives ωf ([η]) = [ζ ] and thus, ωf is surjective. Conversely, suppose now that ωf is surjective. This implies that ωf (N) + tf (M) + (f ∗ m)L = L.

(3.56)

We define L = L/tf (M) and denote by π : L → L the canonical projection. Note that L is a finitely generated S-module (since in fact L ∼ = S p ). Then (3.56) may be rewritten as π ◦ ωf (N) + (f ∗ m)L = L .

(3.57)

Considering L as an R-module via f , (3.57) becomes π ◦ ωf (N) + mL = L .

(3.58)

Since π ◦ ωf (N) is finitely generated over R, it follows that L /mL is also finitely generated over R, and hence finitely generated over F = R/ m (since m annihilates it). Therefore by the Preparation Theorem D.2, L is finitely generated over R. Now from (3.58) it follows by Nakayama’s Lemma that π ◦ ωf (N) = L and hence, ωf (N) + tf (M) = L.   We recall the definition of regular intersection of subspaces E1 , . . . , Er of a vector space F of finite dimension. Definition 3.11 We say that E1 , . . . , Er have regular intersection (or meet in general position) if codim(E1 ∩ · · · ∩ Er ) = codim(E1 ) + · · · + codim(Er ).

3.6 Stability of Multi-Germs

95

An equivalent definition is that the canonical mapping F −→ (F /E1 ) ⊕ · · · ⊕ (F /Er ), is surjective. This follows easily from the fact that the kernel is precisely E1 ∩ · · · ∩ Er . Theorem 3.3 A multi-germ f : (Fn , S) → (Fp , 0) is stable if and only if each branch fi : (Fn , si ) → (Fp , 0) is stable and τ (f1 ), . . . , τ (fr ) meet in general position. Proof We continue with the notation introduced in the proof of Lemma 3.6. By the lemma, we have f is stable if and only if ωf : T0 Fp −→

L (f ∗ m)L + tf (M)

is surjective, (3.59)

and for each i = 1, . . . , r fi is stable if and only if ωfi : T0 Fp −→

Li is surjective, (fi∗ m)Li + tfi (Mi ) (3.60)

where now Mi = θFn ,si and Li = θ (fi ). Note that the kernel of ωfi is τ (fi ). Thus, if fi is stable we have T0 F p ∼ Li . = ∗ τ (fi ) (fi m)Li + tfi (Mi )

(3.61)

On the other hand, we also have an isomorphism L ∼ = (f ∗ m)L + tf (M)

r i=1

Li , (fi∗ m)Li + tfi (Mi )

(3.62)

from which it follows, by (3.61), that we can write ωf in the form ωf : T0 Fp −→ (T0 Fp /τ (f1 )) ⊕ · · · ⊕ (T0 Fp /τ (fr )).

(3.63)

It is now immediate that ωf is surjective if and only if each ωfi is surjective and the τ (fi ) have regular intersection. By (3.59) and (3.60), this proves the theorem.  

Exercises for Sect. 3.6 1 Show that if f : (Fn , S) → (Fn+1 , 0) is stable then |S| ≤ n + 1. 2 If f : (Fn , S) → (Fn+k , 0) is stable, how big can |S| be?

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3 Left-Right Equivalence and Stability

3 If f : (Fn , S) → (Fp , 0) is stable, with n ≥ p, and each point of S is a critical point (i.e., f is not a submersion), how big can |S| be? 4 Find the complete list of stable singularities of multi-germs f : (Fn , S) → (Fp , 0), where all the points si ∈ S are critical points of f , in the following cases: (i) (ii) (iii) (iv)

p = 1 (scalar functions), n = 1, p = 2 (plane curves), n = 2, p = 2 (plane-to-plane singularities), n = 2, p = 3 (surfaces to 3-space).

5 (i) Suppose that f : (Fn , S) → (Fp , 0) is a stable multi-germ with branches f (i) : (Fn , si ) → (Fp , 0). Show that τ (f ) =

|S| !

τ (f (i) ).

i=1

(ii) Suppose that f1 : (Fn , S1 ) → (Fp , 0) and f2 : (Fn , S2 ) → (Fp , 0) are stable multi-germs, with S1 ∩ S2 = ∅. Show that the multi-germ {f1 , f2 } : (Fn , S1 ∪ S2 ) → (Fp , 0) is stable if and only if τ (f1 ) −  τ (f2 ).

Chapter 4

Contact Equivalence

4.1 The Contact Tangent Space Our calculations in Chap. 3 are somewhat atypical. Calculating T Ae f is generally rather complicated. Checking that a given map-germ is stable, however, is made much easier by a theorem of John Mather, which makes use of an auxiliary module known as the contact tangent space, denoted T Ke f and defined by T Ke f = tf (θFn ,S ) + f ∗ mp θ (f ).

(4.1)

Note that f ∗ mp is simply the ideal in O Cn ,0 generated by the component functions of f . Of course, we have already met T Ke f , though not by that name, in Theorem 3.3, in the proof of Proposition 3.8, and, implicitly, in the definition of τ (f ) in Definition 3.10. When p = 1, T Ke f is just the ideal (f, ∂f/∂x1 , . . ., ∂f/∂xn ) of O Fn ,S . In any case it is always an O Fn ,S -module, which makes calculating with it very much easier than calculating T Ae f . Like T Ae f , T Ke f is the “extended” tangent space to the orbit of f under the action of a group, in this case the contact group K , which we will discuss below. The role of T Ke f here does not involve this geometrical interpretation, so it can safely be left until we have seen the significance of T Ke f . Mather’s theorem is Theorem 4.1 For any multi-germ f : (Fn , S) → (Fp , 0), the following are equivalent: 1. T Ae f = θ (f ) (i.e. TA1 e f = 0, so f is stable). 2. T Ke f + SpF {∂/∂y1 , . . ., ∂/∂yp } = θ (f ) p+1 3. T Ke f + SpF {∂/∂y1 , . . ., ∂/∂yp } + mFn ,S θ (f ) = θ (f ).

© Springer Nature Switzerland AG 2020 D. Mond, J. J. Nuño-Ballesteros, Singularities of Mappings, Grundlehren der mathematischen Wissenschaften 357, https://doi.org/10.1007/978-3-030-34440-5_4

97

98

4 Contact Equivalence

Proof (1) ⇒ (2) and (2) ⇒ (3) are trivial, since the left-hand sides of the equalities increase from each statement to the next. The statement that (2) ⇒ (1) is just Lemma 3.6. To see that (3) ⇒ (2), first we simplify the notation. We will write as follows: O n = O Fn ,S ,

mn = mFn ,S ,

θn = θFn ,S ,

O p = O Fp ,0 ,

mp = mFp ,0 ,

θp = θFp ,0 .

Now suppose that (3) holds and let α1 , . . ., αp ∈ mn . We will show that p+1 p α1 · · ·αp ∂/∂yi ∈ T Ke f + mn θ (f ). Because every member of mn θ (f ) is a sum of such elements, it will follow that p

p

mn θ (f ) ⊂ T Ke f + mn mn θ (f ), and therefore, by Nakayama’s Lemma, that p

mn θ (f ) ⊂ T Ke f. p+1

To see that α1 · · ·αp ∂/∂yi ∈ T Ke f + mn dimC

θ (f ), observe that because, by (3),

θ (f ) p+1

T Ke f + m n

θ (f )

≤ p,

the p + 1 elements ∂/∂yi , α1 ∂/∂yi , . . ., α1 · · ·αp ∂/∂yi cannot be linearly independent. Thus there exist c0 , . . ., cp ∈ F, not all zero, such that c0 ∂/∂yi + c1 α1 ∂/∂yi + · · · + cp α1 · · ·αp ∂/∂yi = 0 p+1

in θ (f )/T Ke f + mn can be rewritten as

(4.2)

θ (f ). Let cj be the first of the ci to be non-zero. Then (4.2) p+1

(cj α1 · · ·αj + · · · + cp α1 · · ·αp )∂/∂yi ∈ T Ke f + mn

θ (f ).

The left-hand side here is an O n -unit times α1 · · ·αj ∂/∂yi , and thus α1 · · ·αj ∂/∂yi , p+1   and hence also α1 · · ·αp ∂/∂yi , are members of T Ke f + mn θ (f ). Corollary 4.1 Whether or not f : (Fn , S) → (Fp , 0) is stable is determined by its (p + 1)-jet.

4.1 The Contact Tangent Space

99

Proof If j p+1 f = j p+1 g then p+1

T Ke f + m n

p+1

θ (f ) = T Ke g + mn

θ (g).  

So (3) holds for f if and only if it holds for g.

Example 4.1 In the following calculations, we use the letter L to denote the subspace SpF { ∂y∂ 1 , . . ., ∂y∂p }. (1) We apply Theorem 4.1 to the map-germ f (u, x) = (u, x 2 , ux) of Example 3.5(2). We have T Ke f = tf (θ2 ) + f ∗ m3 θ (f ) = O 2 ·{∂f/∂a, ∂f/∂x} + (u, x 2 )θ (f ) ⎧⎛ ⎞ ⎛ ⎞⎫ 0 ⎬ ⎨ 1 ⎝ ⎠ ⎝ = O2 · 0 , 2x ⎠ + (u, x 2 )θ (f ) ⎩ ⎭ x u It is easy to show that T Ke f + L = θ (f ); in particular, since (x, y 2 )θ (f ) ⊂ T Ke f , it is necessary only to check for terms of the form x∂/∂yi . We have ⎛ ⎞ ⎛ ⎞ x 1 ⎝ 0 ⎠ = x ⎝0⎠ mod (u, x 2 )θ (f ) 0 x ⎛ ⎞ ⎛ ⎞ 0 0 1 ⎝x ⎠ = ⎝2x ⎠ mod (u, x 2 )θ (f ) 2 0 u

and so lies in T Ke f

and so lies in T Ke f

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 1 1 ⎝ 0 ⎠ = ⎝ 0⎠ − ⎝0⎠ ∈ T Ke f + L x x 0 (2) The same theorem can easily be used to show that the map-germs 1. f : (F3 , 0) → (F3 , 0), f (a, b, x) = (a, b, x 4 + ax 2 + bx) 2. f : (F4 , 0) → (F5 , 0) f (a, b, c, x) = (a, b, c, x 3 + ax, bx 2 + cx) 3. f : (F6 , 0) → (F7 , 0) f (x, y, a, b, c, d) = (x 2 + ay, xy + bx + cy, y 2 + dx, a, b, c, d) 4. f : (F5 , 0) → (F4 , 0) f (a, b, c, x, y) = (a, b, c, x 3 + y 3 + axy + bx + cy, a, b, c)

100

4 Contact Equivalence

are stable. We do the calculation for the first of these and leave the rest as exercises. We have ⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫ 0 0 ⎬ ⎨ 1 ⎠ + f ∗ m3 θ (f ) T Ke f = O 3 · ⎝ 0 ⎠ , ⎝ 1 ⎠ , ⎝ 0 ⎭ ⎩ 2 x 4x 3 + 2ax + b x ⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫ 0 0 ⎬ ⎨ 1 = O 3 · ⎝ 0 ⎠ , ⎝ 1 ⎠ , ⎝ 0 ⎠ + f ∗ m3 θ (f ). ⎩ 2 ⎭ x x 4x 3 Since L generates θ (f ) over O 3 , to show that T Ke f + L is equal to θ (f ), it is enough to show that it is an O 3 -module. Because f ∗ m3 contains the variables a and b, we need to check only that x k ∂/∂yi lies in T Ke f + L for each k and i. We have ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 1 0 1 x ⎝0⎠ = x · ⎝ 0 ⎠ − ⎝ 0 ⎠ 4 0 x2 4x 3 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 1 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ x 1 = x 1 − 0 + 0⎠ 0 x x2 0 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 0 x ⎝0⎠ = ⎝ 1⎠ − ⎝1⎠ 1

x

0

The first of these shows that x∂/∂y1 ∈ T Ke f , and since T Ke f is an O 3 module, all multiples of this by powers of x are therefore contained in T Ke f + L. The second and third terms, on the other hand, each have a contribution from L, so it is not immediate that multiples by powers of x are in T Ke f + L, and further checks are required. We have ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 0 1 2⎝ ⎠ 2⎝ ⎠ ⎝ x 1 =x 1 − 0 ⎠ ∈ T Ke f 4 0 x 4x 3 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 1 1 x 2 ⎝0⎠ = ⎝ 0 ⎠ − ⎝0⎠ . 1 x2 0 Again, the second once again uses a contribution from L, so a further check is required. But it is obvious that x 3 ∂/∂y3 ∈ T Ke f , so the proof is complete.

4.1 The Contact Tangent Space

101

(3) Let F : (F42 , 02 ) → (F3 , 0) be defined by 

f (1) (x1 , x2 , x3 , x4 ) = (x1 , x2 , x32 + x42 ) f (2) (x1 , x2 , x3 , x4 ) = (x1 , x32 + x42 , x2 ).

Each branch of F is a fold and hence stable, and the two 2-dimensional isosingular loci, F2 × {0} and F × {0} × F, are in general position, so by Theorem 3.3 F is stable. Here we calculate directly that condition 2 of Theorem 4.1 holds, partly as a means of familiarising the reader with the notation. We have T Ke F + L = T Ke f (1) ⊕ T Ke f (2) + L,

(4.3)

where T Ke f (1) = O F4 ,01 {∂y1 , ∂y2 , 2x3 ∂y3 , 2x4 ∂y3 } + (x1 , x2 , x32 + x42 )θ (f (1) ) T Ke f (2) = O F4 ,02 {∂y1 , ∂y3 , 2x3 ∂y2 , 2x4 ∂y2 } + (x1 , x2 , x32 + x42 )θ (f (2) ). Recall that we write elements of θ (F ) as 3 × 2 matrices, whose left column is in θ (f (1) ) and right column in θ (f (2) ). In this notation ∂y1 , ∂y2 , ∂y3 become ⎛ 1 ⎝0 0

⎞ 1 0⎠ ,

⎛ ⎞ 00 ⎝1 1⎠ , 00

0

⎛

⎞ 00 ⎝ 0 0⎠ 11

respectively. Recall also that O F4 ,02 = O F4 ,01 ⊕ O F4 ,02 and that we write its 2

elements as ordered pairs in the usual way.1 As in the previous example, it is necessary to check only that the left-hand side of (4.3) is an O F4 ,02 -module. It is easy to see that for any h in m01 ⊕ m02 , 2 the Jacobson radical of O F4 ,02 , h∂yi ∈ T Ke F . The elements (1, 0) and (0, 1) 2 are not in m01 ⊕ m02 , so we check: ⎛

1 ⎜ (1, 0) ⎝0 0

1 However,

⎞ ⎛ 1 1 ⎟ ⎜ 0⎠ = ⎝0 0 0

⎞ 0 ⎟ 0⎠ ∈ T Ke f (1) ⊕0, 0

⎛

1 ⎜ (0, 1) ⎝0 0

⎞ ⎛ 0 1 ⎟ ⎜ 0⎠ = ⎝0 0 0

⎞ 1 ⎟ 0⎠ ∈ 0⊕T Ke f (2) , 0

recall the warning about this notation, immediately before Example 3.7.

102

4 Contact Equivalence

and similarly (1, 0)∂y2 ∈ T K f (1) ⊕ 0 and (0, 1)∂y3 ∈ 0 ⊕ T K f (2) . Finally ⎛ ⎞ ⎛ 00 0 ⎝ ⎝ ⎠ (0, 1) 1 1 = 1 00 0 ⎛ ⎞ ⎛ 00 0 (1, 0) ⎝0 0⎠ = ⎝0 11 1

⎞ ⎛ ⎞ 0 00 ⎝ ⎠ 1 − 1 0⎠ ∈ L + (T Ke f (1) ⊕ 0), 0 00 ⎞ ⎛ ⎞ 0 00 0⎠ − ⎝0 0⎠ ∈ L + (0 ⊕ T Ke f (2) ). 1 01

♦

4.2 Using T Ke f to Calculate T Ae f The contact tangent space turns out to be useful in calculating the Ae tangent space of unstable germs as well as in proving stability. In Chap. 3 we used only direct calculation. The following result gives a handle on more complex calculations. Recall that given a map-germ f : (Fn , S) → (Fp , 0), composition with f defines a ring homomorphism f ∗ : O p → O n , making O n , and therefore any O n -module, into a module over O p . Theorem 4.2 Given a map-germ f : (Fn , S) → (Fp , 0), suppose that T Ke f ⊇ mn θ (f )

(4.4)

for some finite  and that C is an O p -submodule of θ (f ) such that C ⊇ mkn θ (f ),

(4.5)

∗ C ⊆ T Ae f + mk+ n θ (f ) + f mp C.

(4.6)

Then C ⊆ T Ae f if and only if

and this equivalence holds also with equality in place of inclusion. Proof We need to only prove “if”. We have   f ∗ mp C + tf (θn ) ⊇ f ∗ mp mkn θ (f ) + tf (θn )   ⊇ mkn tf (θn ) + f ∗ mp θ (f ) + tf (θn ) ⊇ mkn m θ (f ) + tf (θn ) = mk+ n θ (f ) + tf (θn ).

4.2 Using T Ke f to Calculate T Ae f

103

Therefore if (4.6) holds, it follows that C ⊆ T Ae f + f ∗ mp C.

(4.7)

If C were a finitely generated O p -module, we would be able to apply Nakayama’s Lemma. Since we cannot assume this, we argue as follows: let M = θ (f )/tf (θn ). Then θ (f ) M = , f ∗ mp M tf (θn ) + f ∗ mp θ (f ) and by (4.4), this is finite-dimensional over F. It follows by the preparation theorem that M is finitely generated over O p . Now (C + tf (θn ))/tf (θn ) is a submodule of M. It is itself finitely generated over O p , for mkn θ (f ) + tf (θn ) /tf (θn ) = mkn M is finitely generated over O p (Exercise 3.2.5(i)) and the quotient of (C +   tf (θn ))/tf (θn ) by mkn θ (f ) + tf (θn ) /tf (θn ) is a finite-dimensional vector space. So we can apply Nakayama’s Lemma to the inclusion T Ae f C + tf (θn ) ⊆ + f ∗ mp tf (θn ) tf (θn )



C + tf (θn ) tf (θn )

which we obtain by quotienting (4.7) by tf (θn ). This proves that C ⊆ T Ae f . The proof with equality in place of inclusion follows exactly the same lines.

 

Example 4.2 We calculate T Ae f for the germ f : (F2 , 0) → (F4 , 0) defined by f (x1 , x2 ) = (x1 , x1 x2 + x23 , x1 x22 + x25 , x24 ), from the list of Ae -simple singularities of maps from surfaces to 4-space in [KPR07]. Take coordinates y1 , . . ., y4 on F4 . To simplify notation, we write ∂k in place of ∂y∂ k . It is easy to see that T Ke f ⊃ m32 θ (f ). Step 1 It seems reasonable to guess that T Ae f ⊃ m52 θ (f ). To prove that this is so, it is enough, by the theorem, to show that T Ae f + m82 θ (f ) + f ∗ m4 · m52 θ (f ) ⊃ m52 θ (f ).

(4.8)

To check this, one works downwards in degree, checking first that modulo m82 θ (f ), the left-hand side in (4.8) contains m72 θ (f )—i.e. that we get all j monomials x1i x2 ∂k with i + j = 7. Once this is checked, we can work modulo m72 θ (f ) to check that we get all monomials of degree 6, and so on down. Since x1 ∈ f ∗ m4 O 2 , it follows that f ∗ m4 · m52 θ (f ) contains all monomials j x1i x2 ∂k with i + j ≥ 6 and i ≥ 1. So the only monomials of degree 7 that we need to find are x27 ∂k for k = 1, . . ., 4. Working modulo m82 θ (f ) and those terms we have

104

4 Contact Equivalence

already found, we have ωf (y2 y4 ∂k ) = x27 ∂k . Now we repeat the procedure, working modulo m72 θ (f ). Once again, we need to find only x26 ∂k for 1 ≤ k ≤ 4. This is easy: modulo terms we have already found, ωf ((y22 − 2y1y5 − y1 y3 )∂k = x26 ∂k . Now we work modulo m62 θ (f ) to find the monomials of degree 5. Since, modulo m62 θ (f ), f ∗ (y15 ) = x15 , f ∗ (y2 y3 ) = x12 x23 ,

f ∗ (y13 y2 ) = x14 x2 ,

f ∗ (y12 y3 ) = x13 x22 ,

f ∗ (y1 y4 ) = x1 x24 ,

we get all degree 5 monomials except x25 ∂k from ωf (θ4 ). Then working cumulatively and modulo obvious terms such as x24 which we can obtain using ωf ,

∂ ∂ x25 we get x25 ∂x1 ∂y1





∂ ∂ 2 ∂ 3 ∂ from tf x1 x2 , ωf y3 , tf x2 we get x25 ∂x1 ∂y1 ∂x2 ∂y2







∂ ∂ ∂ ∂ ∂ from tf x1 , tf x2 , ωf y2 and ωf y3 we get x25 ∂x1 ∂x2 ∂y2 ∂y3 ∂y3









∂ ∂ ∂ 2 ∂ 3 ∂ from tf x2 , tf x2 , tf x1 x2 , ωf y3 and ωf y2 ∂x2 ∂x1 ∂x1 ∂y2 ∂y1 from tf

we get x25

∂ . ∂y4

These calculations can be quite entertaining. They are best done using a score sheet to cross off all the monomials already obtained, which can then be ignored in further calculations. Now that we have shown that T Ae f ⊃ m52 θ (f ), by such means we can quickly proceed to calculate that T Ae f is equal to ⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎫ ⎞ ⎞⎛ O 2 −{x2 } 0 0 x2 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ x ⎟ ⎜ x 2 ⎟ ⎜ ⎜ ⎟⎪ ⎬ ⎟ ⎟ ⎜ 3x 2 ⎟ ⎜ O 2 −{x2 , x22 } 0 0 ⎜ 2⎟ ⎜ 2 ⎟ ⎜ ⎜ ⎟ ⎟ ⎟⎜ 2 ⎟⎜ ⎜ ⎟⎜ ⎟ ⎟ + SpF ⎜ 2 ⎟ , ⎜ 3 ⎟ ⎜ ⎟⎜ 3 3 2 ⎪ ⎝x ⎠ ⎝x2 ⎠ ⎝x1 x2 + x2 ⎠ ⎝2x1 x2 ⎠ ⎝ ⎝O 2 −{x2 , x1 x2 , x2 , x2 }⎠ ⎠⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩ 2 ⎭ O 2 −{x2 , x1 x2 , x22 , x23 } 0 0 0 4x23 x1 x2 + x23 ⎛

(4.9) ♦

4.3 Construction of Stable Germs as Unfoldings

105

It is important to note that the first step in the calculation was to guess that T Ae f ⊃ m52 θ (f ). This “strategy of guessing,” which we learned of from Terry Gaffney, works for many calculations: make a sensible guess and then check it using Theorem 4.2. The more ambitious the guess, the easier it may be to check it, since a lower k in (4.5) means a lower k +  in (4.6). And one can learn as much from an incorrect guess as from a correct one. The theorem can also be used to check a conclusion such as (4.9) (Exercise 1 below).

Exercises for Sect. 4.2 1 Use Theorem 4.2 to check equality in (4.9). 2 Use Theorem 4.2 to calculate T Ae f for the following germs. (i) f (x) = (x 3 , x 4 ). (ii) f : (F, 0) → (F2 , 0), f (x) = (x 2 , x 2k+1 ). (iii) f : (F2 , 0) → (F3 , 0), f (x1 , x2 ) = (x1 , x23 , x1 x2 +x25 ) (H2 in the classification of [Mon85]). (iv) f : (F2 , 0) → (F3 , 0), f (x, y) = (x, y 2 , y 3 + x k+1 y) (Sk in the classification of [Mon85]). 2 in the classification of [RR91]) (v) f (x1 , x2 ) = (x12 − x22 + x15 , x1 x2 ) (I I2,2 2 2 (vi) f (x1 , x2 , x3 ) = (x1 , x2 , x1 x3 +x2 x3 +x35, x32 +x22 x3 ) (S1,2 in the classification of [HK99]). (vii) f : (F5 , 0) → (F4 , 0), f (x1 , x2 , u1 , u2 , u3 ) = (x12 x2 + 14 x24 + 2u1x1 + u2 x2 + u3 x22 , u1 , u2 , u3 ) (unstable unfolding of D5 from [Fuk99, Section 5]) 3 Show that the statement of Theorem 4.2 continues to hold if we replace T Ae f by any O p submodule N of θ (f ), providing N ⊃ tf (θn ). 4 Ditto, but now with the assumption that N ⊇ tf (msn θn ) for some finite s, and that tf (msn θn ) + f ∗ mp θ (f ) ⊇ mn θ (f ). At this point, the most important application of Theorem 4.2 is where N = T Ae f . Later we will make use of it with N = tf (msn θn ) + ωf (msp θp ) for some s ∈ N.

4.3 Construction of Stable Germs as Unfoldings The reader will note that each of the germs listed in Example 4.1(2) is itself an unfolding of a germ of rank 0 (i.e. whose derivative at 0 vanishes). Of course, thanks to the inverse function theorem any germ can be put in this form, in suitable coordinates (see Lemma 2.3 and Exercise 2.3.6). In fact there is a general procedure

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for finding all stable map-germs as unfoldings of lower dimensional germs of rank zero, due to Mather in [Mat69b]. The procedure is as follows: 1. Given f : (Fn , 0) → (Fp , 0) of rank 0, calculate T Ke f . Because f0 has rank 0 at 0, T Ke f0 ⊂ mn θ (f0 ). 2. Find a basis for the quotient mn θ (f )/T Ke f . 3. If g1 , . . ., gd ∈ θ (f ) project to a basis for the quotient mn θ (f )/T Ke f , then the unfolding F : (Fn × Fd , (0, 0)) → (Fp × Fd , (0, 0)) defined by F (x, u1 , . . ., ud ) = (f (x) +



uj gj (x), u1 , . . ., ud )

(4.10)

j

is stable. In Chap. 7 we explain this construction (see Theorem 7.2). Example 4.3 We believe the following example is self-explanatory. Let f0 : (C2 , 0) → (C3 , 0) be given by f (x1 , x2 ) = (x12 , x23 , x1 x2 ). We show how to use Macaulay 2 to find a stable unfolding following the recipe explained. Our script is deliberately pedestrian—it is of course possible to combine many of the steps into one. Step 1 We find a basis for θ (f )/T Ke f . S0=QQ[x1,x2] T0=QQ[X,Y,Z] f=map(S0,T0,matrix{{x1∧ 2,x2∧ 3,x1*x2}}) J=transpose jacobian matrix f I=ideal matrix f V=(module S0)∧3 W=mingens I*V TT=J|W basis cokernel TT The last step returns the matrix ⎛

⎞ 1 x2 x22 0 0 0 0 0 0 ⎝ 0 0 0 1 x1 x2 0 0 0 ⎠ 0 0 0 0 0 0 1 x1 x2 whose columns form a basis for the quotient θ (f )/T Ke f . Step 2 It follows that F (x, u) = (x12 + u1 x2 + u2 x22 , x23 + u3 x1 + u4 x2 , x1 x2 + u5 x1 + u6 x2 , u1 , . . ., u6 ) is a minimal stable unfolding of f .

♦

4.3 Construction of Stable Germs as Unfoldings

107

Mather showed that given n and p, a stable map-germ (Fn , S) → (Fp , 0) is determined, up to A -equivalence, by its local algebra, so the procedure described here gives the unique stable map-germ for each algebra type and dimension-pair. We prove this theorem in Chap. 7.

Exercises for Sect. 4.3 1 Find an expression for a stable map-germ of corank 1, with local algebra of dimension k, from Fn to Fn . These germs are called Morin singularities. Suggestion: the Whitney cusp map (x, u) → (x 3 + ux, u) is the case k = 3. Try k = 4 next. 2 Find an expression for a stable map-germ of corank 1, with local algebra of dimension k, from Fn to Fn+1 . The Whitney umbrella map (x, u) → (x 2 , ux, u) is the case k = 2. 3 Use the procedure of this subsection to find stable unfoldings of (i) (ii) (iii) (iv)

f (x1 , x2 ) = (x12 , x1 x2 , x22 ), f (x1 , x2 ) = (x12 , x24 , x1 x2 ), f (x1 , x2 ) = (x12 , x23 , x1 x22 ), f (x1 , x2 , x3 ) = (x12 + x2 x3 , x22 + x1 x3 , x32 + x1 x2 , x1 x2 x3 ).

4 (i) Show that the singularities of stable maps M n → N n+1 are all of corank 1 when n ≤ 5. (ii) For which n are all the singularities of stable maps M n → N n of corank 1? (iii) For which n are all the singularities of stable maps M n → N n of corank ≤ 2? (iv) For which n are all the stable maps M n → N n+2 immersions? (v) For which n are all stable map-germs (Fn+1 , 0) → (Fn , 0) unfoldings of function germs (F2 , 0) → (F, 0)? 5 (i) What is the lowest possible Ke -codimension of a corank 3 map-germ (F3 , 0) → (F3 , 0)? Give an example of such a germ, and use Macaulay 2 to find a stable unfolding. (ii) Ditto, but now with germs (F3 , 0) → (F4 , 0). 6 What is the smallest value of n for which there can be a stable map-germ M n → N n+1 consisting of two non-immersive branches? Give a formula for such a germ.

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4.4 Contact Equivalence Contact equivalence of map-germs (Fn , S) → (Rp , 0) is induced by the action of the group K of diffeomorphisms of (Fn × Fp , S × {0}) of the form (x, y) = (ϕ(x), ψ(x, y)) where ψ(x, 0) = 0 for all x. It is obvious that this is a subgroup of Diff(Fn ×Fp , S × {0}). It acts on the set of multi-germs of maps (Fn , S) → (Fp , 0) via its action on their graphs: if f : (Fn , S) → (Fp , 0) and  ∈ K , then  · f is the multi-germ (Fn , S) → (Fp , 0) whose graph is (graph(f )). Since graph(f ) = {(x, f (x)) : x ∈ S}, this means that   graph ( · f ) = {(ϕ(x), ψ(x, f (x)) : x ∈ S} and thus ( · f )(ϕ(x)) = ψ(x, f (x)), so that   ( · f )(x) = ψ ϕ −1 (x), f (ϕ −1 (x)) .

(4.11)

For any map h : N → P , let gr(h) : N → N × P be the graph embedding of N in N × P: gr(h)(x) = (x, h(x)). Remark 4.1 By (4.11), the following diagram commutes: (Fn , S) gr(f )

(Fn × Fp , S × {0})

ϕ

(Fn , S) gr

·f )

(Fn × Fp , S × {0})

(4.12)

Thus two maps (Fn , S) → (Fp , 0) are K -equivalent if and only if their graph embeddings are left-right equivalent via the pair (ϕ, ) ⊂ Diff(Fn , S) × Diff(Fn × Fp , S × {0}). This will be used in the proof of finite determinacy for K -equivalence. ♦

4.4 Contact Equivalence

109

We will see shortly that multi-germs are contact-equivalent if and only if their fibres are isomorphic, and so contact equivalence has a clear geometric significance. Nevertheless its significance for the theory of singularities of mappings goes much further than this. Theorem 4.1 has already given a glimpse of this. One simplifying feature of K -equivalence is that for multi-germs we can work independently on each branch: multi-germs (Fn , S) → (Fp , 0) are contactequivalent if and only if their mono-germ branches are contact-equivalent. Because of this, from now on in this section, we will consider only the case of mono-germs f : (Fn , 0) → (Fp , 0), unless otherwise stated. We define the subgroup C of K as set of all those  = (ϕ, ψ) ∈ K such that ϕ is the identity. Thus by (4.11),  = (id, ψ) ∈ C acts by ( · f )(x) = ψ(x, f (x)). Observe that R and L (and therefore R × L = A ) are also naturally embedded in K : given ϕ ∈ R and η ∈ L , define ϕ and η by ϕ (x, y) = (ϕ(x), y), η (x, y) = (x, η(y); then by (4.11) (ϕ · f )(x) = f (ϕ −1 (x)),

(η · f )(x) = η ◦ f (x).

Proposition 4.1 K is the semi-direct product of R and C . Proof First we show that K = C R. Given  = (ϕ, ψ) ∈ K , define ϕ ∈ R ⊂ K by ϕ (x, y) = (ϕ(x), y), and 1 ∈ C ⊂ K by 1 (x, y) = (x, ψ(ϕ −1 (x), y)). Then  = 1 ◦ ϕ . In view of this, to show that C is normal, we need to only show that if G ∈ C and ϕ ∈ R ⊂ K , then ϕ −1 G ϕ ∈ C . This is straightforward: 

   ϕ −1 G ϕ (x, y) = ϕ −1 G (φ(x), y) = ϕ −1 (φ(x), ψ(ϕ(x), y)) = (x, ψ(ϕ(x), y)).

 

Let Glp (O n ) be the group of invertible p×p matrices over O n = O Fn ,0 . It acts on the space of germs of maps (Fn , 0) → (Fp , 0): if A ∈ Glp (O n ) and f : (Fn , 0) → (Fp , 0), then (A · f )(x) = A(x)f (x). In fact the map (x, y) → (x, A(x)y) is a diffeomorphism of (Fn × Fp , 0) and maps Fn × {0} to itself, and as such lies in the group C . We will denote by CL the subgroup of C consisting of all such maps. Proposition 4.2 Map-germs f, g : (Fn , 0) → (Fp , 0) are C -equivalent only if they are CL -equivalent.

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Proof Let  ∈ C with (x, y) = (x, ψ(x, y)), and let ψ have components ψ1 , . . ., ψp . Because ψ(x, 0) = 0, for each i = 1, . . ., p we have ψi (x, y) =

p 

yj ψij (x, y)

j =1

for some functions ψij . It follows that for any f : (Fn , 0) → (Fp , 0) with components f1 , . . ., fp , ( · f ) (x) = ψ(x, f (x))   = ψ1 (x, f (x)), . . ., ψp (x, f (x)) ⎛ ⎞ p p   =⎝ ψ1j (x, f (x))fj (x), . . ., ψpj (x, f (x))fj (x)⎠ . j =1

(4.13)

j =1

Let aij (x) = ψij (x, f (x)) and define a p × p matrix A by A = (aij ). Then A ∈ Glp (O n ) since the matrix of the linear isomorphism d0  is equal to

In 0 . 0 A(0) Now let A be the corresponding element of CL . Then by (4.13), we have A · f =  · f.   It is an odd feature of this proof that the element A ∈ CL that we construct depends on the map-germ f ; we have not defined a retraction C → CL . Recall that a parameterised family t ∈ G , where G is a group of diffeomorphisms of (Fm , x0 ), is smooth if the resulting map (F × Fm , {0} × {x0 }) → (Fm , x0 ) defined by (t, x) → t (x) is smooth. Lemma 4.1 Let f : (Fn , 0) → (Fp , 0) be a smooth germ. Then

 dt · f |t =0 : t ∈ K is smooth , 0 = id = tf (mn θn ) + f ∗ mp θ (f ). dt (4.14)

Proof Let t be a smooth path in K . By Proposition 4.2, we may suppose that t · f = At f ◦ ϕt , where At ∈ Glp (O n ), A0 = Ip×p and ϕ0 = id. Then dt · f dAt |t =0 = f + tf dt dt



dϕt |t =0 . dt

(4.15)

The first term on the right lies in f ∗ mn θ (f ) and the second in tf (mn θn ), since, as n t we have seen, dϕ dt |t =0 is a vector field on (F , 0) which vanishes at 0 as t (0) = 0

4.4 Contact Equivalence

111

for all t. This shows that the left-hand side of (4.14) is contained in the right. For the opposite inclusion, observe that for any p × p matrix B over O n , the matrix Ip×p + tB is invertible for small t, and thus Bf is in the left-hand side. Similarly, for any vector field ξ ∈ mn θn , the germ ϕt (x) = id + tξ(x) is the germ of a diffeomorphism for small enough t, and thus the left-hand side contains tf (mn θn ).   The space T Ke f = tf (mn θn ) + f ∗ mp θ (f ) is thus the (heuristic) tangent space to the K -orbit of f , and T Ke f = tf (θn ) + f ∗ mp θ (f ) is the “extended K tangent space.” Definition 4.1 Let f : (Fn , 0) → (Fp , 0). Then codimK f = dimF

mn θ (f ) TK f

codimKe f = dimF

θ (f ) T Ke f

τ  (f ) = dimF

On Jf + f ∗ mp O n

Here Jf is the ideal generated by the p × p minors of the matrix of df . The Ke -codimension of f is also known as its Tjurina number, and denoted by τ (f ). Proposition 4.3 With the above notation, we have: τ (f ) < ∞

⇐⇒

τ  (f ) < ∞.

Proof In the analytic case, this follows from the Hilbert–Rückert Nullstellensatz (see Theorem E.3 below), since both modules have the same support, namely the intersection of f −1 (0) with the critical space C(f ). In fact an explicit algebraic argument proves the result also for the C ∞ case. If τ < ∞, then T Ke f ⊃ mkn θ (f ) for some finite k. Now work modulo f ∗ mp O n . For every monomial x α with |α| ≥ k and for i = 1, . . ., p, x α ∂/∂yi is in the image of the Jacobian matrix [df ]. It follows that diag(x α1 , . . ., x αp ) = [df ][B] for some n × p matrix [B]. The determinant of the left- hand side is a product of p × p minors of the two matrices on the right-hand side. Hence, Jf + f ∗ mp O n ⊃ mkp , so τ  (f ) < ∞. Conversely, it is easy to show that Jf θ (f ) ⊂ tf (θn ), so that if τ  (f ) < ∞, so is τ (f ).   (k)

Let Ke f be the set of k-jets in J k (N, P ) of germs (N, x0 ) → (P , y0 ) which are K -equivalent to f . A discussion like that preceding Proposition 3.9 shows Proposition 4.4 Let F : N × U → P be a family of mappings, and suppose that (k) j k F /U : N × U → J k (N, P ) is transverse to Ke f . Then if f is k-determined

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4 Contact Equivalence

for K , the codimension in U of the projection of (j k F /U )−1 (Ke f ) is equal to codimKe f . That is, for a generic family F (x, u) = (fu (x), u) on a parameter space U , the set of parameter values for which fu has a singularity in N which is K equivalent to f has codimension in U equal to codimKe f . (k)

Proposition 4.5 Let f, g : (Fn , 0) → (Fp , 0) be map-germs and suppose that the ideals (f1 , . . ., fp ) and (g1 , . . ., gp ) of O n are equal. Then the map-germs f and g are C -equivalent. Proof Because the two ideals are equal, there exist aij ∈ O n and bij ∈ O n , for 1 ≤ i, j ≤ p, such that fi =

 j

aij gj and gi =



bij fj for 1 ≤ i ≤ p.

(4.16)

j

Defining matrices A = (aij ) and B = (bij ) and writing f and g for the column vectors (f1 , . . ., fp )t and (g1 , . . ., gp )t , (4.16) becomes Af = g and Bg = f, so that BAf = f and ABg = g. Unfortunately, despite this, A and B need not be invertible (consider for example the case where f1 = f2 and g1 = g2 ; it is easy to find non-invertible A and B such that (4.16) holds); to find a suitable element of C transforming f to g we modify A to ensure its invertibility. Let Ip be the p × p identity matrix.   Lemma 4.2 Let A0 , B0 : Fp → Fp be linear maps. There exists a linear map Q0 : Fp → Fp such that A0 + Q0 (Ip − B0 A0 ) is invertible.   Proof of Lemma Let W be a complement to the image of A0 in Fp , and choose Q0 : Fp → W such that Q0 | : ker A0 → W is an isomorphism. Define C0 = A0 + Q0 (Ip − B0 A0 ). Then C0 is injective and therefore an isomorphism.   We apply the lemma by taking A0 and B0 to be A(0) and B(0), respectively. Define the p × p matrix C by C = A + Q0 (Ip − BA). Then C(0) is the matrix C0 of the lemma, so C is invertible. Clearly (Ip − BA) annihilates f, so C · f = g, and f and g are C -equivalent (indeed, CL -equivalent), as required. An immediate consequence of Proposition 4.5 is that the C -class of a germ f uniquely determines the C -class of all its unfoldings. Corollary 4.2 Let F be a d-parameter unfolding of a germ f : (Fn , 0) → (Fp , 0). Then (1) F is C -equivalent to the constant unfolding f × id. (2) codimKe (F ) = codimKe f .

4.4 Contact Equivalence

113

Proof (2): it is an easy calculation that codimKe (f × 1) = codimKe (f ). Because F is C -equivalent, and therefore K -equivalent, to f × 1, codimKe (F ) = codimKe (f × 1).   Definition 4.2 An analytic algebra is an F-algebra of the form A = On /I for some ideal I ⊂ On . We say that two analytic algebras A = On /I and B = On /J are induced isomorphic if there exists a diffeomorphism ϕ : (Fn , 0) → (Fn , 0) such that ϕ ∗ (I ) = J (in particular, ϕ ∗ induces an isomorphism of local algebras between A and B). Definition 4.3 For a map-germ f : (Fn , 0) → (Fp , 0), the analytic algebra Q(f ) =

On On = f ∗ mp (f1 , . . . , fp )

is called the local algebra of f . Proposition 4.6 Two map-germs f, g : (Fn , 0) → (Fp , 0) are K -equivalent if and only if their local algebras Q(f ), Q(g) are induced isomorphic. Proof Suppose that f, g are K -equivalent. There exists  = (ϕ, ψ) ∈ K which transforms the graph of f to that of g. Then g ◦ ϕ and f are C -equivalent and hence  the ideals (g ◦ ϕ)1 , . . ., (g ◦ ϕ)p and C  L -equivalent.  It follows immediately that f1 , . . ., fp are equal. In other words, ϕ ∗ (g ∗ mp ) = f ∗ mp , and thus ϕ ∗ induces an isomorphism between Q(f ), Q(g). Conversely, if Q(f ), Q(g) are induced isomorphic, there exists a diffeomorphism ϕ : (Fn , 0) → (Fn , 0) such that ϕ ∗ (f ∗ mp ) = g ∗ mp . This implies (f ◦ ϕ)∗ mp = g ∗ mp and by Proposition 4.5, f ◦ ϕ and f are C -equivalent and hence K -equivalent. Since f is also R-equivalent to f ◦ ϕ, it follows that f and g are K -equivalent.   When F = C, Q(f ) is nothing but the algebra of germs on the space-germ (f −1 (0), 0), possibly with non-reduced structure. In fact, Proposition 4.6 can be improved as follows: Theorem 4.3 Two map-germs f, g : (Cn , 0) → (Cp , 0) are K -equivalent if and only if their local algebras Q(f ), Q(g) are isomorphic. Proof In view of Proposition 4.6 we only need to prove that if Q(f ) and Q(g) are isomorphic, then they are induced isomorphic. We first show that it is enough to prove it in the case that both f and g have rank 0. In fact, if Q(f ), Q(g) are isomorphic, then f and g have the same rank d by Exercise 4.4.1. Hence, we can choose coordinates in which both f and g are written as d-parameter unfoldings of germs f0 , g0 of rank 0. Then Q(f0 )  Q(f )  Q(g)  Q(g0 ). By Corollary 4.2, it will be enough to show that f0 and g0 are K -equivalent. Assume now that f, g both have rank 0 and let α : Q(f ) → Q(g) be an isomorphism of local algebras. Then, α(m) = n, where m, n are the maximal

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ideals of Q(f ), Q(g), respectively. Both m, n are generated by the classes of the coordinate functions x1 , . . . , xn in their corresponding quotients. Suppose that for each i = 1, . . . , n, α([xi ]) = [ϕi ], for some ϕi ∈ On and let ϕ : (Cn , 0) → (Cn , 0) be the map-germ ϕ = (ϕ1 , . . . , ϕn ). As α(m) = n, in the algebra On we have: mn ⊂ ϕ ∗ mn + g ∗ mp ⊂ ϕ ∗ mn + m2n , where the second inclusion follows from the fact that g has rank 0. By Nakayama’s Lemma C.1, we get mn ⊂ ϕ ∗ mn , hence xi =

n 

aij ϕj ,

i = 1, . . . , n,

i=1

for some aij ∈ On . By taking derivatives with respect to xk and evaluating at x = 0, we get the following matrix equality In = (aij (0)) ·

∂ϕj (0) , ∂xk

so ϕ is a diffeomorphism. Consider the induced isomorphism ϕ ∗ : On → On and the diagram: ϕ∗ n

Q(f )

n

α

Q(g),

where the columns are the quotient maps. Let us see that this diagram is commutative. In fact, for the coordinate functions, we have α([xi ]) = [ϕi ] = [ϕ ∗ (xi )], for all i = 1, . . . , n. From this, we deduce that α([q]) = [ϕ ∗ (q)], for any polynomial function q ∈ On . For any function h ∈ On , let j k h denote its k’th Taylor polynomial. For each k ∈ N, h − j k h ∈ mk+1 n , hence α([h]) − [ϕ ∗ (h)] ∈ nk+1 . Since Q(g) is a Noetherian local ring, ∩k nk+1 = 0 (by Krull’s Intersection Theorem) and thus, α([h]) = [ϕ ∗ (h)]. From the commutativity of the diagram, we deduce that ϕ ∗ (f ∗ mp ) = g ∗ mp , so Q(f ) and Q(g) are induced isomorphic. By Proposition 4.6, f and g are K equivalent.   '∞ k The proof here relies on the fact that k=1 n = 0, and thus does not work in the real C ∞ case. Nevertheless, the result is also true with the additional hypothesis of finite K -determinacy.

4.4 Contact Equivalence

115

Theorem 4.4 Let f, g : (Rn , 0) → (Rp , 0) be two C ∞ map-germs and assume that f is finitely K -determined. Then f, g are K -equivalent if and only if their local algebras Q(f ), Q(g) are isomorphic. Proof Let α : Q(f ) → Q(g) be an isomorphism of local algebras. We proceed as in the proof of Theorem 4.3. We construct a C ∞ -diffeomorphism ϕ : (Rn , 0) → (Rn , 0) such that α([xi ]) = [ϕi ] = [ϕ ∗ (xi )], for all i = 1, . . . , n and hence, α([q]) = [ϕ ∗ (q)], for any polynomial function q ∈ On . Assume that f is k-K -determined. We have ϕ ∗ (f ∗ mp ) + mk+1 = g ∗ mp + mk+1 n n . Let f  = f ◦ ϕ, which is K -equivalent to f . The above equality gives = g ∗ mp + mk+1 (f  )∗ mp + mk+1 n n . For each i = 1, . . . , p, we can write fi + hi =

p 

aij gj ,

j =1

and aij ∈ On . Let fi = fi + hi and f  = (f1 , . . . , fp ). Since for some hi ∈ mk+1 n k  k  j f = j f and f  is k-K -determined, f  and f  are K -equivalent. But now (f  )∗ mp = g ∗ mp and thus f  and g are C -equivalent by Proposition 4.5.   Remark 4.2 For a multi-germ f : (Fn , S) → (Fp , 0), the algebra Q(f ) can be defined analogously as follows: Q(f ) =

O Fn ,S O Fn ,S = . ∗ f mp (f1 , . . . , fp )

The main difference is that Q(f ) is not local, but semi-local. In fact, we have an isomorphism Q(f ) ∼ = Q(f1 ) ⊕ · · · ⊕ Q(fr ), where f1 , . . . , fr are the branches of f . Proposition 4.6 holds also for multigerms: two multi-germs f, g : (Fn , S) → (Fp , 0) are K -equivalent if and only if their algebras Q(f ), Q(g) are induced isomorphic. The obvious versions of Theorems 4.3 and 4.4 also hold. ♦

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Exercises for Sect. 4.4 1 Since the isomorphism type of the local algebra Q(f ) determines f up to contact equivalence, algebraic properties of Q(f ) must reflect contact-invariant properties of f , and one should be able to determine invariants of f from Q(f ) alone. (i) Let f : (Fn , 0) → (Fp , 0) be a map-germ. Show that rank d0 f = n − dimF

m , m2

where m is the maximal ideal of Q(f ). The number dimF m/m2 is the embedding dimension of (f −1 (0), 0)): it is the dimension of the smallest affine space over F containing a germ isomorphic to f −1 (0). (ii) Characterise the local algebra Q(f ) when f is an immersion, and when f is a submersion. 2 Given K -equivalent germs f and g, find a natural isomorphism θ (f ) → θ (g) which passes to the quotient to define an isomorphism θ (f )/T Ke (f ) → θ (g)/T Ke (g). See Exercise 3.3.15 for hints. 3 Show that, as stated above, multi-germs (Fn , S) → (Fp , 0) are contactequivalent if and only if their mono-germ branches are contact equivalent.

4.5 Geometric Criterion for Finite Ae -Codimension We give here a very simple geometric criterion due to Mather and Gaffney for a multi-germ f : (Fn , S) → (Fp , 0) to have finite Ae -codimension. Here the real C ∞ and complex analytic theories diverge: the criterion is necessary in the first case, and necessary and sufficient in the second. Roughly speaking, in the complex analytic case, a multi-germ has finite Ae -codimension if and only if it has an isolated instability. To make this precise, we recall first the notion of locally stable mapping. Definition 4.4 Let f : N → P be a smooth map between manifolds, let C(f ) be its critical set and let (f ) = f (C(f )). We say that f is locally stable if: 1. the restriction f : C(f ) → P is finite (i.e. closed and finite-to-one see Definition 4.5 below), 2. for any y ∈ (f ), the multi-germ f : (N, S) → (P , y) is stable, where S = f −1 (y) ∩ C(f ). Note that this is not the same as the definition of stability of germs, in Definition 3.4(4). Below we prove a Proposition 4.8, linking the two notions.

4.5 Geometric Criterion for Finite Ae -Codimension

117

Notation Given a map f : N → P as in the definition, and y ∈ (f ), we denote by (f )y the multi-germ f : (N, S) → (P , y), where S = f −1 (y) ∩ C(f ). We recall that the critical set C(f ) is by definition the set of points x ∈ N such that dx f is not surjective, so that C(f ) = N when dim N < dim P , but C(f ) coincides with the singular set (f ) otherwise. The fact that we only need to check the stability of the mapping at the critical points is based on the following fact: A germ f : (Fn , S) → (Fp , 0) is stable if and only if the germ at the smaller set, S ∩ C(f ), namely f : (Fn , S ∩ C(f )) → (Fp , 0), is stable (Exercise 4.5.1 below). For this reason, from now on we consider only multi-germs f : (Fn , S) → (Fp , 0) for which S ⊂ C(f ).

4.5.1 Sheafification Let f0 : (Cn , S) → (Cp , 0) be a germ, and f : X → Y a representative. In order to describe what happens in the neighbourhood of S in the source and 0 in the target, we extend the module TA1 e f0 to a coherent sheaf, TA1e f , on a possibly smaller neighbourhood Y  of 0 with Y  ⊆ Y , and then use the properties of coherent sheaves to derive the information we want. The proof of the geometric criterion uses the finite coherence theorem of Grauert, (Theorem E.2 in Appendix E) which says that the push-forward of a coherent sheaf by a finite map is once again coherent, and a corollary of the Hilbert–Ruckert Nullstellensatz (Theorem E.3 in Appendix E), which says that that if S is a coherent sheaf on an analytic space X, then a point x ∈ X is isolated in the support of S if and only if the C-vector space dimension of its stalk at x is finite. Definition 4.5 A continuous map f : X → Y is finite if it is closed (f (C) is closed in Y whenever C is closed in X) and has finite fibres. The following property will be also discussed in Theorem D.5. Proposition 4.7 ([GR04, I.3.2]) Let f : X → Y be a holomorphic map of complex spaces, such that x0 is an isolated point in f −1 (f (x0 )). Then there exist neighbourhoods X and Y  of x0 in X and f (x0 ) in Y such that f (X ) ⊂ Y  and f : X → Y  is finite.  We fix some notation which slightly extends what we have used previously. Let f : (Cn , S) → (Cp , 0) be a holomorphic multi-germ and fix a representative f : X → Y , where X, Y are open sets in Cn , Cp , respectively. We consider the following sheaves and morphisms of holomorphic vector fields: • θX , θ (f ) are the sheaves of OX -modules of germs of vector fields on X and of vector fields along f |X , respectively, and tf : θX → θ (f ) is the morphism defined by composition with df . • θY is the sheaf of OY -modules of germs of vector fields on Y and ωf : θY → f∗ θ (f ) is the induced morphism defined by composition with f .

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4 Contact Equivalence

Proposition 4.8 If the germ f0 : (Fn , S) → (Fp , 0) is stable, in the sense of Definition 3.4(4), then for every representative f : X → Y , there is a subrepresentative which is locally stable in the sense of Definition 4.4. Proof Suppose first that F = C. By Theorem 4.1, dimC

θ (f ) ≤ p. tf (θn ) + f ∗ mp θ (f )

The support of this quotient is C(f ) ∩ f −1 (0), and it follows, by Theorem E.3, that S is isolated in this support. By Proposition 4.7, after shrinking X and Y , we may assume that the restriction of f to its critical set is finite, and that f −1 (0)∩C(f ) ⊆ S (see Lemma 4.3 below). We define a sheaf TR1e f on X by TR1e f =

θ (f ) . tf (θX )

As cokernel of the morphism tf : θX → θ (f ), this is a coherent sheaf. It is supported on C(f ). By the finite mapping Theorem E.2, f∗ (TR1e f ) is a coherent sheaf of O Y -modules. The morphism ωf : θY → f∗ (θ (f )) induces a morphism of coherent sheaves θY → f∗ TR1e (f ) . We define

TA1e f :=

  f∗ TR1e (f ) ωf (θY )

.

As the cokernel of a morphism of coherent sheaves, this too is coherent. Since f −1 (0) ∩ X = S, the stalk of TA1e f at 0 ∈ Y is naturally identified with TA1 e f0 , and is thus equal to 0. Since the support of a coherent sheaf is closed (in fact this is a special case of Theorem E.3), there is a neighbourhood Y  of 0 in Y which does not meet the support of TA1e f . Let X = f −1 (Y  ). Then the representative f : X → Y  is locally stable. Now suppose that F = R, and that f is a stable C ∞ germ. By the finite determinacy Theorem 6.2 below, f is right-left equivalent to a polynomial mapgerm. Since the assertion of the proposition is invariant under right-left equivalence, we may assume that f is this polynomial germ. It therefore has a complexification, fC : (Cn , S) → (Cp , 0), defined by the same real formula, now with complex variables. This formula also defines a complexification of any real analytic repre , Y sentative of f . By what we have shown, there are open neighbourhoods XC C   p of S and 0 ∈ C such that fC : XC → YC is locally stable. Let x1 , . . ., xr ∈  ∩ Rn share the same image. The stability of the complex analytic multi-germ XC  , {x , . . ., x }) → (Y  , y) implies the stability of the real C ∞ multifC : (XC 1 r C  ∩ Rn , {x , . . ., x }) → (Y  ∩ Rp , y), by Proposition 3.8. Thus the germ f : (XC 1 r C  ∩ Rn → Y  ∩ Rp has the required property. representative f : XC   C

4.5 Geometric Criterion for Finite Ae -Codimension

119

Lemma 4.3 If f : (Cn , S) → (Cp , 0) has finite Ae -codimension, then for a sufficiently small representative f : U → V , f −1 (0) ∩ C(f ) ⊂ S. Proof By the hypothesis that dimC TA1 e f < ∞, there exist h1 , . . . , hr ∈ OCn ,S such that T Ae (f ) + SpC {h1 , . . . , hr } = θ (f ). Since ωf (θCp ,0 ) ⊂ (f ∗ mCp ,0 )θ (f ) + {∂/∂y1 , . . . , ∂/∂yp }, dimC θ (f )/T Ke (f ) ≤ r + p. Therefore for each x ∈ S we also have dimC θ (fx )/T Ke (fx ) < ∞, where fx denotes the mono-germ of f at x. Let f : U → V be any representative of the multi-germ and consider the sheaf of OU -modules TK1 e f =

θ (f ) , tf (θU ) + (f ∗ mV ,0 )θ (f )

where mV ,0 is the ideal sheaf in OV of functions vanishing at 0. Since all sheaves appearing in the definition of TK1 e f are coherent (in fact, θU , θ (f ) are locally free of rank n, p, respectively), TK1 e f is also coherent. The stalk of TK1 e f at x ∈ S is θ (fx ) θ (fx ) = . tfx (θU,x ) + (fx∗ mV ,0)θ (fx ) T Ke (fx )   Hence, for each x ∈ S, dimC TK1 e f < ∞, and therefore by the Nullstellensatz x

Theorem E.3, x is isolated in the support of TK1 e f . Thus we can find an open neighbourhood U  of S in U such that Supp S ∩ U  ⊂ S. To finish the proof, it only remains to show that Supp TK1 e f = C(f ) ∩ f −1 (0). In fact, if x ∈ C(f ) ∩ f −1 (0), then f (x) = 0 and dx f is not surjective. This implies that fx∗ mV ,0 ⊂ mU,x and that ∂/∂yi ∈ / tfx (θU,x ) for some i = 1, . . . , p. So  TK1 e f

x

= 0.

Conversely, if x ∈ / C(f ) ∩ f −1 (0), then either f (x) = 0 or dx f is surjective. It ∗ followsthat either  fx mV ,0 = OU,x or ∂/∂yi ∈ tfx (θU,x ) for all i = 1, . . . , p and

hence, TK1 e f

x

= 0 in both cases.

 

Theorem 4.5 (Mather–Gaffney Criterion) A holomorphic multi-germ f : (Cn , S) → (Cp , 0) (with S ⊂ C(f )) has finite Ae -codimension if and only if there is a small enough representative f : X → Y such that: 1. f −1 (0) ∩ C(f ) = S; 2. the restriction f : X \ f −1 (0) → Y \ {0} is locally stable.

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4 Contact Equivalence

Proof Suppose first that f0 has finite Ae -codimension. By Lemma 4.3, we can choose a representative f satisfying condition (1). By Proposition 4.7, after shrinking the neighbourhoods X, Y if necessary, we can assume that the restriction to the critical set f : C(f ) → Y is finite. The coherent sheaf TA1e f introduced in Proposition 4.8 has stalk at y ∈ Y equal to  (f )∗ (TR1e f )y ωf (θY,y )

=

x∈S 

θ (fx ) tfx (θX,x )



ωf (θY,y )

∼ =

θ (fS  ) = TA1 e (fS  ), tfS  (θX,S  ) + ωfS  (θY,y )

where S  = f −1 (y) ∩ C(f ) and fS  denotes the multi-germ of f at S  . Now we prove the equivalence of finite Ae -codimension with condition (2). The stalk at 0 ∈ Y of TAe f is equal to TA1 e f0 . If this has finite dimension over C, then  by the corollary to the Nullstellensatz, Theorem E.3, there is an open  set Y , with 0 ∈ Y  ⊂ Y , such that Supp TA1e f ∩ Y  ⊂ {0}. But this means that TA1e f =0 y

for every y ∈ Y  \ {0} and hence the restriction of f to X \ S is locally stable, where X = f −1 (Y  ). Conversely, assume there is an open set X , S ⊂ X ⊂ X, such that the restriction of f to X \ S is locally stable. Since f : C(f ) → Y is finite, there is an open neighbourhood Y  of 0, contained in Y , suchthat f (X ∩ C(f )) = Y  ∩ f (C(f )). = 0. Thus, Supp TA1e f ∩ Y  ⊂ {0} Then for each y ∈ Y  \ {0} we have TA1e f y

and again by Theorem E.3 we get   dimC TA1e f = dimC TA1 e (fS ) < ∞. 0

  An argument using the complexification and the finite determinacy theorem, as in the last part of the proof of Proposition 4.8, shows that if f : (Rn , S) → (Rp , 0) is A -finite, then it has isolated instability. The converse does not hold: for example, the germ (R2 , 0) → (R, 0) defined by (x, y) → (x 2 + y 2 )2 is not A -finite despite having isolated instability (in R2 though not in C2 ). Corollary 4.3 Let f : (Rn , S) → (Rp , 0) be A -finite. Then there exists a representative f : X → Y such that: 1. f −1 (0) ∩ C(f ) = S; 2. the restriction f : X \ f −1 (0) → Y \ {0} is locally stable. Proof By the Finite Determinacy Theorem 6.2 we can assume that f is a polynomial and by Proposition 3.8, its complexification fC is also A -finite. Hence, there exists a representative fC : Xˆ → Yˆ such that 1. fC−1 (0) ∩ C(fC ) = S; 2. the restriction fC : Xˆ \ fC−1 (0) → Yˆ \ {0} is locally stable.

4.5 Geometric Criterion for Finite Ae -Codimension

121

Let X and Y be the projections of Xˆ and Yˆ on Rn and Rp , respectively, and consider the representative f : X → Y . For each y ∈ Y , the set S  = f −1 (y) ∩ C(f ) is contained in Sˆ  = fC−1 (y) ∩ C(fC ). For y = 0, we get f −1 (0) ∩ C(f ) = S. For y = 0, the complexification of the germ f : (Rn , S  ) → (Rp , y) is obtained as the restriction of fC : (Cn , Sˆ  ) → (Cp , y) to the branches with real base-point. Since the restriction of a stable germ is also stable, f : (Rn , S  ) → (Rp , y) is stable, again by Proposition 3.8.   Example 4.4 Consider the case of a holomorphic function f : (Cn , 0) → (C, 0). If f has finite Ae -codimension, then by the Mather–Gaffney criterion Theorem 4.5, there is a small enough representative f : X → Y such that 1. f −1 (0) ∩ C(f ) ⊂ {0}; 2. the restriction f : X \ f −1 (0) → Y \ {0} is locally stable. By Exercise 5, all the critical points of f on X \ {0} are non-degenerate, and so are isolated. Since C(f ) is a closed analytic subset of X, it follows that C(f ) cannot accumulate at the origin. Thus, after shrinking the neighbourhood X if necessary, we can assume that C(f ) = {0}. The converse is obviously true, so we have shown that f has finite Ae -codimension if and only if it has an isolated critical point. ♦

Exercises for Sect. 4.5 1 Let f : (Cn , S) → (Cp , 0) be a holomorphic multi-germ, with S a finite set. Show that f is stable if and only if the germ at the smaller set, S ∩ C(f ), namely f : (Fn , S ∩ C(f )) → (Fp , 0), is stable. 2 Use an analogous argument to that of Example 4.4 to show that a germ f : (C, 0) → (C2 , 0) is A -finite if and only if it is one-to-one. 3 Suppose that f : (Fn , 0) → (Fn+1 , 0) is A -finite. Show that (i) the set of points where f is not an immersion has codimension ≥ 2. Hint: J 1 (n, n + 1) can be identified with the space L(n, n + 1) of linear maps Fn → Fn+1 . Apply Proposition A.1(2) to each of the strata Wi = {σ ∈ J 1 (n, n + 1) : dim ker σ = i}. (ii) if F = C, then the codimension of this set is exactly 2, unless it is empty (see Sect. C.5). 4 Suppose that f : (Fn , 0) → (Fp , 0), with n ≥ p, is A -finite. Show that (i) the set C(f ) of points in the domain where f is not a submersion has codimension ≥ n − p + 1. (ii) if F = C, then the codimension of this set is exactly n − p + 1, unless it is empty.

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4 Contact Equivalence

5 Use Theorem 4.5 to give a necessary and sufficient geometrical criterion for a germ (C2 , 0) → (C2 , 0) to be A -finite. 6 Use an analogous argument to that of Example 4.4 to show that a germ f : (C2 , 0) → (C3 , 0) has finite Ae -codimension if and only if f is an immersion with only transverse double points away from the origin. Equivalently, codimAe (f ) = ∞ if and only if either: (i) there is a curve of non-immersive points in the domain; (ii) there is a curve of non-transverse self-intersection in the target; (iii) there is a curve of triple points in the target. 7 Show that the germ f : (C2 , 0) → (C3 , 0) given by f (x, y) = (x, y 2 , y(x k +y 2 )) has finite Ae -codimension. (Hint: Use the fact that the mapping (x, y) → (x, y 2 ) has no triple points to deduce that the same is true for f ). 8 Let f : (C2 , 0) → (C3 , 0) be the germ fc (x, y) = (x, y 3 +xy, y 4 +cxy 2), where c ∈ C is a constant. Show that fc has infinite Ae -codimension when c = 2/3, 1 or 2 by showing explicitly that the image contains a curve of one of the types listed in Exercise 6. Hint: in fact each of the three degeneracies (a), (b) and (c) of Exercise 2 occurs for one of these three values of c. When c = 1 the first coordinate axis in the target is triple-covered. 9 Prove the following geometric criterion for K -finiteness: the germ f : (Cn , 0) → (Cp , 0) is K -finite if and only if there is a representative which is finite-to-one on its critical set.

4.6 Transversality The aim of this section is to prove that a map-germ is stable if and only if its multijet extension map is transverse to its A -orbit in the jet space, if and only if it is transverse to its K -orbit. Let us denote by r J k (n, p) the space of k-jets of smooth map germs f : (Fn , S) → (Fp , 0), where r = |S|. Evidently r J k (n, p) is isomorphic to (1 J k (n, p))r . In what follows we generally omit the initial subscript, since the value of r will be clear from the context. Since J k (n, p) is a finite-dimensional vector space, it has a natural structure of manifold. There is an epimorphism j k : mn θ (f ) → J k (n, p), k whose kernel is mk+1 n θ (f ), which allows us to identify J (n, p) with the quotient k+1 mn θ (f )/mn θ (f ). In fact, this quotient can be thought as the tangent space of J k (n, p) at σ = j k f . We can also describe the tangent vectors to J k (n, p) at σ by means of k-jets of 1-parameter unfoldings of f . Let F : (Fn × F, S × {0}) → (Fp × F, 0) be any

4.6 Transversality

123

origin-preserving unfolding of f , given by F (x, t) = (ft (x), t). This gives rise to a smooth curve γ : (F, 0) → J k (n, p) defined by γ (t) = j k ft , such that γ (0) = σ . The associated tangent vector is:   d k  dft   k =j γ (0) = {j ft } , dt dt t =0 t =0 where the last equality comes by changing the order of differentiation. In fact, it is easy to see that any tangent vector to J k (n, p) at σ can be obtained in this way. For each  ≥ k, there is a natural projection πk : J  (n, p) → J k (n, p), defined by πk (j  f ) = j k f . This map is obviously an epimorphism of vector spaces and hence a submersion when we consider the manifold structures. For each group G = R, L , C , A or K , denote by r G (k) the space of k-multijets of germs in G . As above, we will in general omit the subindex r. In each case, G (k) is an open set of some k-jet space, so it has again a natural structure of manifold. Moreover, composition induces a group structure on G (k) so that it becomes a Lie group and the action of G on the set of map germs also induces a Lie group action of G (k) on J k (n, p). For G = R, C and K , the group r G (k) acting on r J k (n, p) is just (1 G(k) )r , with the product action, but note that r L (k) is just a single copy of (k) with the diagonal action, and A (k) = R (k) × L (k) . 1L r r r If G is a Lie group acting smoothly on a manifold M, the orbit of a point x ∈ M is denoted by Gx, and is the image of the smooth map αx : G → M given by αx (g) = gx. We consider the differential de αx : Te G → Tx M at the identity e of G. The tangent space Te G is called the Lie algebra of G and is denoted by g. There is an induced map g × M → T M given by η · x = de αx (η), for each η ∈ g and x ∈ M, known as the infinitesimal action of g on M. It is well known that orbits are injectively immersed submanifolds, so that they always have a well-defined tangent space at each point. We include here a proof of this result. Lemma 4.4 Let G be a Lie group with Lie algebra g acting smoothly on a manifold M. For each x ∈ M, the orbit Gx is an injectively immersed submanifold whose tangent space at x is gx. Proof Let Gx be the isotropy group of x, i.e. Gx = {g ∈ G : gx = x}. This is a closed subgroup of G, hence G/Gx has a unique smooth manifold structure and moreover G acts transitively on G/Gx by left multiplication. We factor the map αx as a composition π

G αx

G/Gx i

M

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4 Contact Equivalence

where π is surjective and i is injective. Since π and i are both equivariant with respect to the G action, they have constant rank. By the constant rank theorem, π must be a submersion and i an immersion. Thus, Gx is the image of i which is an injective immersion and its tangent space at x is the image of the differential of i, that is, Tx (Gx) = d[e] i(T[e] (G/Gx )) = d[e] i(de π(Te G)) = de αx (Te G) = gx.

 

We remark that in general the orbits may not be (embedded) submanifolds, unless they are closed subsets of M. In our case, the manifold M = J k (n, p) is a vector space, the Lie group G = G (k) is a semialgebraic subset of a vector space and the action is given by a semialgebraic map (see Appendix B for the definitions of semialgebraic sets and maps). The next proposition shows that in that case, the orbits are in fact submanifolds. Proposition 4.9 Let G be a Lie group acting smoothly on a manifold M. If the action G × M → M is semialgebraic, then the orbits are submanifolds of M. Proof For each x ∈ M, the map αx : G → M is semialgebraic and hence, its image Gx is also semialgebraic by the Tarski–Seidenberg Theorem B.4. By Theorem B.5, Gx has at least one regular point y ∈ Gx of dimension d = dim Gx. Let z ∈ Gx be any other point and assume that y = gx and z = hx, for some g, h ∈ G. The map φ : M → M given by u → hg −1 u is a diffeomorphism such that φ(Gx) = Gx and φ(y) = z. Thus, z is also a regular point of Gx of dimension d. This proves that Gx is a submanifold of M.   Lemma 4.5 For each σ = j k f ∈ J k (n, p), the orbit G (k) σ is a submanifold of J k (n, p) whose tangent space at σ is Tσ (G (k) σ ) = j k (T G (f )). Proof By Lemma 4.4 and Proposition 4.9, G (k) σ is a submanifold of J k (n, p) whose tangent space at σ is Tσ (G (k) σ ) = de ασ (Te G (k) ), where ασ : G (k) → J k (n, p) is the map ασ (ϕ) = ϕσ . Thus, all tangent vectors in Tσ (G (k) σ ) can be obtained as follows: 

   dϕt  d k  d k k de ασ j {j ϕt } {ασ (j ϕt )} = de ασ =  dt t =0 dt dt t =0 t =0     d k d(ϕt f )  {j (ϕt f )} = = jk , dt dt t =0 t =0

4.6 Transversality

125

where ϕt is any origin-preserving unfolding of the identity in G . To complete the proof, it only remains to show that T G (f ) coincides with the set of all vector fields obtained as d(ϕt f )/dt|t =0 , where ϕt is any origin-preserving unfolding of the identity in G . Assume G = A . Let φt and ψt be origin-preserving unfoldings of the identity in (Fn , S) and (Fp , 0), respectively. As we saw in Sect. 3.2, we have     d dφt−1  −1  {ψt ◦ f ◦ φt } = tf  dt dt  t =0



+ ωf

t =0

 dψt  ∈ T A (f ). dt t =0

Conversely, if ϑ ∈ T A (f ), there exist ξ ∈ mn θn and η ∈ mp θp such that ϑ = tf (ξ ) + ωf (η). Let φt and ψt be origin-preserving unfoldings of the identity in (Fn , S) and (Fp , 0), respectively, such that ξ = dφt−1 /dt|t =0 and η = dψt /dt|t =0 , Then,      dψt  dφt−1  d −1  + ωf ϑ = tf {ψ = ◦ f ◦ φ }  t t   . dt  dt t =0 dt t =0 t =0

 

The proof for the other groups is analogous and is left as an exercise.

Let f : (Fn , S) → (Fp , 0) be a smooth map-germ and let σ = j k f be its k-jet. Take a representative f : X → Y , where X, Y are open neighbourhoods of S and 0 in Fn , Fp , respectively, and denote by r j k f : X(r) → r J k (X, Y ) the multi-jet extension map of f . Recall that r J k (X, Y ) is a smooth fibre bundle over X(r) × Y r with fibre (J k (n, p))r , so that its tangent space at σ can be described as mn θ (f ) ∼ rn θ (f ) Tσ (r J k (X, Y )) ∼ . = F ⊕ k+1 = Frn ⊕ Frp ⊕ k+1 mn θ (f ) mn θ (f )

(4.17)

For any A (k) -invariant subset W ⊂ J k (n, p), there is an associated subbundle W (X, Y ) ⊂ r J k (X, Y ) with fibre W . In particular, this applies when W is an orbit A (k) or K (k) orbit, in which case W (X, Y ) is a submanifold of r J k (X, Y ). Lemma 4.6 Assume S = {s1 , . . . , sr } and set z = (s1 , . . . , sr ) ∈ X(r) . Then: 1. For W = A (k) σ , r j k f is transverse to W (X, Y ) at z if and only if T Ae (f ) + mk+1 n θ (f ) = θ (f ). 2. For W = K

(k) σ ,

rj

kf

is transverse to W (X, Y ) at z if and only if

T Ke (f ) + T Ae (f ) + mk+1 n θ (f ) = θ (f ).

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4 Contact Equivalence

Proof Consider first the case W = A (k) σ . By Lemma 4.5 and via the identification (4.17), the tangent space to W (X, Y ) at σ is Frn ⊕ Frp ⊕

tf (mn θn ) + ωf (θp ) + mk+1 T A (f ) + mk+1 n θ (f ) n θ (f ) ∼ rn F ⊕ . = k+1 k+1 mn θ (f ) mn θ (f ) (4.18)

Since r j k f is the r-fold product of j k f , the image of dz (r j k f ) is the sum of the images of the differentials dsi (j k f ), for i = 1, . . . , r. Take coordinates x = (x1 , . . . , xn ) in X. Then, for each j = 1, . . . , n   d k ∂j k f {j ft (si )} , (si ) = ∂xj dt t =0 where ft (x) = f (x + tej ). By linearity of the k-jet, we get ∂j k f (si ) = j k ∂xj





dft  ∂f k ) = j (s (si ). i dt t =0 ∂xj

By means of the identification (4.17), we deduce that the image of dz (r j k f ) is equal to Frn ⊕

tf (SpF {e1 , . . . , en }) + mk+1 n θ (f ) mk+1 n θ (f )

(4.19)

.

The sum of (4.18) plus (4.19) gives Frn ⊕ which proves part 1. In the case W = K

(k) σ ,

Frn ⊕ Frp ⊕

T Ae (f ) + mk+1 n θ (f ) mk+1 n θ (f )

,

the tangent space to W (U, V ) at σ is

tf (mn θn ) + f ∗ mp θ (f ) + mk+1 n θ (f ) mk+1 n θ (f )

.

(4.20)

Since Frp ⊕ f ∗ mp θ (f ) = ωf (θp ) + f ∗ mp θ (f ), the sum of (4.20) plus (4.19) now gives Frn ⊕ so we have part 2.

T Ae (f ) + T Ke (f ) + mk+1 n θ (f ) mk+1 n θ (f )

,  

4.6 Transversality

127

Theorem 4.6 Let f : (Fn , S) → (Fp , 0) be a smooth map-germ, where S = {s1 , . . . , sr }, and let f : X → Y be a representative. Write z = (s1 , . . . , sr ) ∈ X(r) , and σ = j p+1 f . The following statements are equivalent: 1. f is stable; 2. r j p+1 f is transverse to W (X, Y ) at z, for W = A (p+1) σ ; 3. r j p+1 f is transverse to W (X, Y ) at z, for W = K (p+1) σ . Proof By Theorem 3.2 and Lemma 4.6, we have to prove that the following statements are equivalent: (a) T Ae (f ) = θ (f ), p+2 (b) T Ae (f ) + mn θ (f ) = θ (f ), p+2 (c) T Ke (f ) + T Ae (f ) + mn θ (f ) = θ (f ). The implications (a) ⇒ (b) ⇒ (c) are obvious. Assume (c). Then since T Ae (f ) ⊂ T Ke (f ) + SpF {e1 , . . . , ep }, the classes of {e1 , . . . , ep } generate θ (f )/(T Ke (f ) + p+2 mn θ (f )) and hence, dimF

θ (f ) p+2

T Ke (f ) + mn

θ (f )

≤ p.

The submodules Bk = T Ke (f )+mkn θ (f ), with k = 0, . . . , p+2, form a decreasing chain. Since dimF B0 /Bp+2 ≤ p, they cannot be all distinct. Thus, mkn θ (f ) ⊂ T Ke (f ) + mk+1 n θ (f ), for some k = 0, . . . , p + 1. By Nakayama’s Lemma C.1, p+2 mkn θ (f ) ⊂ T Ke (f ), and hence mn θ (f ) ⊂ T Ke (f ). Thus T Ke (f )+T Ae (f ) = θ (f ), or equivalently, T Ae (f ) + f ∗ mp θ (f ) = θ (f ).

(4.21)

  Because θ (f )/ tf (θn ) + f ∗ mp θ (f ) is finite-dimensional as F-vector space, it follows by the preparation theorem that θ (f )/tf (θn ) is finitely generated over O p . Hence so is θ (f )/T Ae (f ). But now from (4.21) it follows by Nakayama’s Lemma that T Ae (f ) = θ (f ).   Corollary 4.4 Let f : N → P be a smooth map between manifolds, with dim P = p and such that the restriction f : C(f ) → P is finite. The following statements are equivalent: 1. f is locally stable; 2. for all r ≥ 1, r j p+1 f is transverse to every submanifold W (N, P ) in p+1 (N, P ), where W ranges over all of the A (p+1) -orbits; rJ 3. for all r ≥ 1, r j p+1 f is transverse to every submanifold W (N, P ) in p+1 (N, P ), where W ranges over all the K (p+1) -orbits. rJ

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4 Contact Equivalence

4.7 Thom–Boardman Singularities The Thom–Boardman singularities were introduced by Thom in [Tho56] for generic mappings f : X → Y as a way to find a partition of X which is finer than the rank partition and to take into account higher order derivatives. Recall that given an integer i1 ≥ 0,  i1 (f ) is the set of points x ∈ X where dx f has kernel rank i1 . Clearly,  i1 (f ) = (j 1 f )−1 ( i1 ), where  i1 ⊂ J 1 (X, Y ) is the submanifold X × Y × {kernel rank i1 matrices} and j 1 f : X → J 1 (X, Y ) is the jet extension map. If f is generic, then j 1 f is transverse to  i1 , so  i1 (f ) is a submanifold of X. For each i2 ≥ 0 we can then define  i1 ,i2 (f ) as  i2 (f | i1 ). In general, if we have constructed  i1 ,...,ik−1 (f) and assume  that it is a submanifold of X, then we can define  i1 ,...,ik (f ) as  ik f | i1 ,...,ik−1 . The unsatisfactory point of this construction is that we must ensure that f is sufficiently generic that at each step,  i1 ,...,ik (f ) is a submanifold of X, in order to proceed to the next step. Boardman solved this problem in [Boa67] by constructing subsets  i1 ,...,ik ⊂ J k (X, Y ) which allow to define  i1 ,...,ik (f ) as (j k )−1 ( i1 ,...,ik ), for any smooth mapping f : X → Y (not necessarily generic). The main results of [Boa67] are that the  i1 ,...,ik are in fact submanifolds of J k (X, Y ) and if j k f is transverse to  i1 ,...,ik , then f  is sufficiently  generic for the earlier construction to work, and  i1 ,...,ik (f ) =  ik f | i1 ,...,ik−1 . Example 4.5 Let fu : R2 → R2 be the mapping fu (x, y) = (x, y 3 +x 2 y+uy), with u ∈ R. When u = 0, f0 has an isolated instability at the origin known as the “lips” singularity. For all u, fu has only kernel rank 1 singularities and the singular set  1 (fu ) is given by 3y 2 + x 2 = −u. If u > 0, this is the empty set so fu is regular and hence, locally stable. Otherwise, if u < 0,  1 (fu ) is an ellipse. To compute 1 the singularities of the restriction √ of fu to  (fu ), we take the parameterisation √ x(θ ) = u cos θ and y(θ ) = u/3 sin θ , which gives η(θ ) = fu (x(θ ), y(θ )) =

√



2 −u cos θ, √ u cos θ sin2 θ . 3 3

The  1 -points of the restriction correspond to η (θ ) = 0. We obtain (x, y) ∈  1,1 (fu ) when θ = 0, π and (x, y) ∈  1,0 (fu ) otherwise. If (x, y) ∈  1,0 (fu ), η is regular at θ and hence, the germ of fu at (x, y) is A -equivalent to the Whitney fold (Exercise 3.3.6). When (x, y) ∈  1,1 (fu ), we have η (θ ) = 0 but also η (θ ) = 0 and the germ of fu at (x, y) is A -equivalent to the Whitney cusp (Exercise 3.3.7). Furthermore, η is one-to-one, so fu is locally stable. Notice that the partition of the source R2 given by the types  0 ,  1,0 and  1,1 distinguishes between all the A -types which appear in fu . Figures 4.1 and 4.2 show the picture of  1 (fu ) and its image, which explains the name “lips” for the singularity of f0 at the origin. ♦

4.7 Thom–Boardman Singularities

129

Fig. 4.1 Stable perturbation of the “lips” singularity. Points inside the lips on the right have three preimages, one inside the ellipse on the left and two outside

fu

C(fu)

D(f u)

Fig. 4.2 The short vertical lines on the left show the kernel of dfu . At the right and left extremes of the critical set, and only there, the kernel of dfu is tangent to C(fu ), and there the germ of fu has Boardman symbol  1,1 . At the other points of C(fu ), it has Boardman symbol  1,0

Given any q × p matrix λ with entries in a ring R, we adopt the following conventions: ⎧ ⎪ if  ≤ 0; ⎪ ⎨R, min(λ) = ideal in R generated by the  × minors of λ, if 1 ≤  ≤ min{p, q}; ⎪  ⎪ ⎩0, if  > min{p, q}. Definition 4.6 Let I ⊂ On be a proper ideal generated by f1 , . . . , fp ∈ mn . The rank of I is defined as the rank at 0 of Jf , the Jacobian matrix of f = (f1 , . . . , fp ). For each integer s, the sth-Jacobian extension of I is the ideal s I = I + min (Jf ). n−s+1

This gives a chain of inclusions I = 0 I ⊂ 1 I ⊂ · · · ⊂ n I ⊂ n+1 I = On ,

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4 Contact Equivalence

and n − rank(I ) is the last s such that s I is a proper ideal. The above definitions depend neither on the set of generators of I nor the choice of coordinates in Fn . The proof is easy and is left as an exercise (see Exercises 4.7.1 and 4.7.2). Definition 4.7 Let I ⊂ On be a proper ideal generated by f1 , . . . , fp ∈ mn and let k ≥ 1. The k’th Boardman symbol of I is the sequence i = (i1 , . . . , ik ) such that 1. I has rank n − i1 ; 2. for each  = 2, . . . , k, i−1 . . . i1 I has rank n − i . Given a smooth germ f : (Fn , 0) → (Fp , 0), the k’th Boardman symbol of f is defined as the k’th Boardman symbol of the ideal If generated by the components of f . The same definition can also be given in terms of critical Jacobian extensions. By definition, the ideal s I = I + minn−s+1 Jf is the critical Jacobian extension of I if s I = On and s+1 I = On . The k’th Boardman symbol of an ideal I is i = (i1 , . . ., ik ) if i1 I is the critical jacobian extension of I , i2 (i1 I ) is the critical Jacobian extension of i1 I and so on. Proposition 4.10 The k’th Boardman symbol of a smooth germ f : (Fn , 0) → (Fp , 0) is determined by j k f , for each k ≥ 0. Proof Let I = If . For  = 1, the first term i1 of the Boardman symbol is determined by j 1 f and i1 I is generated by polynomials in the partial derivatives of order ≤ 1 of the components fj , 1 ≤ j ≤ p. Assume that for  − 1, the terms i1 , . . . , i−1 are determined by j −1 f and that i−1 . . . i1 I is generated by combinations of partial derivatives of order ≤  − 1 of the fj . The term i only depends on the rank of the generators and hence is determined by j  f . Moreover, i . . . i1 I is generated by polynomials in the partial derivatives of order ≤  of the fj .   Definition 4.8 A Boardman symbol of length k is any sequence of integers i = (i1 , . . . , ik ). For each Boardman symbol i, we define the Thom–Boardman set  i as the subset of J k (n, p) of jets σ = j k f such that f has Boardman symbol i. Example 4.6 Consider the case k = 1. Then f has Boardman symbol i1 if and only if it has kernel rank i1 . Thus  i1 coincides with the definition given in Appendix A. In this case, J 1 (n, p) can be identified with Matp×n (F), the space of p × n matrices with coefficients in F, and  i1 is the subset of matrices of rank n − i1 . This is in fact a locally closed semialgebraic submanifold of codimension of J 1 (n, p) of codimension (p − n + i1 )i1 (see Exercise 1). ♦ It is clear from the construction that all the Thom–Boardman sets  i are locally closed and semialgebraic in J k (n, p). The next proposition gives a necessary and sufficient condition on i for  i to be non-empty.

4.7 Thom–Boardman Singularities

131

Proposition 4.11 For each Boardman symbol i,  i is non-empty if and only if 1. n ≥ i1 ≥ · · · ≥ ik ≥ 0; 2. i1 ≥ n − p; 3. if i1 = n − p then i1 = · · · = ik . In particular, the sets  i provide a finite partition of J k (n, p). Proof Assume that  i = ∅ and let σ = j k f ∈  i . Since i1 is the kernel rank of f , we have n ≥ i1 ≥ n − p. Moreover, rank I ≤ rank s I for any s and for any ideal I . This shows that i−1 ≥ i for all  = 2, . . . , k. Finally, if s = n − p then s I = I and hence i1 = n − p implies that i1 = · · · = ik . Conversely, let i be a Boardman symbol satisfying conditions 1, 2 and 3. It is enough to find a germ f with Boardman symbol i. If i1 = n − p, we take f1 = x1 , . . . , fp = xn−p . Otherwise, if i1 > n − p, we take

fi =

⎧ ⎪ xi , ⎪ ⎪ ⎪ ⎪ ⎨ n−i 2 ⎪ ⎪ j =n−i1 +1 ⎪ ⎪ ⎪ ⎩0,

1 ≤ i ≤ n − i1 , n−i 3

xj2 +

xj3 + . . ., i = n − i1 + 1,

j =n−i2 +1

n − i1 + 2 ≤ i ≤ p.

It is an easy computation to check that in both cases f has Boardman symbol i.

 

For each Boardman symbol i, we define the number ν(i) = (p − n + i1 )μ(i1 , . . . , ik ) − (i1 − i2 )μ(i2 , . . . , ik ) + · · · + (ik−1 − ik )μ(ik ), where μ(i1, . . . , ik ) is the number of sequences (j1 , . . . , jk ) such that j1 ≥ · · · ≥ jk ≥ 0, i ≥ j for all  = 1, . . . , k and j1 > 0. Example 4.7 When k = 1, we have μ(i1 ) = i1 and hence ν(i1 ) = (p − n + i1 )i1 . For k = 2, we have μ(i1, i2 ) = i1 (i2 + 1) − j2 (i2 − 1)/2, so ν(i1 , i2 ) = (p − n + i1 )i1 +

 i2  (p − n + i1 )(2i1 − i2 + 1) − 2i1 + 2i2 ) . 2

Another interesting particular case is when i1 = · · · = ik = 1. We have μ(1, . . . , 1) = k and hence ν(1, . . . , 1) = (p − n + 1)k. ♦ Now we can state the main result of Boardman in [Boa67]: Theorem 4.7 For every Boardman symbol i,  i is a submanifold of J k (n, p) of codimension ν(i). We refer to the paper of Mather [Mat73] for a proof simpler than Boardman’s original proof. A consequence of the theorem is that the submanifolds  i provide a stratification of J k (n, p). It is known as the Thom–Boardman stratification.

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4 Contact Equivalence

Lemma 4.7 If f, g : (Fn , 0) → (Fp , 0) are K -equivalent, then they have the same k’th Boardman symbol, for all k ≥ 1. Proof By Theorem 4.3, there exists a diffeomorphism φ : (Fn , 0) → (Fn , 0) such that φ ∗ If = Ig . The result is now a consequence of Exercise 4.7.2.   This lemma implies that  i is K -invariant (and hence A -invariant) in J k (n, p). Thus, for any pair of smooth manifolds X, Y , there exists a unique subbundle  i (X, Y ) ⊂ J k (X, Y ) whose fibre is  i . If f : X → Y is any smooth mapping, we have the k-jet extension map j k f : X → J k (X, Y ) and we define  i (f ) = (j k )−1 ( i (X, Y )). By construction,  i (f ) is the subset of X of points x such that the germ of f at x has k’th Boardman symbol i. Analogously, if f : (Fn , S) → (Fp , 0) is any smooth germ, then we define  i (f ) = (j k )−1 ( i ). where j k f : (Fn , S) → J k (n, p) is the k-jet extension. Again,  i (f ) is the setgerm in (Fn , S) of points x such that the germ of f at x has k’th Boardman symbol i. Definition 4.9 A smooth mapping f : X → Y is called Thom–Boardman generic   i , for all Boardman symbols i and for all k ≥ 1. if j k f − A germ f : (Fn , S) → (Fp , 0) is called Thom–Boardman generic if there exists a representative f : X → Y which is Thom–Boardman generic. The following are direct consequences of Theorem 4.7 and of Corollary A.1: Corollary 4.5 Let f : X → Y be a smooth mapping. 1. If f is Thom–Boardman generic, then  i (f ) is a submanifold of X of codimension ν(i). 2. If f is locally stable it is Thom–Boardman generic. We state without proof another important result of Boardman [Boa67] which says that his definition of Thom–Boardman singularities coincides with Thom’s original construction when f is generic. Theorem 4.8 Let f : X → Y be Thom–Boardman generic. For all i,  i (f ) =  ik (f | i1 ,...,ik−1 ).



Example 4.8 Let f : R2 → R2 be the mapping f (x, y) = (x 2 , y 2 ). The Jacobian determinant is 4xy, so we have

0}, •  0 (f ) = {(x, y) ∈ R2 : xy = •  1 (f ) = {(x, y) ∈ R2 : xy = 0}  {0} and •  2 (f ) = {0}.

4.7 Thom–Boardman Singularities

133

The Jacobian extension 1 If is generated by x 2 , y 2 , xy, which has rank 2 at every point (x, y) ∈  1 (f ). Hence  1 (f ) =  1,0 (f ). Observe that the germ of f at (x, y) ∈  1 (f ) is A -equivalent to the Whitney fold, which is stable and hence Thom–Boardman generic. Analogously, 2 If is generated by x, y and has also rank 2. So,  2 (f ) =  2,0 (f ). However, f is not Thom–Boardman generic at the origin, since if it was,  2 (f ) should have codimension 4. Now let F : R4 → R4 be the unfolding F (x, y, u, v) = (x 2 + uy, y 2 + vx, u, v). It is a locally stable mapping, so it is Thom–Boardman generic. In fact, the Jacobian determinant is 4xy − uv, so • • • •

 0 (F ) = {(x, y, u, v) ∈ R4 : 4xy − uv =

0},  1 (F ) = {(x, y, u, v) ∈ R4 : 4xy − uv = 0}  {0},  2 (F ) = {0} and  3 (F ) =  4 (F ) = ∅.

Since F is an unfolding of f we deduce that  2 (F ) =  2,0 (F ). But 1 IF is generated by x 2 + uy, y 2 + vx, u, v, 4xy − uv. Taking the Jacobian matrix of this, we get •  1,1 (F ) = {(x, y, u, v) ∈  1 (F ) : 8x 2 − 4uy = 8y 2 − 4vx = 0} and •  1,0 (F ) =  1 (F )   1,1 (F ). We can proceed one step more: 1,1 IF is generated by 1 F and the two new equations 8x 2 − 4uy, 8y 2 − 4vx. Again by taking the Jacobian of this, we arrive at •  1,1,1 (F ) = {(x, y, u, v) ∈  1,1 (F ) : x = y = uv = 0} and •  1,1,0 (F ) =  1,1 (F )   1,1,1 (F ). In the last step, we just check that  1,1,1,0 (F ) =  1,1,1 (F ), which gives the complete Thom–Boardman stratification of F . When uv = 0, fu,v : R2 → R2 is Thom–Boardman generic, as one can check easily. We have  2 (fu,v ) =  1,1,1 (fu,v ) = ∅ and the equations for  1,0 (fu,v ) and  1,1 (fu,v ) are the same as those of F . At a point (x, y) ∈  1,0 (fu,v ) the germ of f is A -equivalent to the Whitney fold. The equations of  1,1 (fu,v ) give exactly three solutions (one is real and the other two are complex) and at these points the germ of f is A -equivalent to the Whitney cusp. Figure 4.3 shows a picture of the image this mapping. ♦ As the above example shows, the Thom–Boardman singular sets of a mapping can be obtained by means of the iterated Jacobian extensions. The procedure can be systematised easily by using the ideals Ji (f ) introduced by Bernard Morin in [Mor75]. These ideals are more convenient if we want to compute not only the Boardman symbol at a specific point but also the sets  i (f ). Definition 4.10 Let f : (Fn , 0) → (Fp , 0) be a smooth map-germ and let I ⊂ On be an ideal generated by g1 , . . . , gr . For each integer s, the sth-Jacobian extension

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4 Contact Equivalence

Fig. 4.3 The image of fu,v , with uv = 0

of the pair (f, I ) is the ideal s (f, I ) = I + min (J(f,g)). n−s+1

For each Boardman symbol i = (i1 , . . . , ik ), the iterated Jacobian extension of f is the ideal Ji (f ) defined recursively as follows:

Ji (f ) =

⎧ i1 ⎪ ⎪ ⎨ (f, {0}), if k = 1, ⎪ ⎪ ⎩ik (f, J

i1 ,...,ik−1 (f )),

if k > 1.

The following lemma gives the relationship between Ji (f ) and the ideals ik . . . i1 If used to define the Boardman symbol of f . Lemma 4.8 For any germ f : (Fn , 0) → (Fp , 0) and for any Boardman symbol i = (i1 , . . . , ik ), ik . . . i1 If = If + Ji (f ).

(4.22)

In particular, ik . . . i1 If is a proper ideal if and only if Ji (f ) is. Proof The equality (4.22) is clear for k = 1. Assume that (4.22) is true for k − 1. If Ji (f ) = (g1 , . . . , gr ), then ik−1 . . . i1 If = (f1 , . . . , fp , g1 , . . . , gr ). By construction and by the induction hypothesis, ik . . . i1 If = ik−1 . . . i1 If + min (J(f,g)) n−ik +1

= If + Ji1 ,...,ik−1 (f ) + min (J(f,g) ) n−ik +1

= If + Ji (f ).

 

4.7 Thom–Boardman Singularities

135

We consider the lexicographical order in the set of all Boardman symbols: if i = (i1 , . . . , ik ) and j = (j1 , . . . , j ), we write i % j if either i = j or ir0 < jr0 , where r0 is the least r such that ir = jr . Lemma 4.9 Let f : (Fn , 0) → (Fp , 0) be a germ and let i, j be Boardman symbols such that i % j. Then Ji (f ) ⊂ Jj (f ). Proof We first claim that for any s, t and for any ideal I ⊂ On , we have t (f, s (f, I )) ⊂ s+1 (f, I ).

(4.23)

In fact, let I = (g1 , . . . , gr ). We denote by h1 , . . . , hm the n − s + 1-minors of J(f,g). Then t (f, s (f, I )) is generated by g1 , . . . , gr , h1 , . . . , hm and by the n − t + 1-minors of J(f,g,h). Each hi can be written as a linear combination of elements in s+1 (f, I ). It follows from the rule of derivation of the determinants, that any n − t + 1-minor of J(f,g,h) also belongs to s+1 (f, I ). This proves the claim. Assume now that i % j, with i = (i1 , . . . , ik ) and j = (j1 , . . . , j ) and i = j. We have ir0 < jr0 , where r0 is the least r such that ir = jr . By (4.23), Ji (f ) = Ji1 ,...,ir0 −1 ,ir0 ,...,ik (f ) ⊂ Ji1 ,...,ir0 −1 ,ir0 ,...,ik−1 +1 (f ) ⊂ Ji1 ,...,ir0 −1 ,ir0 +1 (f ) ⊂ Ji1 ,...,ir0 −1 ,jr0 (f ) ⊂ Ji1 ,...,ir0 −1 ,jr0 ,...,j (f ) = Jj (f ).   Definition 4.11 Given a Boardman symbol i of length k, the successor i is the least Boardman symbol j of length ≤ k such that i % j and i = j. In the non-trivial cases, we only consider decreasing Boardman symbols, that is, Boardman symbols i = (i1 , . . . , ik ) such that i1 ≥ · · · ≥ ik . Let r such that ir > ir+1 = · · · = ik . Then the successor of i is i = (i1 , . . . , ir , ir+1 + 1). Proposition 4.12 Let f : (Fn , 0) → (Fp , 0) be a smooth map-germ. Then, for each Boardman symbol i,  i (f ) = V (Ji (f ))  V (Ji (f )). Proof Given a point x in a neighbourhood of 0 in Fn , we must show that f has k’th Boardman symbol i at x if and only if x ∈ V (Ji (f ))  V (Ji (f )). We can assume without loss of generality that x = 0. So, we must prove that the germ f has k’th Boardman symbol i if and only if Ji (f )) = On and Ji (f ) = On . Again we proceed by induction on k. The equivalence is clear for k = 1. Assume that the equivalence is true for k − 1. Suppose that f has k’th Boardman symbol

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4 Contact Equivalence

i. In particular, f has (k − 1)th Boardman symbol j := (i1 , . . . , ik−1 ) and by the hypothesis induction, Jj (f ) = On and Jj (f ) = On . Moreover, ik−1 . . . i1 If has rank n − ik or equivalently, ik . . . i1 If = On ,

ik +1 ik−1 . . . i1 If = On .

By (4.22), this implies that Ji (f ) = On ,

Ji1 ,...,ik−1 ,ik +1 (f ) = On .

If ik−1 = ik , then i = j and otherwise, if ik−1 > ik , then i = (i1 , . . . , ik−1 , ik + 1). In both cases, we get Ji (f ) = On . Conversely, suppose that Ji (f ) = On and Ji (f ) = On and let j := (i1 , . . . , ik−1 ). We have Jj (f ) ⊂ Ji (f ) = On . Assume first that ik−1 = ik . Then i = j and hence Jj (f ) = On . By the induction hypothesis, j is the (k − 1)’th Boardman symbol of f . In particular, ik−2 . . . i1 If has rank n − ik−1 and hence, ik−1 . . . i1 If has rank ≥ n − ik−1 = n − ik . On the other hand, from (4.22) we also see that ik . . . i1 If = On and thus, ik−1 . . . i1 If has rank ≤ n − ik . We deduce that ik−1 . . . i1 If has rank n − ik , so f has k’th Boardman symbol i. Otherwise, if ik−1 > ik , then i = (i1 , . . . , ik−1 , ik + 1). Let r be such that ir > ir+1 = · · · = ik−1 , then j = (i1 , . . . , ir , ir+1 + 1). We have i % j and hence, On = Ji (f ) ⊂ Jj (f ) by Lemma 4.9. By the induction hypothesis, j is the (k −1)’th Boardman symbol of f . Moreover, by (4.22), ik . . . i1 If = On ,

ik +1 ik−1 . . . i1 If = On .

This is equivalent to ik−1 . . . i1 If having rank n − ik . Therefore, f has k’th Boardman symbol i.   Example 4.9 Consider the stable map F : R4 → R4 given by F (x, y, u, v) = (x 2 + uy, y 2 + vx, u, v) of Example 4.8. The first order Jacobian extensions give J1 (F ) = (4xy − uv),

J2 (F ) = (x, y, u, v),

J3 (F ) = J4 (f ) = (1),

and the iterated Jacobian extensions are J1,1 (F ) = (4xy − uv, 8x 2 − 4uy, 8y 2 − 4vx), J1,1,1,1(F ) = (1),

J1,1,1 (F ) = (x, y, uv),

J2,1 (F ) = (1).

This allows us to compute the Thom–Boardman stratification easily by Proposition 4.12: •  1 (F ) = V (J1 (F ))  V (J2 (F )), •  2 (F ) = V (J2 (F )),  3 (F ) =  4 (F ) = ∅, •  1,0 (F ) = V (J1 (F ))  V (J1,1 (F )),

4.7 Thom–Boardman Singularities

• • • • •

137

 1,1 (F ) = V (J1,1 (F ))  V (J2 (F )),  1,1,0 (F ) = V (J1,1 (F ))  V (J1,1,1 (F )),  1,1,1 (F ) = V (J1,1,1(F ))  V (J2 (F )),  1,1,1,0 (F ) = V (J1,1,1 (F )),  1,1,1,1 (F ) = ∅,  2,0 (F ) = V (J2 (F )),  2,1 (F ) =  2,2 (F ) = ∅.

In the last part of this section we restrict ourselves to Thom–Boardman generic germs f : (Fn , 0) → (Fp , 0), with n ≤ p and of type  1 . These are called Morin singularities, after Bernard Morin [Mor65]. We will see that in this class of singularities, the Thom–Boardman stratification is enough to characterise stability and the A -class. For each k ≥ 1 we denote by 1k the Boardman symbol (1, . . . , 1, 0) with 1 repeated k-times. If f has (k + 1)’st Boardman symbol 1k , then its ’th Boardman symbol is (1, . . . , 1, 0, . . . , 0), for all  ≥ k + 1. In such a case we will say that f has Boardman type 1k . Lemma 4.10 Let f : (Fn , 0) → (Fp , 0) be K -finite, with n ≤ p and type  1 . Then there exists k ≥ 1 such that has Boardman type 1k . Moreover, any other germ with Boardman type 1k is K -equivalent to f . Proof Since f has type  1 , we can choose coordinates (x, u) in Fn = F × Fn−1 such that f (x, u1 , . . . , un−1 ) = (f1 (x, u), . . . , fp−n+1 (x, u), u), with fi ∈ m2n , for all i = 1, . . . , p − n + 1. Thus, Q(f ) ∼ = O1 /I , where I is the ideal generated by the functions hi (x) = fi (x, 0), i = 1, . . . , p − n + 1. But Q(f ) has finite F-dimension, for f is K -finite. Hence, Q(f ) ∼ = O1 /(x k+1 ), for some k ≥ 1. By Theorem 4.3 (see also Exercise 4.4.1), h : (F, 0) → (Fp−n+1 , 0) is K -equivalent to the germ (x k+1 , 0, . . . , 0), which has Boardman type 1k (see Exercise 4.7.5). But since f is an unfolding of h, f also has Boardman type 1k , by Exercise 4.7.3 . Let g : (Fn , 0) → (Fp , 0) be any germ with Boardman type 1k . Then g has type 1  and as above, Q(g) ∼ = O1 /I , for some ideal I . If I = 0, then g should be K -equivalent to an unfolding of the constant germ (F, 0) → (Fp−n+1 , 0) which has ’th Boardman symbol (1, . . . , 1) for all . Hence, I = 0 and necessarily I = (x m+1 ), for some m ≥ 1. Again by the above argument, this implies that g has Boardman type 1m , so m = k and hence g is K -equivalent to f .   An immediate consequence of the lemma is the following: Corollary 4.6 Let σ ∈ J k+1 (n, p), with n ≤ p and assume that σ ∈  1k . Then the K (k+1)-orbit of σ is  1k . Definition 4.12 A Thom–Boardman generic germ f : (Fn , 0) → (Fp , 0) with n ≤ p and type  1 is called a Morin singularity. A Morin map is a Thom–Boardman generic map f : X → Y with dim X ≤ dim Y and with only  1 -singularities.

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Corollary 4.7 Any Morin singularity f : (Fn , 0) → (Fp , 0) is stable. Moreover, two Morin singularities f, g : (Fn , 0) → (Fp , 0) with the same Boardman type are A -equivalent. The first part of the corollary follows from Corollary 4.6 and Theorem 4.6. For the second part, we also need the fact that if two stable germs are K -equivalent, then they are also A -equivalent. This will be proved in Chap. 7 (see Theorem 7.1). The above corollary is not true for multi-germs, since we need that the images of the Thom–Boardman submanifolds meet in general position in the target. This motivates the next definition. Definition 4.13 Let f : X → Y be a Morin map. We say that f satisfies the normal crossing condition if given distinct points x1 , . . . , xr ∈ X such that f (x1 ) = · · · = f (xr ) = y and f has Boardman type 1ki at each xi , then the subspaces dxi f Txi  1ki (f ) meet in general position in Ty Y .

Now we have the global version of the above corollary, which follows easily from Corollary 4.4. Corollary 4.8 Let f : X → Y be a Morin map satisfying the normal crossing condition and such that the restriction f : C(f ) → Y is finite. Then f is locally stable.

Exercises for Sect. 4.7 1 Suppose that f1 , . . . , fp ∈ mn and g1 , . . . , gq ∈ mn generate the same ideal I in On . Show that rank Jf = rank Jg and that I + min (Jf ) = I + min (Jg ), for all . 2 Let I be a proper finitely generated ideal in On and φ : (Fn , 0) → (Fn , 0) a diffeomorphism. Show that rank φ ∗ I = rank I and that φ ∗ (s I ) = s (φ ∗ I ), where φ ∗ : On → On is the induced isomorphism. 3 Let f : (Fn , 0) → (Fp , 0) be smooth and let F be an unfolding of f . Show that F and f have the same k’th Boardman symbol, for all k ≥ 1. Show also that f and the germ (f, 0) obtained from f by adding p + 1’st component 0 have the same k’th Boardman symbol for each k. 4 Let f : (Fn , 0) → (F, 0) be smooth. Show that f has Boardman symbol (n, 0) if and only if it has a non-degenerate critical point. 5 Let f : (F, 0) → (Fp , 0) be the germ f (x) = (x k+1 , 0, . . . , 0). Show that its (k + 1)’st Boardman symbol is (1, . . . , 1, 0). 6 Show that the k’th Boardman symbol of f is (i1 , . . ., ik−1 , 0) if and only if the k − 1’st symbol is (i1 , . . ., ik−1 ) and ik−1 · · ·i1 I is the maximal ideal m.

4.7 Thom–Boardman Singularities

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7 Show that if f : (Fn , 0) → (Fp , 0) is K -finite then for some finite k, the k’th Boardman symbol (i1 , . . ., ik ) has ik = 0. We call the first such k the length of the Boardman symbol of f . 8 Place bounds on the length of the Boardman symbol of a stable map-germ (Fn , 0) → (Fp , 0). 9 Let f : F6 → F5 be the mapping f (x, y, u, v, w, t) = (x 2 y + y 4 + ux + vy + wx 2 + ty 2 , u, v, w, t). This is a stable unfolding of the function x 2 y + y 4 . Find the iterated Jacobian ideals of f and deduce the Thom–Boardman stratification by means of Proposition 4.12 as in Example 4.9. 10 Let f : (F4 , 0) → (F4 , 0) be Thom–Boardman generic. Show that f is stable. 11 Let f : X → Y be a smooth mapping with dim X = dim Y = 4. State and prove a result analogous to Corollary 4.8 but without the assumption that f has only  1 -singularities.

Chapter 5

Versal Unfoldings

Consider the parameterised plane curve f (x) = (x 3 , x 4 ). The 1-parameter deformations ft(1) (x) = (x 3 − tx, x 4 − 0.6tx 2 ),

ft(2) (x) = (x 3 − tx, x 4 − 0.8tx 2 ),

ft(3) (x) = (x 3 − tx, x 4 − 1.2tx),

ft(4) (x) = (x 3 − tx, x 4 − 0.8tx 2 + 0.4tx),

have the following images for different values of the parameter t (Fig. 5.1). Are these the only possible pictures we can get by deforming this curve? Can adding judicious combinations of other monomials produce curves with different images and other singularities? Which other singularities can appear? The main theorem of this chapter, Theorem 5.1, tells us that if f has finite Ae -codimension, then there is a family fu , where u varies in a finite-dimensional parameter space, which contains all possible deformations, up to a suitable notion of parameterised left-right equivalence. This needs some clarification: we are interested only in phenomena which are “infinitely close to f ”, that is, which appear in germs of deformations of f . This will be made clear in what follows. This notion is captured by the definition of germ of unfolding. As already introduced in Definition 3.4, an unfolding of a germ f0 : (Fn , S) → p (F , 0) is a germ F : (Fn × Fd , S × {0}) → (Fp × Fd , 0) of the form F (x, u) = (f˜(x, u), u) where f˜(_, 0) = f0 . Evidently, from any germ of deformation fu one can obtain an unfolding F (x, u) = (fu (x), u), and vice versa. Unfoldings are often written in the form F (x, u) = (fu (x), u) to emphasise that the first term is a deformation of some f0 , but it is important to realise that for u = 0, fu is not a germ, since it does not have a base-point. If f0 has finite Ae -codimension then (with some care) it will be possible to view each fu as a mapping on a more-or-less well-defined domain and target. We explore this further in Sect. 5.5. © Springer Nature Switzerland AG 2020 D. Mond, J. J. Nuño-Ballesteros, Singularities of Mappings, Grundlehren der mathematischen Wissenschaften 357, https://doi.org/10.1007/978-3-030-34440-5_5

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