Topological Complexity and Related Topics [1 ed.] 9781470444051, 9781470434366

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702

Topological Complexity and Related Topics Mini-Workshop Topological Complexity and Related Topics February 28–March 5, 2016 Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, Germany

Mark Grant Gregory Lupton Lucile Vandembroucq Editors

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Topological Complexity and Related Topics Mini-Workshop Topological Complexity and Related Topics February 28–March 5, 2016 Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, Germany

Mark Grant Gregory Lupton Lucile Vandembroucq Editors

EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss

Kailash Misra

Catherine Yan

2010 Mathematics Subject Classification. Primary 55-06, 20F36, 52C35, 55M30, 55P62, 55P91, 55Q25, 57M15, 68T40, 93C85. Library of Congress Cataloging-in-Publication Data Names: Grant, Mark, 1980– editor. | Lupton, Gregory, 1960– editor. | Vandembroucq, Lucile, 1971– editor. Title: Topological complexity and related topics / Mark Grant, Gregory Lupton, Lucile Vandembroucq, editors. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Contemporary mathematics; volume 702 | “Mini Workshop on Topological Complexity and Related Topics, February 29-March 2, 2016, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, Germany.” | Includes bibliographical references. Identifiers: LCCN 2017042708 | ISBN 9781470434366 (alk. paper) Subjects: LCSH: Algebraic topology. | Topology. | AMS: Algebraic topology – Proceedings, conferences, collections, etc. msc | Group theory and generalizations – Special aspects of infinite or finite groups – Braid groups; Artin groups. msc | Convex and discrete geometry – Discrete geometry – Arrangements of points, flats, hyperplanes. msc | Algebraic topology – Classical topics – Ljusternik-Schnirelman (Lyusternik-Shnirel’man) category of a space. msc | Algebraic topology – Classical topics – None of the above, but in this section. msc | Algebraic topology – Homotopy theory – Rational homotopy theory. msc | Algebraic topology – Homotopy theory – Equivariant homotopy theory. msc | Algebraic topology – Homotopy groups – Hopf invariants. msc | Algebraic topology – Fiber spaces and bundles – Discriminantal varieties, configuration spaces. msc | Algebraic topology – Operations and obstructions – Sectioning fiber spaces and bundles. msc | Manifolds and cell complexes – Low-dimensional topology – Relations with graph theory. msc | Computer science – Artificial intelligence – Robotics. msc | Systems theory; control – Control systems – Automated systems (robots, etc.). msc Classification: LCC QA612 .T6525 2018 | DDC 514/.2–dc23 LC record available at https://lccn.loc.gov/2017042708 DOI: http://dx.doi.org/10.1090/conm/702 Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2018 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

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Contents

Preface

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Survey Articles Equivariant topological complexities ´ Andr´ es Angel and Hellen Colman

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Rational methods applied to sectional category and topological complexity Jos´ e Carrasquel

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Topological complexity of classical configuration spaces and related objects Daniel C. Cohen

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A topologist’s view of kinematic maps and manipulation complexity ´ Petar Paveˇ sic

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Research Articles On the cohomology classes of planar polygon spaces Donald M. Davis

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Sectional category of a class of maps Jean-Paul Doeraene, Mohammed El Haouari, and Carlos Ribeiro

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Q-topological complexity ´ ndez Sua ´ rez and Lucile Vandembroucq Luc´ıa Ferna

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Topological complexity of graphic arrangements Nathan Fieldsteel

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Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes ´ s Gonza ´ lez, Mark Grant, and Lucile Vandembroucq Jesu

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Topological complexity of collision-free multi-tasking motion planning on orientable surfaces ´ s Gonza ´ lez and Ba ´ rbara Guti´ Jesu errez

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Topological complexity of subgroups of Artin’s braid groups Mark Grant and David Recio-Mitter

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Preface This is the proceedings volume of a conference on Topological Complexity and Related Topics, held at the Mathematisches Forschungsinstitut Oberwolfach (MFO) from February 28 to March 5, 2016, under the auspices of their Mini-Workshop program. There were 16 participants. The talks were a mixture of presentations of research results and surveys, as reflected in the contents of this volume. Details about the conference, including a list of participants as well as short abstracts of the talks presented, are available from the corresponding Oberwolfach Report [Report No. 15/2016, Mini-Workshop: Topological Complexity and Related Topics, Oberwolfach Reports, Vol. 13, No. 1 (2016), pp. 705–740]. The notion of topological complexity (TC) was introduced by Farber in 2003. This is a numerical homotopy invariant of a space, of Lusternik–Schnirelmann type, which provides a topological approach to the study of the complexity of the motion planning problem in robotics. If X is the configuration space of a mechanical system, that is, the space of all the possible states of the system, then a motion planner in X is a (not necessarily continuous) function that assigns to each pair (x, y) ∈ X ×X a path in X from x to y prescribing the motion of the system from the initial state x to the final state y. The point of departure for this notion as a topic of interest is a basic result that says a global continuous motion planner is possible only when the configuration space X is contractible. Roughly speaking, then, TC(X) corresponds to the minimum number of local continuous motion planners needed to determine a complete motion planner in X. It soon becomes apparent that TC(X) is determined by the topology of the space X in ways that are often not well-understood, making it a delicate invariant to compute. By specifying some intermediate states in the motion, the concept was generalized to the notion of higher topological complexity by Rudyak in 2010. These invariants have been intensively studied in the last decade, and their values have been determined for numerous interesting spaces. Much work has been done in particular on the case of classical configuration spaces, whose points consist of ordered n-tuples of distinct points in a given space, and which model the collision-free motion planning problem with n agents. Some of theses spaces are Eilenberg-Mac Lane spaces K(G, 1). It remains a challenge to understand the topological complexity of such spaces in terms of the properties of the group G. The survey of Cohen presents the methods of determination of the topological complexity of the configuration spaces of various classical spaces (such as Euclidean spaces and compact orientable surfaces). The discussion is also extended to some related spaces such as orbits of configuration spaces and some Eilenberg-Mac Lane spaces K(G, 1). In the same direction, the article of Gonz´alez and Guti´errez determines the higher topological complexity of the configuration spaces of ordered distinct v

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PREFACE

points in an orientable surface. The article of Grant and Recio-Mitter studies the (higher) topological complexity of Eilenberg-Mac Lane spaces K(G, 1) where G is a certain type of subgroup of Artin’s full braid group, while the article by Fieldsteel is dedicated to the (higher) topological complexity of the complement in Cn of some arrangements of hyperplanes associated to a graph. In most of these works, an important tool is the cohomological lower bound of TC given by the zero-divisors cup-length. As a potential useful step towards the computation of this cohomological lower bound, the article of Davis analyses the top cohomology class of the space of isometry classes of polygons whose side lengths satisfy some condition. As mentioned above, TC is related to the Lusternik–Schnirelmann category (LS-category, or cat), the theory of which progressed rapidly in the last decade of the twentieth century. Many tools and techniques of rational homotopy theory and of classical homotopy theory have been developed in order to study the properties of this invariant, and to solve outstanding problems such as the Ganea conjecture on cat(X × S p ). Topological complexity (and its higher versions) and LS-category are special cases of the Schwarz genus, or sectional category, of a fibration. Several studies have been dedicated to this unifying notion in the recent past, with efforts being made to generalize the ideas, techniques, and approximations developed in the context of LS-category to sectional category, in order to make them available for the study of TC. The survey of Carrasquel presents the algebraic tools and results which have been developed in rational homotopy theory to study approximations of sectional category and TC using Sullivan models. In a more topological approach, the article by Doeraene, El Haouari and Ribeiro develops a notion of sectional category of a class of maps which generalizes the classical notion of sectional category, while the article of Fern´andez-Su´arez and Vandembroucq studies a notion of Q-topological complexity inspired by the notion of Q-category introduced by Scheerer, Stanley and Tanr´e. In the last decade, some work has also been done to understand the (close) relationship between TC(X) and the LS-category of the cofibre of the diagonal map Δ : X → X × X. The article by Gonz´alez, Grant and Vandembroucq pursues this investigation in the special case of two-cell complexes, making use of the Berstein-Hilton-Hopf invariants, which played an important role in Iwase’s resolution of the Ganea conjecture for LS-category. Since the introduction of TC, many extensions and variations of the concept have been conceived that take into account some additional aspects of the motion planning problem. For instance, various authors have developed versions of equivariant TC in order to study the complexity of the motion planning problem when the configuration space is given with the action of a group. The survey of ´ Angel and Colman reviews these different approaches, and discusses the properties and advantages of each invariant. In another direction, the notion of topological complexity of a map has recently been introduced, in order to take into account the information encoded by the so-called kinematic map associated to a mechanical system. The survey of Paveˇsi´c reviews various aspects of robotics which are relevant in the study of the motion planning problem, discussing in particular the properties of kinematic maps and the study of this new extension of topological complexity. ´ Finally, we note that the articles of Angel–Colman, Cohen, Davis and Paveˇsi´c use Farber’s original definition of topological complexity for which the topological complexity of a contractible space is 1, whereas the other articles have adopted

PREFACE

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the normalized version which assigns 0 to a contractible space (as is usual in more homotopical approaches to invariants of Lusternik–Schnirelmann type). The AMS publications department has been very encouraging and supportive throughout the preparation of this volume. We would like to thank Christine Thivierge, especially, for her guidance at each stage. The editors, who were also the conference organisers, thank the MFO for support with overall organization of the conference, and for providing an ideal location with outstanding facilities for our activity. This activity was partially supported by a grant from the Simons Foundation (#209575 to Gregory Lupton), and by funds from the (Portuguese) Funda¸c˜ ao para a Ciˆencia e a Tecnologia, through the Project UID/MAT/0013/2013. The MFO and the editors thank the National Science Foundation for supporting the participation of junior researchers in the workshop under the grant DMS-1049268, “US Junior Oberwolfach Fellows.” Mark Grant Gregory Lupton Lucile Vandembroucq

Contemporary Mathematics Volume 702, 2018 http://dx.doi.org/10.1090/conm/702/14103

Equivariant topological complexities ´ Andr´es Angel and Hellen Colman Abstract. The aim of this article is to review different generalizations of the the notion of topological complexity to the equivariant setting. In particular, we review the relation (or non-relation) between these notions and the topological complexity of the quotient space and the topological complexity of the fixed point sets. We give examples of calculations and stress the question: When the action is free, do we recover the topological complexity of the quotient?

1. Introduction The topological complexity of spaces with group actions was first introduced by Colman and Grant [7] as a navigational complexity quantifier of certain mechanical problems best described by groups acting on spaces and on the other hand as a tool for obtaining useful information in classical nonequivariant topological complexity. Since then there have been several approaches to defining other equivariant versions of topological complexity emphasizing in different degrees one or the other of these two broad objectives. Namely, • (Colman-Grant) Equivariant topological complexity. • (Lubawski-Marzantowicz) Invariant topological complexity. • (Dranishnikov) Strongly equivariant topological complexity. • (Blaszczyk-Kaluba) Effective topological complexity. In this paper we give a general view of the different definitions as well as a summary of their properties. Most results considered in our exposition are based on the original articles by the respective authors [5, 7, 8, 13] and calculations from [4, 12]. We also include some new results. The layout of the paper is as follows. In section 2 we review some general equivariant notions and we define the equivariant Lusternik-Schnirelmann category of a group action. In section 3 we introduce the Farber’s topological complexity of a topological space and discuss some of its properties. Section 4 is the core of the article and provides a survey of the four definitions of equivariant versions of topological complexity and its properties. In section 5 we finish with comments 2010 Mathematics Subject Classification. Primary 55M99, 57S10; Secondary 55M30, 55R91. The work of the second author was supported in part by Simons Foundation. The work of A.A. was supported in part by the FAPA funds from Vicerrector´ıa de Investigaciones de la Universidad de los Andes. c 2018 American Mathematical Society

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on the role of the notion of Morita equivalence of group actions in the study of topological complexity of spaces with symmetries. We propose the study of Morita invariance for the invariant topological complexity. Moreover, we aim to develop a theory of topological complexity for orbifolds [1] which will generalize and give as particular cases the classical topological complexity of spaces and a topological complexity for group actions. In general, an orbifold topological complexity will give a version of topological complexity for all contexts modeled by classes of equivalence of group actions such as topological spaces, group actions, foliations and stacks. 2. Equivariant Notions In this paper, G will always denote a compact Hausdorff topological group acting continuously on a Hausdorff space X on the left. In this case, we say that X is a G-space. For each x ∈ X the isotropy group Gx = {h ∈ G | hx = x} is a closed subgroup of G. The set Gx = {gx | g ∈ G} ⊆ X is called the orbit of x. The orbit space X/G is the set of equivalence classes determined by the action, endowed with the quotient topology. Since G is compact and X is Hausdorff, X/G is also Hausdorff, and the orbit map p : X → X/G sending a point to its orbit is both open and closed. If H is a closed subgroup of G, then X H = {x ∈ X| hx = x for all h ∈ H} is called the H-fixed point set of X. We call x a global fixed point if x ∈ X G . A G-space X is said to be G-connected if the H-fixed point set X H is pathconnected for every closed subgroup H of G. 2.1. G-homotopy. Let X and Y be G-spaces. Two G-maps φ, ψ : X → Y are G-homotopic, written φ G ψ, if there is a G-map F : X × I → Y with F0 = φ and F1 = ψ, where G acts trivially on I and diagonally on X × I. If there exist G-maps φ : X → Y and ψ : Y → X such that φψ G idY and ψφ G idX , then φ and ψ are G-homotopy equivalences, and X and Y are G-homotopy equivalent, written X G Y . Definition 2.1. We say that a G-invariant subset U ⊆ X is G-compressible into a G-invariant subset A ⊆ X, if the inclusion map iU : U → X is G-homotopic to a G-map c : U → X with c(U ) ⊆ A. Definition 2.2. A G-invariant subset U ⊆ X is called G-categorical if U is G-compressible into a single orbit. We say that the space X is G-contractible if X is G-categorical. The following two lemmas emphasize the importance of G-connectedness and the existence of global fixed points in relation to G-homotopy. They will be repeatedly used in many subsequent arguments. Lemma 2.3 (Conservation of isotropy). Let X be a G-connected G-space, and let x, y ∈ X such that Gx ⊆ Gy . Then there exists a G-homotopy F : Gx × I → X such that F0 = iGx : Gx → X and F1 (Gx) ⊆ Gy. Lemma 2.4 (G-categorical is G-compressible into a point). Let X be a Gconnected G-space with X G = ∅ , and let x ∈ X G . Then every G-categorical subset U of X is G-compressible to x.

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2.2. Equivariant Lusternik-Schnirelmann category. Marzantowicz [14] studied the equivariant LS-category as a generalization of the usual notion of Lusternik-Schnirelmann category adapted to the equivariant context. Definition 2.5. Given a G-space X, the equivariant LS-category, catG (X), of X is the least integer k such that X may be covered by k G-invariant open sets {U1 , . . . , Uk }, each one G-categorical. By definition we have that catG (X) = 1 iff X is G-contractible. Like in the nonequivariant version, if M is a manifold, we have that catG (M ) is a lower bound for the number of critical points of invariant differentiable functions f : M → R. Proposition 2.6. Let X and Y be G-spaces. (1) If X G Y then catG (X) = catG (Y ). (2) If catG (X) = 1 and X G = ∅ then X is G-compressible to a point. (3) |π0 (X G )| ≤ cat(X G ) ≤ catG (X). (4) cat(X/G) ≤ catG (X). If G is a compact Lie group, Marzantowicz proved that the equivariant LScategory of a free space is just the LS-category of the quotient. The main ingredient of the proof is the use of the Palais covering homotopy theorem [15] which gives conditions to find equivariant lifts of homotopies between quotient spaces. Theorem 2.7. [14] If a compact Lie group G acts freely on a metrizable space X, then catG (X) = cat(X/G). More generally, the result holds if X has one orbit type. For example, from this theorem follows the calculation of the equivariant LScategory of the antipodal action of Z2 on S n : catZ2 (S n ) = cat(RPn ) = n + 1. A cohomological lower bound for catG (X) is given in [14] using the nilpotency of reduced equivariant cohomologies. An equivariant cohomology theory is called ∗ (G/H) = 0 for ∗ > 0. singular if for every closed subgroup H ⊆ G we have that HG ∗ Theorem 2.8. [14] Let HG (·) be a singular multiplicative G-cohomology theory. ˜ ∗ (X), then catG (X) ≥ If we denote by Z the reduced equivariant cohomology H G nil(Z).

For any group G we have the universal example of free G-space EG. It is a free G-space that is contractible and such that the projection to the quotient space BG = EG/G is a principal G-bundle. When G is a finite group we have that cat(BG) = ∞ because BG has infinite cohomological dimension. When G is an infinite discrete group, Palais covering homotopy theorem cannot be applied. Instead, by using the homotopy lifting property of covering spaces we have a similar result as in the previous theorem: catG (EG) = cat(EG/G) = cat(BG) which is equal to cd(G) + 1 where cd(G) is the cohomological dimension of G. A product inequality catG (X × Y ) ≤ catG (X) + catG (Y ) − 1 is treated in theorem 3.15 in [7], but the theorem as stated is not true. A counterexample and

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the correct hypotesis for the result to be true, are discused in theorem 2.23 and example 6.4 in [2]. The assumptions that X is G-connected and X G = ∅ are enough to have that catG (X × X) ≤ 2 catG (X) − 1. We have also the generalization to the equivariant case of the Clapp-Puppe category [6] that is relevant to the study of invariant topological complexity and strongly equivariant topological complexity. This generalization is defined in [14]. Definition 2.9. Let A ⊆ X be a G-invariant subset of X. The equivariant A-category, A catG (X), is the least integer k such that X may be covered by k G-invariant open sets {U1 , . . . , Uk }, each G-compressible into A. If ∗ ∈ X G then catG (X) ≤{∗} catG (X) and if X is G-connected, using lemma 2.4, we have that every G-categorical set is G-compressible to the point ∗ and thus {∗} catG (X) ≤ catG (X). Note that in this case we have cat(X) ≤ {∗} catG (X) = catG (X). 3. Topological complexity Let X be the space of configurations of a mechanical system. A motion planning algorithm is a set of rules that to each pair of configurations (initial and final) assigns a path between them. Let X I be the space of all paths in a space X. Consider the evaluation map ev : X I → X × X given by   ev (γ) = γ(0), γ(1) . A motion planner on an open subset U ⊆ X × X is a section of the evaluation map ev over U , i.e. a (continuous) map s : U → X I such that the following diagram commutes: XI x; x s xx ev xx x  xx / X ×X U The topological complexity of a space X, denoted TC(X), is the least integer k such that there exists an open cover of X × X by k open sets on each of which there is a motion planner [9, 10]. Proposition 3.1. Let X and Y be spaces. (1) If X  Y then TC(X) = TC(Y ). (2) TC(X) = 1 iff X is contractible. (3) TC(X) = ∞ if X is not connected. Theorem 3.2. Let X be a connected space. (1) cat(X) ≤ TC(X) ≤ cat(X × X). (2) TC(X) ≤ 2 dim X + 1 where dim denotes the covering dimension.

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Topological complexity can be defined equivalently in terms of the sectional category of a map [16] and in terms of the Clapp-Puppe category: Theorem 3.3. For a space X, the following statements are equivalent: (1) TC(X) ≤ n. (2) secat(ev) ≤ n: there exist open sets U1 , . . . , Uk which cover X × X and sections si : Ui → X I such that ev ◦ si is homotopic to the inclusion map Ui → X × X. (3) Δ(X) cat(X × X) ≤ n: there exist open sets U1 , . . . , Uk which cover X × X such that each is compressible into Δ(X). Cohomological lower bounds are given by the nilpotency of the kernel of zero divisors of X. Let H ∗ (−) be the cohomology with coefficients in a field and Z = ker(∪) where ∪ : H ∗ (X) ⊗ H ∗ (X) → H ∗ (X) is the cup product homomorphism. Proposition 3.4. TC(X) > nil(Z) where Z is the kernel of zero divisors. Example 3.5. Topological complexity of spheres is given by: (1) TC(S 0 ) = ∞. (2) TC(S n ) = 2 for n odd. (3) TC(S n ) = 3 for n > 0 even. Theorem 3.6. For the trivial F -bundle B × F → B, we have that TC(B × F ) ≤ TC(B) + TC(F ) − 1. 4. Equivariant versions of TC We review in this section the different approaches to define a topological complexity in the context of group actions. From now on, X will be always a G-space. All four versions reduce to the ordinary (nonequivariant) TC(X) when the action of G on X is trivial. Moreover, all four versions are invariant under G-homotopy type. 4.1. Equivariant topological complexity (Colman-Grant). The evaluation map ev : X I → X × X is a G-fibration with respect to the actions G × XI → XI, g(γ)(t) = g(γ(t)),

G × (X × X) → X × X, g(x, y) = (gx, gy).

Definition 4.1. The equivariant topological complexity of X, TCG (X), is the least integer k such that X × X may be covered by k G-invariant open sets {U1 , . . . , Uk }, on each of which there is a G-equivariant map si : Ui → X I such that the diagram commutes: XI w; w si ww ev ww w  ww / X ×X Ui In other words, the equivariant topological complexity of a G-space X is the minimum number of G-invariant open sets needed to cover X ×X, on each of which the free path fibration ev admits a local G-equivariant section. If no such integer exists then we set TCG (X) = ∞.

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Proposition 4.2. Let X and Y be G-spaces. (1) For a G-connected space X with X G = ∅, TCG (X) = 1 iff X is Gcontractible. (2) TCG (X) = ∞ if X is not G-connected. Moreover there are inequalities relating TCG (X) to the equivariant and nonequivariant topological complexities of the various fixed point sets. Proposition 4.3. If H is a closed subgroup of G, then (1) TC(X H ) ≤ TCG (X). (2) TCH (X) ≤ TCG (X), in particular TC(X) ≤ TCG (X). The next results describe the basic relationship of equivariant topological complexity with equivariant Lusternik-Schnirelmann category. Proposition 4.4. Let X be a G-connected space. (1) TCG (X) ≤ catG (X × X). (2) If X G = ∅, then catG (X) ≤ TCG (X). The assumptions that X is G-connected and X G = ∅ are enough to have that catG (X × X) ≤ 2 catG (X) − 1 and it follows that TCG (X) ≤ 2 catG (X) − 1. s Since an invariant open subset U ⊆ X ×X has a G-equivariant section U → X I of ev : X I → X × X if and only if the inclusion iU : U → X × X is G-homotopic to a map with values in the diagonal Δ(X) ⊆ X ×X, we can reformulate the definition of equivariant topological complexity in terms of equivariant deformations to the diagonal. See lemma 3.5 in [13] for the following: Theorem 4.5. For a G-space X, the following statements are equivalent: (1) TCG (X) ≤ n. (2) secatG (ev) ≤ n: there exist G-invariant open sets U1 , . . . , Uk which cover X × X and G-equivariant sections si : Ui → X I such that ev ◦ si is G-homotopic to Ui → X × X. (3) Δ(X) catG (X ×X) ≤ n: there exist G-invariant open sets U1 , . . . , Uk which cover X × X which are G-compressible into Δ(X). We have the following cohomological lower bound for TCG (X), using equivari∗ ant cohomology theory. Denote by HG (X) the Borel G-equivariant cohomology of X, with coefficients in an arbitrary commutative ring. ∗ Proposition 4.6. Let Z be the kernel of the homomorphism HG (X × X) → ∗ (X) induced by the diagonal, then TCG (X) > nil(Z). HG

Example 4.7. (1) For the free S 1 -action on itself by rotations, TCS 1 (S 1 ) = 2 while TC(S 1 /S 1 ) = TC(∗) = 1, see example 5.10 in [7]. (2) For the antipodal action of Z2 on S n , we have  3 if n is even n TCZ2 (S ) = 2 if n is odd while TC(S n /Z2 ) = TC(RPn ) ≥ n + 1. See lemma 4.1 in [12] where equivariant vector fields are used to construct the G-equivariant motion planners. In general TC(RPn ) for n = 1, 3, 7 is equal to the smallest k such that there is an immersion of RPn in Rk−1 , see [11].

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(3) For the reflection action of Z2 on S n with n = 0, TCZ2 (S n ) ≤ catZ2 ×Z2 (S n × S n ) ≤ 2 catZ2 (S n ) − 1 = 2 × 2 − 1 = 3. • For n = 1 odd, TCZ2 (S n ) = 3, because (S n )Z2 = S n−1 and 3 = TC(S n−1 ) ≤ TCZ2 (S n ) by proposition 4.3. • For n even, TCZ2 (S n ) = 3, because 3 = TC(S n ) and TC(S n ) ≤ TCZ2 (S n ) by proposition 4.3. We have therefore shown that  ∞ if n = 1 n TCZ2 (S ) = 3 if n ≥ 2. while TC(S n /Z2 ) = TC(Dn ) = 1. (4) Let X = S 1 \{N, S} where N = (0, 1) and S = (0, −1). For the (Z2 ×Z2 )action on X given by the reflection across the x-axis and the y-axis, we have TCZ2 ×Z2 (X) = ∞. The action has no global fixed points and the space X is (Z2 × Z2 )-compressible into an orbit that is disconnected, see example 2.10 in [4]. From the previous examples we see that if G acts freely on X, not necessarily TCG (X) = TC(X/G) and there are no inequalities of the type TCG (X) ≤ TC(X/G) or TC(X/G) ≤ TCG (X) for general actions. Examples show that TCG (X) can be equal to TC(X), or at the other extreme, one can be finite and the other infinite as shows the case in which X is a G-manifold which is connected but not G-connected. For a group acting on itself by left translations, we have that TCG (G) = cat(G), so that category of topological groups is obtained as a special case of equivariant topological complexity. Theorem 4.8. Let G be a connected metrizable group acting on itself by left translation. Then TCG (G) = cat(G). Note that this shows that even for spaces that are G-compressible to an orbit the equivariant topological complexity can be arbitrarily high. Proposition 4.9. Let X be a G-connected topological group. Assume that G acts on X by topological group homomorphisms. Then TCG (X) = catG (X). The next result relates equivariant and nonequivariant topological complexity for G-bundles. Theorem 4.10. Let E → B be a numerable principal G-bundle and X a Gspace, then TC(XG ) ≤ TCG (X)TC(B) where XG = E ×G X is the total space of the associated bundle over B. Theorem 4.11. [13] (1) Let X be a H-space and Y a K-space. Consider X ×Y as a (H ×K)-space. Then TCH×K (X × Y ) ≤ TCH (X) + TCK (Y ) − 1. (2) From the previous, for G-spaces X and Y and X × Y equipped with the diagonal action, we have that TCG (X × Y ) ≤ TCG (X) + TCG (Y ) − 1. See also theorem 4.1 in [12] for the special case of G-manifolds.

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4.2. Invariant topologicalcomplexity (Lubawski-Marzantowicz). Con = Gβ(0) . The map sider the space X I ×X/G X I = (α, β) ∈ X I ×X I : Gα(1)  π : X I ×X/G X I → X × X given by π(α, β) = α(0), β(1) is a (G × G)-fibration with respect to the obvious actions. Definition 4.12. The invariant topological complexity of X, TCG (X), is the least integer k such that X × X may be covered by k (G × G)-invariant open sets {U1 , . . . , Uk }, on each of which there is a (G × G)-equivariant section si : Ui → X I ×X/G X I such that the diagram commutes: X I ×X/G X I s9 si sss π s s ss s  s / X ×X Ui Proposition 4.13. Let X and Y be G-spaces. (1) If X G = ∅, then X is G-compressible to a global fixed point if and only if TCG (X) = 1. See corollary 2.8 in [4]. (2) TCG (X) can be finite even when X is not G-connected. By restricting a (G × G)-equivariant deformation c : U × I → X × X to the (G × G)-fixed point set, we have Proposition 4.14. TC(X G ) ≤ TCG (X). Note that in general there are no inequalities of the type TC(X H ) ≤ TCG (X) for other subgroups H ≤ G as item (5) of example 4.18 below shows. Next we show the relationship of invariant topological complexity with equivariant Lusternik-Schnirelmann category. Proposition 4.15. Let x ∈ X and Gx be the orbit of x. If X is a G-connected space then TCG (X) ≤Gx×Gx catG×G (X × X). If x ∈ X G and X is G-connected then Gx×Gx

catG×G (X × X) =x×x catG×G (X × X) = catG×G (X × X)

and therefore TCG (X) ≤ catG×G (X × X). Proposition 4.16. [4] If X G = ∅, then catG (X) ≤ TCG (X). Let (X) be the saturation of the diagonal Δ(X) with respect to the (G × G)action. Theorem 4.17. For a G-space X the following are equivalent: (1) TCG (X) ≤ n. (2) secatG×G (π) ≤ n: there exist (G×G)-invariant open sets U1 , . . . , Uk which cover X × X and (G × G)-equivariant sections si : Ui → X I ×X/G X I such that ev ◦ si is (G × G)-homotopic to the inclusion Ui → X × X. (3) (X) catG×G (X×X) ≤ n: there exist (G×G)-invariant open sets U1 , . . . , Uk which cover X × X which are (G × G)-compressible into (X).

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Example 4.18. 1 (1) TCS (S 1 ) = 1 for the free S 1 -action on itself by rotations but TC(S 1 ) = 2. (2) For the antipodal action of Z2 on S n , we have TCZ2 (S n ) = TC(RPn ). (3) For the reflection action of Z2 on S n with n = 1, we have TCZ2 (S n ) ≤ catZ2 ×Z2 (S n × S n ) ≤ 2 catZ2 (S n ) − 1 = 2 × 2 − 1 = 3. If n is odd, TCZ2 (S n ) = 3, since we have that (S n )Z2 = S n−1 and 3 = TC(S n−1 ) ≤ TCZ2 (S n ) by proposition 4.14. (4) TCZ2 (S 1 ) = ∞ for the reflection action of Z2 on S 1 . (5) Let X = S 1 \ {N, S} where N = (0, 1) and S = (0, −1). The (Z2 × Z2 )action on X given by the reflection across the x-axis and the y-axis has no global fixed points. The space X is G-compressible to an orbit that is disconnected. Therefore TCZ2 ×Z2 (X) = 1 but for one of the copies of Z2 inside Z2 × Z2 we have TC(X Z2 ) = TC(S 0 ) = ∞. See example 2.10 in [4]. The first two and last examples show that in general there are no inequalities of the form TC(X) ≤ TCG (X) or TCG (X) ≤ TC(X). Also the last example shows that there are no inequalites of the form TC(X H ) ≤ TCG (X) for H ≤ G. The case of TCZ2 (S n ) for n even is still open. This case is treated in example 4.2 in [13] but there is a mistake that is explained in remark 3.7 in [4], it does not follow like in item (3) of example 4.7 because in general we do not have TC(X) ≤ TCG (X). Since the quotient space of X × X by the (G × G)-action is X/G × X/G, we have that (G × G)-invariant open sets in X × X give open sets in X/G × X/G. Moreover (G × G)-equivariant local sections of X I ×X/G X I give local sections of (X/G)I and therefore, Proposition 4.19. TC(X/G) ≤ TCG (X). There is no analogue of this inequality for equivariant topological complexity as the examples in the previous section show. For a free action, we can lift a deformation c : U × I → X/G × X/G to the diagonal Δ(X/G), to an equivariant one, by using the Palais covering homotopy theorem. Since π −1 (Δ(X/G)) = (X) under the projection π : X × X → X/G × X/G, we have that the lift is a (G × G)-equivariant deformation to (X). This proves TCG (X) ≤ TC(X/G) and gives the following fundamental result. Proposition 4.20. If G is a compact Lie group that acts freely on a metrizable space X then TCG (X) = TC(X/G). More generally, the result holds if X has one orbit type. In general it is not known if for a general compact Lie group (X) ⊂ X × X is a (G × G)-cofibration, it is only known for G finite group and X a compact G-ANR [13]. This is relevant because of the following product formula. Proposition 4.21. Let X be an H-space and Y be a K-space. If the inclusion (X) ⊂ X ×X is a (H ×H)-cofibration and (Y ) ⊂ Y ×Y is a (K ×K)-cofibration, then TCH×K (X × Y ) ≤ TCH (X) + TCK (Y ) − 1.

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From the previous, for G-spaces X and Y we do not obtain the inequality TCG (X × Y ) ≤ TCG (X) + TCG (Y ) − 1 1

as shown by the example of the free S 1 -action on S 1 satisfying TCS (S 1 ) = 1 and 1 TCS (S 1 × S 1 ) = 2. See remark 3.20 in [13]. 4.3. Strongly equivariant topological complexity (Dranishnikov). Let G×G act on X×X where G acts in each component. Strongly equivariant topological complexity of X, TC∗G (X), is defined like the equivariant topological complexity, only that X × X is now viewed as a (G × G)-space and the open cover is required to be (G × G)-invariant. Definition 4.22. The strongly equivariant topological complexity of X, TC∗G (X), is the least integer k such that X × X may be covered by k (G × G)invariant open sets U1 , . . . , Uk , on each of which there is a G-equivariant section si : Ui →X I for the diagonal action on Ui , such that the diagram commutes: XI w; w si ww ev ww w  ww / X ×X Ui Proposition 4.23. Let X and Y be G-spaces. (1) For a G-connected space X with X G = ∅, we have that TC∗G (X) = 1 iff X is G-contractible. (2) TC∗G (X) = ∞ if X is not G-connected. It is obvious that TCG (X) ≤ TC∗G (X) and therefore using the properties of equivariant topological complexity we have the following. Proposition 4.24. If H is a closed subgroup of G, then (1) TC(X H ) ≤ TC∗G (X). (2) TCH (X) ≤ TC∗G (X), in particular TC(X) ≤ TC∗G (X). Proposition 4.25. For a G-connected space with X G = ∅ we have that TC∗G (X) ≤ catG×G (X × X). Since TCG (X) ≤ TC∗G (X) and catG (X) ≤ TCG (X) for a G-connected space X with X G = ∅, we have the following, Proposition 4.26. For a G-connected space with X G = ∅ catG (X) ≤ TC∗G (X). We can characterize strongly equivariant topological complexity in terms of relative sectional category and relative Clapp-Puppe category. Theorem 4.27. For a G-space X the following are equivalent: (1) TC∗G (X) ≤ n. (2) There exist (G × G)-invariant open sets U1 , . . . , Uk which cover X × X and G-equivariant sections si : Ui → X I such that ev ◦ si is G-homotopic to Ui → X × X. (3) There exist (G × G)-invariant open sets U1 , . . . , Uk which cover X × X which are G-compressible into Δ(X).

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Example 4.28. (1) For a Lie group acting on itself by translation, we have that TC∗G (G) ≥ TCG (G) = cat(G) by proposition 4.8 but TCG (G) = 1. (2) For the reflection action of Z2 on S n , similar arguments as in example 4.7, prove that  ∞ if n = 1 TC∗Z2 (S n ) = 3 if n ≥ 2. In case that the group is discrete and the action is free we can describe (X) as (G × G)-space. By freeness (X) ∼ = G × Δ(X) and the action is given by (h, k) · (g, (x, x)) = (hgk−1 , (kx, kx)). Now restricting to the diagonal subgroup, we have a G-action given by (hgh−1 , (hx, hx)). The quotient (X)/G is a disjoint union of copies of the diagonal of X/G indexed by the conjugacy classes of G, this is a subspace of X ×G X. Since the action is free, there is a covering space X ×G X → X/G × X/G. If X is simply connected, the fundamental group of X ×G X is G. Given a deformation H : U × I → X/G × X/G from the inclusion of an open set to a map c : U → X/G × X/G with c(U ) ⊆ Δ(X/G), the lifting property of covering spaces gives maps H : U × I → X ×G X covering H. Since (X)/G is a disjoint union of copies of the diagonal of X/G , the lift H(x, t) can be chosen so H(x, 1) ⊆ Δ(X/G) ⊆ X ×G X. By using the homotopy lifting property of covering spaces, there is a G-equivariant deformation H : U ×I → X × X that covers H with the property that H(x, 1) ⊆ Δ(X). The open set U is π −1 (U ) where π : X × X → X/G × X/G and therefore is (G × G)-invariant. Proposition 4.29. If a discrete group G acts freely on the simply connected space X then TC∗G (X) ≤ TC(X/G). The examples above show that this is not true in general. Theorem 4.30. Let p : X → B be a F -bundle between locally compact metric ANR-spaces with the structure group G acting properly on F . Then TC(X) ≤ TC(B) + TC∗G (F ) − 1. Proposition 4.31. Suppose that a discrete group G acts freely and properly on a simply connected locally compact ANR-space Y . Then TC∗G (Y ) ≤ dim Y + 1. By using the two previous results, Theorem 4.32. Let X be a CW-complex with fundamental group G. Then TC(X) ≤ TC(G) + dim X.

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4.4. Effective topological complexity (Blaszczyk-Kaluba). Definition 4.33. The effective topological complexity of X, TCG,∞ (X), is the minimum of the numbers TCG,n defined as the least integer k such that X ×X may be covered by k open sets U1 , . . . , Uk , on each of which there is a (nonequivariant) section si : Ui →X I ×X/G X I ×X/G · · · ×X/G X I =: Pn (X) such that the diagram commutes: Pn (X) x; si xx x πn x xx  xx / X ×X Ui Proposition 4.34. Let X and Y be G-spaces. (1) TCG,n+1 (X) ≤ TCG,n (X). (2) TCG,n (X) = 1 if X is contractible or G-contractible, but the converse is not true. (3) TCG,n (X) = ∞ if X/G is not path connected. We have the following cohomological lower bound using non-equivariant cohomology theory of the quotient space. Proposition 4.35. Let Z be the kernel of zero divisors in the cohomology of X/G with rational coefficients. If G is finite then TCG,n (X) > nil(Z) for n ≥ 2. When the action on the cohomology is trivial, the previous result gives a very useful method to calculate TCG,n (X). Proposition 4.36. Let G be a finite group acting on X such that the G-action on the rational cohomology is trivial. If TC(X) = nil(Z) + 1, then for n ≥ 2 TC(X) = TCG,n (X). These cohomological bounds do not work with arbitrary coefficients as the last example below shows. Example 4.37. (1) Let Zp act on a sphere S n with p a prime number, p > 2. By using the previous result we have,  3 if n is even TCZp ,∞ (S n ) = 2 if n is odd. (2) If Z2 acts on a sphere S n preserving the orientation, the previous result holds. (3) If Z2 acts freely on S n , then TCZ2 ,∞ (S n ) = 2, while TC(S n /Z2 ) = TC(RPn ) ≥ n + 1. (4) For the reflection action of Z2 on S n , TCZ2 ,∞ (S n ) = 1. (5) In [4] there are examples of Zp -actions (p > 2 prime) on spheres S n (n ≥ 5) with fixed point set (S n )Zp homology spheres that are essential manifolds of dimension n − 2. The category of a (n − 2)-dimensional  closed essential Z n manifold is n − 1 by [3] and therefore catZp (S ) ≥ cat (S n ) p ≥ n − 1 while TCZp ,∞ (S n ) ≤ 3 as in the first example.

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(6) For a finite group G, consider the total space EG of the universal Gbundle with its G-action. Since the space is contractible, we have that TCG,∞ (EG) ≤ TC(EG) = 1, but catG (EG) = cat(BG) = ∞. From the previous examples we see that if G acts freely on X, not necessarily TCG,∞ (X) = TC(X/G) and that the equivariant Lusternik-Schnirelmann category cannot be a lower bound. See [5] for a nice discussion. Since TCG,1 (X) = TC(X) and TCG,2 (X) ≤ TCG (X), we have that TCG,∞ (X) ≤ min{TC(X), TCG (X)}. Together with the fact that for free actions TCG (X) = TC(X/G), we have Proposition 4.38. For a free action of a group G on X TCG,∞ (X) ≤ TC(X/G). Proposition 4.39. If H is a subgroup of G, then TCG,n (X) ≤ TCH,n (X). Proposition 4.40. Let X be a H-space and Y a K-space. Then TCH×K,n (X × Y ) ≤ TCH,n (X) + TCK,n (Y ) − 1. From the previous, for G-spaces X and Y we do not obtain the inequality TCG,n (X × Y ) ≤ TCG,n (X) + TCG,n (Y ) − 1 as shown by the example of Z2 acting on S 1 by a reflection. From the examples above, we know that TCZ2 ,∞ (S 1 ) = 1 but under the diagonal action S 1 × S 1 /Z2 is homeomorphic to S 2 and therefore by the cohomological bound stated above TCZ2 ,∞ (S 1 × S 1 ) > 1, see section 6 in [5]. 5. Comments We believe that all the invariants presented in this survey contribute to the study of classical topological complexity and also provide, in different degrees, a valid invariant to further investigate motion planning problems with symmetries. Much of ordinary topological complexity theory can be adapted and extended to the setting of spaces with an action of a group and each of these notions takes a different approach to that end. If we view a G-space as being described by the diagram of its fixed points X H for the various subgroups H of G, the equivariant and invariant topological complexities provide a way to import this point of view to the motion planning problems with symmetries. From another point of view, we believe that the topological complexity of a G-space should reflect in a certain way the quotient object defined by the action. In particular, in case that the action is free, the invariant topological complexity of a G-space X coincides exactly with the topological complexity of the orbit space X/G. As a future direction, we envision a notion of topological complexity that recreates a version of this property even in the case when the action is not free. The most suitable object to substitute the orbit space in this case is what is called a topological orbifold.

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There has been much recent interest in the study of orbifolds, which can locally be described as the quotient of an open subset of Euclidean space by the action of a finite group. In this setting, different group actions may define the same orbifold. Specifically, representable orbifolds are given by a Morita equivalence class of group actions. If the action is free, it is Morita equivalent to the trivial action on its quotient space. We think that a notion of topological complexity for a G-space should display the property that the invariant topological complexity exhibits for free actions but in greater generality. Namely, if two actions are Morita equivalent, they should have the same topological complexity. Since the representation of an orbifold by a group action is not unique, we need a topological complexity that is a true invariant of the orbifold structure and not of the particular representation, i.e. it needs to be checked that we get the same topological complexity for all Morita equivalent actions. We do not know if the invariant topological complexity satisfies this property, but we believe it is the notion best equipped to be developed to be the topological complexity of the actual object defined by the orbits of the action.

References [1] [2] [3]

[4] [5] [6]

[7] [8] [9] [10] [11]

[12]

[13] [14]

[15] [16]

´ Andr´ es Angel and Hellen Colman, Groupoid topological complexity, talk at the XXIst Oporto Meeting on Geometry, Topology and Physics (2015). Marzieh Bayeh and Soumen Sarkar, Some aspects of equivariant LS-category. part A, Topology Appl. 196 (2015), no. part A, 133–154, DOI 10.1016/j.topol.2015.09.006. MR3422738 Israel Berstein, On the Lusternik-Schnirelmann category of Grassmannians, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 1, 129–134, DOI 10.1017/S0305004100052142. MR0400212 Zbigniew Blaszczyk and Marek Kaluba, On equivariant and invariant topological complexity of smooth Zp -spheres, to appear in Proc. Amer. Math. Soc. , Effective topological complexity of spaces with symmetries, to appear in Publ. Mat. M´ onica Clapp and Dieter Puppe, Invariants of the Lusternik-Schnirelmann type and the topology of critical sets, Trans. Amer. Math. Soc. 298 (1986), no. 2, 603–620, DOI 10.2307/2000638. MR860382 Hellen Colman and Mark Grant, Equivariant topological complexity, Algebr. Geom. Topol. 12 (2012), no. 4, 2299–2316, DOI 10.2140/agt.2012.12.2299. MR3020208 Alexander Dranishnikov, On topological complexity of twisted products, Topology Appl. 179 (2015), 74–80, DOI 10.1016/j.topol.2014.08.017. MR3270927 Michael Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211–221, DOI 10.1007/s00454-002-0760-9. MR1957228 Michael Farber, Instabilities of robot motion, Topology Appl. 140 (2004), no. 2-3, 245–266, DOI 10.1016/j.topol.2003.07.011. MR2074919 Michael Farber, Serge Tabachnikov, and Sergey Yuzvinsky, Topological robotics: motion planning in projective spaces, Int. Math. Res. Not. 34 (2003), 1853–1870, DOI 10.1155/S1073792803210035. MR1988783 Jes´ us Gonz´ alez, Mark Grant, Enrique Torres-Giese, and Miguel Xicot´ encatl, Topological complexity of motion planning in projective product spaces, Algebr. Geom. Topol. 13 (2013), no. 2, 1027–1047, DOI 10.2140/agt.2013.13.1027. MR3044600 Wojciech Lubawski and Waclaw Marzantowicz, Invariant topological complexity, Bull. Lond. Math. Soc. 47 (2015), no. 1, 101–117, DOI 10.1112/blms/bdu090. MR3312969 Waclaw Marzantowicz, A G-Lusternik-Schnirelman category of space with an action of a compact Lie group, Topology 28 (1989), no. 4, 403–412, DOI 10.1016/0040-9383(89)90002-5. MR1030984 Richard S. Palais, The classification of G-spaces, Mem. Amer. Math. Soc. No. 36, 1960. MR0177401 Albert S. Schwarz, The genus of a fiber space, Amer. Math. Soc. Transl. 55 (1966), 49–140.

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´ticas, Universidad de los Andes, Carrera 1 N. 18A - 10 Departamento de Matema ´, Colombia Bogota Email address: [email protected] Department of Mathematics, Wright College, 4300 N. Narragansett Avenue, Chicago, Illinois 60634 Email address: [email protected]

Contemporary Mathematics Volume 702, 2018 http://dx.doi.org/10.1090/conm/702/14108

Rational methods applied to sectional category and topological complexity Jos´e Carrasquel Abstract. This survey is a guide for the non specialist on how to use rational homotopy theory techniques to get approximations of Farber’s topological complexity, in particular, and of Schwarz’s sectional category, in general.

Contents Introduction 1. Sullivan’s rational homotopy theory 1.1. Sullivan models 1.2. The connection with topology 1.3. Models for homotopy pullbacks 1.4. Models for fibrations 1.5. Models for homotopy pushouts 1.6. Models for cofibrations 1.7. Models for the base point inclusion 1.8. Models for the diagonal map 2. Rational Lusternik-Schnirelmann category 2.1. The mapping theorem for LS category 3. The Whitehead and Ganea characterizations 3.1. The Whitehead characterization 3.2. The Ganea characterization 3.3. Whitehead vs. Ganea 3.4. First algebraic characterizations 4. Rational approximations of sectional category 4.1. Module sectional category 4.2. Poincar´e Duality 5. Characterization `a la F´elix-Halperin 5.1. When f admits a homotopy retraction 2010 Mathematics Subject Classification. Primary 55M30, 55P62. Key words and phrases. Rational homotopy, topological complexity, Lusternik-Schnirelmann category, sectional category. The author was supported in part by the Belgian Interuniversity Attraction Pole (IAP) within the framework “Dynamics, Geometry and Statistical Physics” (DYGEST P7/18) and the Polish National Science Centre grant 2016/21/P/ST1/03460 within the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 665778. ©2018 American Mathematical Society

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5.2. Applications to topological complexity 6. A mapping theorem for topological complexity Acknowledgements References

Introduction The sectional category [Sch66] of a continuous map f ∶ X → Y is the least integer m for which there are m + 1 local homotopy sections for f whose domains form an open cover of Y . If X is a path-connected topological space then two important invariants of the homotopy type of X can be described through sectional category. The first one is the Lusternik-Schnirelmann (LS) category of X [LS34], cat(X), which is the sectional category of the base point inclusion cat(X) = secat(∗ ↪ X). The second one is the (higher) topological complexity [Far03,Rud10] of X, TCn (X), which is the sectional category of the n diagonal inclusion TCn (X) = secat(X ↪ X n ). In this paper we will only consider, unless stated otherwise, simply connected CW complexes of finite type. As a consequence, every time we write the word space, we actually mean one of such CW complexes. If X is a space, then there exists a cofibration ρX ∶ X ↪ X0 , called the rationalization of X, which verifies: ● X0 is a rational space, that is, H∗ (X, Z) (or equivalently π∗ (X)) is a rational vector space. ● ρX is a weak rational homotopy equivalence, that is, H∗ (ρX , Q) (or equivalently π∗ (ρX ) ⊗ Q) is an isomorphism. ● Every continuous map f ∶ X → Y , with Y a rational space, factors uniquely up to homotopy: X0 f0

ρX

X

f

Y

The main lower bound for sectional category is cohomological. Namely, if f is a continuous map and R is any ring, then nil ker H∗ (f, R) ≤ secat(f ), where nil I denotes the nilpotency of an ideal I, that is, the longest non trivial product of elements in I. Often, this bound is not good enough. This is mainly because cohomology does not capture all the homotopic information of a map. However, rational homotopy theory offers a whole new set of algebraic lower bounds which are based on the following inequalities: nil ker H∗ (f, Q) ≤ secat(f0 ) ≤ secat(f ).

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More precisely, there is a pair of contravariant adjoint functors APL ∶ Top

cdga ∶ ∣ ⋅ ∣

where ● cdga is the category of simply connected commutative differential graded algebras over Q of finite type (see Section 1). ● Top is the category of simply connected CW complexes of finite type. When restricted to rational spaces, these functors actually yield an equivalence in the associated homotopy categories [BG76, Sul77]. This means that all the rational homotopic information of a continuous map f can be encoded algebraically through APL (f ). From this algebraic object, one can deduce approximations for sectional category which are better than the cohomological lower bound, nil ker H ∗ (f, Q) ≤ Hsecat(f ) ≤ msecat(f ) ≤ secat(f0 ) ≤ secat(f ), by relaxing the algebraic characterization of secat(f0 ). 1. Sullivan’s rational homotopy theory For a deep description of the tools we use, the reader is invited to read the standard reference on rational homotopy theory [FHT01]. We start describing the algebraic objects we use. A commutative graded algebra (cga) is a graded Q-vector space A = ⊕ Ai i≥0

together with a degree zero bilinear map μ∶ A ⊗ A → A, a ⊗ b ↦ ab, called multiplication, verifying for a, b, c ∈ A, ● A0 = Q and a1 = 1a = a, ● (ab)c = a(bc), and ● ab = (−1)∣a∣∣b∣ ba. where ∣a∣ = i means that a ∈ Ai and we say that a is homogeneous of degree i. Observe that, if a has odd degree, then a2 = 0. A morphism of cga f ∶ A → B is a linear map of degree zero such that f (1) = 1, f (ab) = f (a)f (b). A derivation of degree k on A is a degree k linear map θ∶ A → A verifying θ(ab) = θ(a)b + (−1)k∣a∣ aθ(b). A differential on A is a derivation d of degree 1 such that d2 = d ○ d = 0. A commutative differential graded algebra (cdga) is a pair (A, d) where A is a cga and d is a differential on A. A morphism of cdga is a morphism of cga commuting with differentials. We denote by cdga this category. There exists a homology ker d , 0) and H(f )([z]) = [f (z)]. A functor H∶ cdga → cdga defined as H(A, d) ∶= ( d(A) morphism of cdga f is said to be a quasi-isomorphism if H(f ) is an isomorphism. The direct sum of two cdgas is a cdga, (A, d) ⊕ (B, d) ∶= (A ⊕Q B, d), with d(a + b) = d(a) + d(b) and ab = 0 for a ∈ A and b ∈ B. The tensor product of two cdgas is a cdga, (A, d) ⊗ (B, d) ∶= (A ⊗Q B, d), with ′ (a ⊗ b)(a′ ⊗ b′ ) ∶= (−1)∣b∣∣a ∣ (aa′ ) ⊗ (bb′ ) and d(a ⊗ b) ∶= d(a) ⊗ b + (−1)∣a∣ a ⊗ d(b).

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Given a graded Q-vector space V , define T V = ⊕j≥0 T j V where T 0 V = Q and T j V = V ⊗j . We have that T V is a graded algebra with product (a1 ⊗ ⋯ ⊗ aj )(b1 ⊗ ⋯ ⊗ bk ) ∶= a1 ⊗ ⋯ ⊗ aj ⊗ b1 ⊗ ⋯ ⊗ bk . Consider I the ideal of T V generated by elements of the form a ⊗ b − (−1)∣a∣∣b∣ b ⊗ a and define the free commutative graded algebra generated by V : TV . ΛV ∶= I We will write simply ab to denote the element [a ⊗ b] ∈ ΛV . We say that an element v1 ⋯vj ∈ Λj V has word length j and degree ∣v1 ∣ + ⋯ + ∣vj ∣. We will denote by A+ the elements of positive degree of a graded algebra A, and by Λ+ V the elements of positive word length of a free graded algebra. We will also employ notations such as Λ>m V to mean ⊕j>m Λj V , and so on. Observe that there is an isomorphism ΛV ⊗ ΛW ≅ Λ(V ⊕ W ) which we often will use implicitly. The free commutative graded algebra ΛV has the following two universal properties: ● Given A a cga and f ∶ V → A a linear map of degree zero, then there exists an unique cga morphism fˆ∶ ΛV → A verifying fˆ(v) = f (v) for all v ∈ V . ● If θ∶ V → ΛV is a linear map of degree k then there exists a unique derivaˆ such that θ(v) = θ(v) ˆ tion on ΛV of degree k, θ, for all v ∈ V . Let (A, d) be a cdga. A relative Sullivan algebra over (A, d) is a cdga of the form (A ⊗ ΛV, D) such that D(a) = d(a), if a ∈ A, and V = ⊕k≥1 V (k) with D(V (1)) ⊂ A ⊗ 1 and D(V (k)) ⊂ A ⊗ Λ (V (1) ⊕ ⋯ ⊕ V (k − 1)) (nilpotence condition). If, in addition, the differential D satisfies D(V ) ⊂ A+ ⊗ ΛV + A ⊗ Λ≥2 V we say that (A ⊗ ΛV, D) is a minimal relative Sullivan algebra. Observe that the inclusion i∶ (A, d) → (A ⊗ ΛV, D) is a cdga morphism. In fact, the category cdga is a closed model category [Qui67] (if we restrict to cdgas (A, d) for which H0 (A, d) = Q [Hal83]) with the following structure: ≃

● Weak equivalences: quasi-isomorphisms (A, d) → (B, d). ● Fibrations: surjective morphisms (A, d) ↠ (B, d). ● Cofibrations: Inclusions of a cdga (A, d) into a relative Sullivan algebra (A, d) ↪ (A ⊗ ΛV, D). As a consequence we get the following important facts: Existence of relative Sullivan models: Every cdga morphism ϕ can be factored as ϕ

(A, d)

(B, d) ≃

i

θ

(A ⊗ ΛV, D), where either θ is surjective or (A ⊗ ΛV, D) is minimal (but not necessarily both at the same time).

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An (A, d)-module homotopy retraction for ϕ is a chain map r∶ (A ⊗ ΛV, D) → (A, d) such that r(a) = a and r(aξ) = ar(ξ) for all a ∈ A and ξ ∈ A ⊗ ΛV . Moreover, if r is a cdga morphism, we say that it is a (cdga) homotopy retraction for ϕ. The surjective trick: Every cdga morphism ϕ can be factored as ϕ

(A, d)

(B, d)

≃

(A ⊗ ΛV, D), The lifting lemma: For any solid commutative cdga diagram, there exists a dashed arrow completing the commutative cdga diagram (A, d)

(B, d)

(A ⊗ ΛV, D)

(C, d),

provided that at least one of the vertical morphisms is a quasi-isomorphism. Since cdga is a closed model category, there exists a notion of homotopy of maps (ΛV, d) → (B, d) which is nicely described in [FHT01, Chap. 12-14]. Two cdga morphism ϕ1 , ϕ2 are weakly equivalent when there is a homotopy commutative cdga diagram (A1 , d)

≃

(ΛV, d)

≃

(ΛW, d)

≃

ϕ1

(B1 , d)

(A2 , d) ϕ2

≃

(B2 , d),

where (ΛV, d) and (ΛW, d) are Sullivan algebras (see below). 1.1. Sullivan models. In the special case that (A, d) = (Q, 0), the initial object of cdga, we get Sullivan (perhaps minimal ) models for a cdga (B, d), ≃ θ∶ (ΛV, d) → (B, d). These objects are very important as they are the fibrantcofibrant objects in the category cdga. For us, unless stated otherwise, cdgas denoted in the form (ΛV, d) will be assumed to be Sullivan algebras, the same applies to relative Sullivan algebras (A ⊗ ΛV, D). To be more explicit, a Sullivan algebra is a cdga of the form (ΛV, d) verifying the nilpotence condition, that is, V = ⊕k≥1 V (k) with D(V (1)) = 0 and D(V (k)) ⊂ Λ (V (1) ⊕ ⋯ ⊕ V (k − 1)) . If moreover, the differential D verifies D(V ) ⊂ Λ≥2 V , we say that it is a minimal Sullivan algebra. As with relative Sullivan models for cdga morphism, the ≃ construction of Sullivan models for a given cdga, θ∶ (ΛV, d) → (B, d), can be carried out inductively, degree by degree, by adding generators to V and defining their differentials to kill or create the necessary homology classes in order to turn θ into a quasi-isomorphism. This process is very well explained in [FHT01, Pg. 144] when H1 (B, d) = 0.

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Let (ΛV, d) be a Sullivan algebra and write d = d0 + d1 + ⋯ with dk (V ) ⊂ Λk+1 V . The equation d2 = 0 implies that ∑ki=0 di dk−i = 0 for all k ≥ 0. Observe then that d0 is a differential on V , called the linear part of d, which gives a chain complex (V, d0 ). Analogously, given a cdga morphism ϕ∶ (ΛV, d) → (ΛW, d) define the linear part of ϕ as the chain complex morphism Q(ϕ)∶ (V, d0 ) → (W, d0 ) defined by (ϕ − Q(ϕ))(V ) ⊂ Λ≥2 W . We can now state some important facts about Sullivan algebras which will be used throughout this survey[FHT01, Prop. 14.13, Thm. 14.12]. Proposition 1.1. Let ϕ∶ (ΛV, d) → (ΛW, d) be a cdga morphism between Sullivan algebras. ● Then ϕ is a quasi-isomorphism if and only if H(Q(ϕ)) is an isomorphism. ● If (ΛV, d) and (ΛW, d) are minimal, then ϕ is a quasi-isomorphism if and only if ϕ is an isomorphism. 1.2. The connection with topology. Given a space X, we say that a cdga (A, d) is a model for X if APL (X) is weakly equivalent to (A, d), that is, if there is a Sullivan algebra (ΛV, d) and quasi-isomorphisms (A, d)

≃

(ΛV, d)

≃

APL (X).

In this case, we say that (ΛV, d) is a Sullivan model of X (and of (A, d)). These special cofibrant models are suitable for encoding the rational homotopic information of X. In fact, by Proposition 1.1, two minimal models for a space X are isomorphic. Furthermore, we have [FHT01, Cor. 10.10, Thm. 15.11] Theorem 1.2. If (ΛV, d) is the minimal Sullivan model for X, then ● H∗ (X, Q) ≅ H(APL (X)) ≅ H(ΛV, d), and ● for every i ≥ 1 there is a natural linear isomorphism V i → homZ (πi (X), Q). A cdga morphism ϕ is said to be a model for a continuous map f if ϕ is weakly equivalent to APL (f ). We will often use a special notation for describing Sullivan algebras which can be understood by the following example: (Λ(ai , bj , ck ); db = α, dc = γ) means (ΛV, d) with V spanned by a, b, c of degrees i, j, k respectively and with da = 0, db = α ∈ Λ(a) and dc = γ ∈ Λ(a, b). The universal properties of p. 20 show that the previous (ΛV, d) is well defined. Example 1.3. Let us compute the minimal Sullivan models for spheres. We ≃ have to construct a quasi isomorphism θ∶ (ΛV, d) → APL (S n ). Recall that H∗ (APL (S n )) = Q ⟨1, Ω⟩ with Ω the fundamental class of S n . So V must have one generator of degree n which is a cycle, say a. Take ω ∈ APL (S n ) a cycle in degree n representing Ω and define (Λ(a), 0) a

θ

APL (S n ) ω

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Here the dimension is crucial. If n is odd, then a2 = 0 and H(θ) is an isomorphism. Therefore (Λ(a), 0) is the minimal model of an odd sphere. However, if n is even, then H(Λ(a), 0) = Q⟨1, a, a2 , a3 , . . .⟩ and θ cannot be a quasi-isomorphism, this means that we need to add new generators in V that turn a2 , a3 , . . . into boundaries. So let x be a new generator of degree 2n − 1 and define d(x) = a2 . Then in (Λ(a, x), d) we have 1

a2

a d

x

a4 ⋯

a3 d

ax

a2 x

d

a3 x⋯

which shows that H(Λ(a, x), d) ≅ H(APL (S n )). However, we still have to define the quasi-isomorphism θ to make sure that (Λ(a, x), d) is the minimal model of an even sphere. Observe that, in H∗ (APL (S n )), Ω2 = 0, this means that there exists ξ ∈ APL (S n ) of degree 2n − 1 such that d(ξ) = ω 2 . We then define θ as (Λ(a, x), d)

θ

APL (S n )

a

ω

x

ξ.

Observe finally that d(θ(x)) = d(ξ) = ω 2 = θ(a)2 = θ(a2 ) = θ(d(x)). This proves that θ is a quasi-isomorphism showing that (Λ(a, x), d) is the minimal Sullivan model of an even sphere. Notice that, as corollary, we get the Serre finiteness theorem for the homotopy groups of spheres. In the previous example, there exists also a quasi-isomorphism ≃

(ΛV, d) → (H(ΛV, d), 0) defined as a ↦ [a] and x ↦ 0. When this happens we say that the space is formal. More precisely, a cdga (A, d) is said to be formal when (A, d) and (H(A), 0) are weakly equivalent. A space X is said to be formal when APL (X) is a formal cdga. Example 1.4. A cdga which is not formal. Consider (ΛV, d) = (Λ(a3 , b3 , x5 ), dx = ab) and observe that, as vector spaces, ΛV = Q⟨1, a, b, x, ab, ax, bx, abx⟩ and H(ΛV, d) = Q⟨1, [a], [b], [ax], [bx], [abx]⟩. For degree reasons, any cdga morphism ϕ∶ (ΛV, d) → (H(ΛV, d), 0) must satisfy ϕ(x) = 0, but then, we would have H(ϕ)([ax]) = [ϕ(ax)] = [ϕ(a)0] = 0, so ϕ cannot be a cdga quasi-isomorphism. This proves that (ΛV, d) is not the minimal model of the cdga (H(ΛV, d), 0) and therefore (ΛV, d) is not formal. A cdga morphism ϕ is said to be formal when it is weakly equivalent to H(ϕ). A continuous map f is said to be formal if APL (f ) is formal [Vig79,Opr86,FT88] or equivalently, when H∗ (f, Q) is a model for f . Obviously, if a map f ∶ X → Y is formal then both X and Y are formal spaces, however, the converse is not true as we will see in Example 1.5.

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1.3. Models for homotopy pullbacks. One of the most frequent ways in rational homotopy theory to construct models is through homotopy pullbacks and pushouts. Suppose that either f or g is a fibration in the following commutative diagram β

A (1.1)

X

α

f

Y

Z.

g

Then the universal property of pullbacks induces a diagram β

A (α,β)

(1.2)

g ˆ

T α

X

fˆ

f

Y

g

Z.

in which the square is a (homotopy) pullback[Mat76]. We now explain how to construct cdga models for the previous diagram. Suppose we have a commutative cdga square modeling Diagram 1.1 (applying APL for example), f

(A, d)

(B, d)

g

β

(C, d)

(D, d),

α

where the name of a cdga morhpism is the same as the name of the continuous maps it models (for instance, f ∶ (A, d) → (B, d) is a cdga model for f ∶ X → Z). We will often adopt this convention. Now choose f or g and factor it as a cofibration followed by a weak equivalence, say f = θ ○ i with i∶ (A, d) ↪ (A ⊗ ΛV, D) and ≃ θ∶ (A ⊗ ΛV, D) → (B, d). Since C ⊗A (A ⊗ ΛV ) ≅ C ⊗ ΛV , diagram 1.2 is modeled by (B, d) f

(A, d)

i

≃ θ

(A ⊗ ΛV, D)

g

(C, d)

β

g ˆ

fˆ

(C ⊗ ΛV, D) (α,β)

(D, d) α

in which the square is a pushout in cdga, gˆ(a) = g(a), g(v) = v, D(v) = gˆ(Dv) for a ∈ A and v ∈ V . Observe also that the induced morphism (α, β) is given by (α, β)(c) = α(c), (α, β)(v) = β(θ(v)).

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1.4. Models for fibrations. Let p∶ E ↠ B be a fibration with fiber F . Then we have a (homotopy) pullback F

E p

∗

B.

Following previous section, take i∶ (A, d) ↪ (A ⊗ ΛV, D) a relative Sullivan model for p. Then we have a model for the previous diagram (A, d)

i

(A ⊗ ΛV, D) q

(Q, 0)

(ΛV, D),

where the projection q∶ (A ⊗ ΛV, D) ↠ (

A ⊗ ΛV , D) ≅ (ΛV, D) A+ ⊗ ΛV

is a model for the inclusion F ↪ E[FHT01, Theorem 15.3]. Example 1.5 (The Hopf fibration). Consider the Hopf fibration p∶ S 7 → S 4 . A cdga model for p is the projection q∶ (Λ(a4 , x7 ), dx = a2 ) ↠ (Λ(x), 0), since there is a bijection [S07 , S04 ] ≅ [(Λ(a, x), d), (Λ(x), 0)]. Construct now a relative Sullivan model for q (and thus, for p) as follows: Since H(q)([a]) = 0, introduce a generator y of degree 3 and define Dy = a. Now extend q to θ∶ (Λ(a, x) ⊗ Λ(y), D) → (Λ(x), 0) by setting θ(y) = 0. Now observe that H((Λ(a, x) ⊗ Λ(y), D)) = Q⟨1, [x − ay]⟩, therefore, θ is a quasi-isomorphism (Proposition 1.1). This gives a model of the fiber of f (

Λ(a, x) ⊗ Λ(y) , D) ≅ (Λ(y), 0), Λ+ (a, x) ⊗ Λ(y)

which is S 3 . Observe lastly that H(q) is trivial but p is not, this shows that p is a map between formal spaces which is not formal. 1.5. Models for homotopy pushouts. Dually, one can also model homotopy pushouts throught pullbacks provided one of the cdga models is surjective. Consider (Q, d)

(A, d) f

(B, d)

g

(X, d),

where (Q, d) is the sub cdga of the direct sum (A, d) ⊕ (B, d), Q = {(a, b) ∈ A ⊕Q B∶ f (a) = g(b)} .

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1.6. Models for cofibrations. If i∶ C ↪ X is a cofibration modeled by a surjective cdga morphism ϕ∶ (A, d) → (B, d), the projection X → X/C is modeled by the the inclusion (Q ⊕ ker ϕ, d) → (A, d). The weakness of this approach, in contrast with the models of fibration, is that the model for the cofiber, Q ⊕ ker ϕ, need not be a Sullivan model. Example 1.6 (The homotopy cofibre of the Hopf fibration). Recall q the model of the Hopf fibration p in Example 1.5. Then a model for the homotopy cofiber of p is given by Q ⊕ ker q = Q ⊕ (a). One can see that the Sullivan model for this space is given by (Λ(a4 , y11 ), dy = a3 ). 1.7. Models for the base point inclusion. Let (A, d) be a cdga model for a space X with A0 = Q, then the augmentation morphism ∶ (A, d) → (Q, 0) is a cdga model for the base point inclusion ∗ ↪ X. Observe that this inclusion is homotopy equivalent to the based path space fibration e∶ P X → X, α ↦ α(1) whose fiber is ≃ the loop space ΩX. Now let θ∶ (ΛV, d) → (A, d) be the minimal Sullivan model for (A, d) and take a relative Sullivan model for the augmentation (ΛV, d) → (Q, 0) which is of the form (ΛV ⊗ ΛVˆ , D) with D0 ∶ Vˆ → V a degree 1 linear isomorphism. Moreover, since the fiber of e, ΩX, is an H-space, by [FHT01, Pg. 143] we must have that D(Vˆ ) ⊂ Λ+ V ⊗ ΛVˆ . Now the following homotopy pushout diagram gives a relative model for  (ΛV, d) θ ≃

(A, d)

(ΛV ⊗ ΛVˆ , D) θ⊗Id ≃

≃

(A ⊗ ΛVˆ , D) ≃

(Q, 0), 

where D(ˆ v ) = (θ ⊗ Id)(Dˆ v ) ∈ A+ ⊗ ΛVˆ = (ker ) ⊗ ΛVˆ . This construction is called the acyclic closure of (A, d)[FHT01, p. 192]. 1.8. Models for the diagonal map. Let X be a space and ΔnX ∶ X ↪ X n the n diagonal map x ↦ (x, x, . . . , x). If (A, d) is a cdga model for X, then the n-multiplication morphism μn ∶ (A⊗n , d) ↠ A is a surjective cdga model for ΔnX [FHT01, Pg. 142]. Let us now give a model of

Xn , Δn (X) X

the cofibre of ΔnX . If a ∈ A, denote by

ai ∈ A⊗n the corresponding inclusion of a into the i-th factor. Let G be a set of generators for A, that is, A+ = (G), the ideal of A generated by G. By [Car15, Section 6], ker μn is generated by {x1 − xi ∶ x ∈ G, i = 2, . . . , n}. We see then that a cdga n model for ΔnX(X) is Q ⊕ ({x1 − xi ∶ x ∈ G, i = 2, . . . , n}). X

The diagonal inclusion ΔnX is homotopy equivalent to the path fibration πn ∶ X I ↠ 1 2 X , α ↦ α (0, n−1 , n−1 . . . , n−2 , 1) whose fiber is the product of based loops on X, n−1 n−1 (ΩX) . n

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Let us now construct a cdga model for this fibration. Let (ΛV, d) be a Sullivan model for X, then by [FHT01, Pg. 206] a relative Sullivan model for the multiplication μ2 ∶ (ΛV1 ⊗ ΛV2 , d) → (ΛV, d), where V ≅ V1 ≅ V2 , is given by (ΛV1 ⊗ ΛV2 ⊗ ΛVˆ , D), together with Vˆ = sV , Vˆ k = V k+1 and (sD)k Dˆ v = v2 − v1 − ∑ (v1 ). k! k≥1 The map s is the degree −1 derivation defined by s(v1 ) = s(v2 ) = vˆ, s(v) = 0. Here, subscripts are used to distinguish copies of V . Observe that D(Vˆ ) ⊂ (ker μ2 ) ⊕ (ΛV1 ⊗ ΛV2 ⊗ Λ+ Vˆ ) therefore a quasi-isomorphism ≃

θ∶ (ΛV1 ⊗ ΛV2 ⊗ ΛVˆ , D) → (ΛV, d) v) = 0. Now π3 fits in the homotopy pullback is defined by θ(v1 ) = θ(v2 ) = v and θ(ˆ XI × XI

XI

π2 ×π2

π3

X ×X ×X

Id×Δ2 ×Id

X ×X ×X ×X

which is modeled through the pushout (ΛV, d) ⊗ (ΛV, d) μ2 ⊗μ2

≃ θ⊗θ

(ΛV ⊗ ΛV ⊗ ΛVˆ , D) ⊗ (ΛV ⊗ ΛV ⊗ ΛVˆ , D)

(ΛV ⊗ ΛV, d) ⊗ (ΛV ⊗ ΛV, d) Id⊗μ2 ⊗Id

(ΛV1 ⊗ ΛV2 ⊗ ΛV3 ⊗ ΛVˆ1 ⊗ ΛVˆ2 , D).

(ΛV1 ⊗ ΛV2 ⊗ ΛV3 , d)

An inductive argument proves that the relative Sullivan model for μn is given by the inclusion (ΛV1 ⊗ ⋯ ⊗ ΛVn , d) ↪ (ΛV1 ⊗ ⋯ ⊗ ΛVn ⊗ Λ(Vˆ1 ⊕ ⋯ ⊕ Vˆn−1 ), D) where (si D)k (vi ), si (vi ) = si (vi+1 ) = vˆi . k! k≥1

Dˆ vi = vi+1 − vi − ∑

Observe, in particular that the minimal model for the homotopy fibre of πn is given by (ΛsV, 0)⊗n−1 ≅ (ΛsV ⊕n−1 , 0). This corresponds to the model given in [FHT01, Pg. 143] for H-spaces and the fact that πi−1 (ΩX) ≅ πi (X). Example 1.7. Even spheres S m . Let (Λ(a, x), dx = a2 ) be the minimal model for S m . Then, with previous notation, the multiplication morphism is given by μn ∶ (Λ(a1 , . . . , an , x1 , . . . , xn ), d) → (Λ(a, x), d),

28

´ CARRASQUEL JOSE

with μn (ai ) = a, μn (xi ) = x, and dxi = a2i , i = 1, . . . , n. Therefore a relative Sullivan model for μm is given by a1 , . . . , a ˆn−1 , x ˆ1 , . . . x ˆn−1 ), D), (Λ(a1 , . . . , an , x1 , . . . , xn ) ⊗ Λ(ˆ xi ∣ = 2m − 2. Let us now compute the differentials. Since with ∣ˆ ai ∣ = m − 1 and ∣ˆ dai = 0 we have Dˆ ai = ai+1 − ai . We now compute (si D)k (xi ) using the fact that D and si are derivations: ˆi , ● k = 1: (si D)(xi ) = si (a2i ) = 2ai s(ai ) = 2ai a ˆi ) = 2si (ai (ai+1 − ai )) = 2ˆ ai (ai+1 − ai ), ● k = 2: (si D)2 (xi ) = 2(si D)(ai a ai (ai+1 − ai )) = 2si (ai+1 − ai )2 = 0. ● k = 3: (si D)3 (xi ) = 2(si D)(ˆ Therefore ˆi + a ˆi (ai+1 − ai )) = xi+1 − xi − a ˆi (ai+1 + ai ). Dxˆi = xi+1 − xi − (2ai a Observe that D2 = 0 and that the morphism a1 , . . . , a ˆn−1 , x ˆ1 , . . . x ˆn−1 ), D)) → (Λ(a, x), d) θ∶ ((Λ(a1 , . . . , an , x1 , . . . , xn ) ⊗ Λ(ˆ ai ) = θ(ˆ xi ) = 0 is a cdga quasi-isomorphism since H(Q(θ)) θ(ai ) = a, θ(xi ) = x, θ(ˆ is an isomorphism. 2. Rational Lusternik-Schnirelmann category We present this section as an overview of the rational homotopy techniques used to study LS category. The rational methods used to study sectional category are generalizations of the ones used for LS category. We begin stating the F´elix-Halperin theorem [FH82, Theorem VIII] which gives an algebraic characterization of the LS category of rational spaces. This theorem is key in the development of computation methods for LS category. Theorem 2.1. Let X be a space and (ΛV, d) a Sullivan model for X. Then the rational LS category of X, cat(X0 ), is the least m for which the cdga projection ΛV ρm ∶ (ΛV, d) → ( >m , d) Λ V admits a homotopy retraction. Example 2.2. Let X = S 7 and (ΛV, d) = (Λ(a), 0), with ∣a∣ = 7, its minimal model. Since ρ1 = Id and X is not contractible, we see that cat(X0 ) = cat(X) = 1. Now take a cdga model for X of the form A=(

Λ(a4 , x3 ) , dx = a) , (a2 )

the projection ρ1 ∶ (A, d) → ( (AA+ )2 , d) is not homology injective since the fundamental class is represented by ax ∈ (A+ )2 . Therefore ρ1 cannot have a homotopy retraction. This shows that, in previous theorem, it is necessary to take a Sullivan model for the space. One can define new invariants by weakening the requirements on homotopy retractions, namely ● the module LS category of X, mcat(X), as the smallest m such that ρm admits a homotopy retraction as (ΛV, d)-module,

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● the rational Toomer invariant of X, e(X), as the least m such that H(ρm ) is injective. Hence Corollary 2.3. Let X be a space and (A, d) be any cdga model for X, then nil H+ (X, Q) ≤ e(X) ≤ mcat(X) ≤ cat(X0 ) ≤ cat(X), nil A+ . If X is formal then one can take (A, d) = (H(A), 0) and all the above are thus equalities. Proof. For the first inequality, suppose e(X) = m and consider [x0 ], [x1 ], . . . , [xm ] ∈ H+ (X, Q) ≅ H+ (ΛV, d). Since one can identify xi ∈ Λ+ V , we have x0 x1 ⋯xm ∈ Λ>m V and therefore, [x0 x1 ⋯xm ] ∈ ker H(ρm ). Since, by hypothesis, H(ρm ) is injective, we must have [x0 ][x1 ]⋯[xm ] = [x0 x1 ⋯xm ] = 0. Let us now prove the last inequality. Suppose nil A+ = m, and take a quasi-isomorphism θ∶ (ΛV, d) → (A, d). Since (A+ )m+1 = 0, θ factors through ρm as θ ≃

(ΛV, d) ρm

( ΛΛV >m V , d).

(A, d) ρm

Then taking a relative Sullivan model for ρm , the lifting lemma gives a homotopy retraction r for ρm : (ΛV, d)

Id

ρm r

( ΛΛV >m V , d)

≃ ξ

(ΛV ⊗ ΛZ, D)

ρm ○ξ

(ΛV, d) ≃ θ

(A, d) 

Proposition 2.4. If X is a space such that π∗ (X) ⊗ Q is finite dimensional and concentrated in odd degrees, then cat(X0 ) = dim π∗ (X) ⊗ Q. Proof. By Theorem 1.2, the minimal model of X is of the form (ΛV, d) = (Λ(a1 , . . . , an ), d) with ai in odd degree and n = dim π∗ (X) ⊗ Q. Since Λn+1 V = 0 then ρn = Id. Now consider the element ω ∶= a1 ⋯an ∈ Λn V . For degree reasons, d(ω) = 0 and by the nilpotence condition of a Sullivan algebra, ω cannot be a boundary. This shows that ρn has a homotopy retraction and ρn−1 does not as it is not homology injective.  The following is a first example of how rational homotopy theory gives better lower bounds for sectional category than the standard cohomological ones. Example 2.5. Let X be the space of Example 1.4 and recall its minimal model (ΛV, d) = (Λ(a, b, x), d) with ∣a∣ = ∣b∣ = 3 and d(x) = ab. By the previous proposition, cat(X0 ) = 3. However nil H+ (ΛV, d) = 2. The following is a surprising result by Hess which is a second pillar for the study of rational LS category.

30

´ CARRASQUEL JOSE

Theorem 2.6 ([Hes91]). Let X be a space and (ΛV, d) be its minimal model. Then the projection ΛV ρm ∶ (ΛV, d) → ( >m , d) Λ V admits a cdga homotopy retraction if and only if it admits a homotopy retraction as (ΛV, d)-module. In particular mcat(X) = cat(X0 ). Using this result, F´elix-Halperin-Lemaire[FHL98] proved Theorem 2.7. Let X and Y be spaces. Then ● cat(X0 × Y0 ) = cat(X0 ) + cat(Y0 ), ● If X is a Poincar´e duality complex (see Section 4.2), then e(X) = cat(X0 ). In this theorem, the hypothesis of Poincar´e duality is necessary: Example 2.8 ([LS81]). Let X be the space modeled by the cdga (A, d) = (

Λ(a, b, x) , d) , (a4 , ab, ax)

with ∣a∣ = 2, ∣b∣ = 3, d(a) = d(b) = 0 and d(x) = a3 . Observe that, as vector spaces, A = Q⟨1, a, b, a2 , x, a3 , bx⟩ and H(A, d) = Q⟨1, [a], [b], [a]2, [bx]⟩. This shows that 2 ≤ cat(X0 ) ≤ 3. The minimal model of X is of the form (ΛV, d) with V = Q⟨a, b, v, x, w⟩ ⊕ V ≥7 , d(a) = d(b) = 0, d(v) = ab, d(x) = a3 and d(w) = bv. Then H(ΛV, d) = Q⟨1, [a], [b], [a]2, [bx + a2 v]⟩, and there is no element in Λ>2 V representing a non zero class in H(ΛV, d). This shows that e(X) = 2. We now prove by contradiction that cat(X0 ) = 3. Suppose that ρ2 admits a cdga homotopy retraction r. This implies that H(r) ○ H(ρ2 ) = Id but this is absurd since d(x) ∈ Λ3 V and, for degree reasons, (H(r) ○ H(ρ2 ))([bx + a2 v]) = H(r)([bx + a2 v]) = H(r)([b][x]) = H(r)([b])0 = 0. 2.1. The mapping theorem for LS category. Another important consequence of Theorem 5.4 is the so called mapping theorem [FH82, Theorem I]. We include a simple proof based on the proof of [FHT01, Pg. 389] which we will later on extend to Theorem 6.1. Theorem 2.9. Let f ∶ Y → X be a continuous map. If π∗ (f ) ⊗ Q is injective then cat(Y0 ) ≤ cat(X0 ). Proof. Let (ΛV, d) and (ΛW, d) be the minimal models of X and Y respectively. Then, by Theorem 1.2, the hypothesis tells us that f is modeled by a surjective cdga morphism ϕ∶ (ΛV, d) → (ΛW, d). Denote p∶ P X → X the based path fibration on X, so that cat(X) = secat(p). We construct a model of the (homotopy) pullback E

PX

q

Y

p f

X

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31

(using the acyclic closure of (ΛV, d)) as the homotopy pushout (ΛV, d)

ϕ

(ΛW, d)



Q

≃

j

(ΛV ⊗ ΛVˆ , D)

(ΛW ⊗ ΛVˆ , D)

Here, j is a cdga model for q and D verifies D(Vˆ ) ⊂ Λ+ W ⊗ ΛVˆ and D0 ∶ Vˆ → W is surjective. Now write Z ∶= ker D0 and consider the projection π∶ (ΛW ⊗ ΛVˆ , D) → (ΛZ, 0). Observe that Q(π)∶ (W ⊕ Vˆ , D0 ) → (Z, 0) is a quasi-isomorphism. By Proposition 1.1, π is a cdga quasi-isomorphism. Since π ○j is trivial, we deduce that q0 is trivial and thus cat(X0 ) = secat(q0 ). The result follows since rationalization  commutes with limits and secat(q0 ) ≤ secat(p0 ). Observe that in previous proof, the fact that D(Vˆ ) ⊂ Λ+ V ⊗ ΛVˆ was crucial in assuring that π is a cdga morphism. 3. The Whitehead and Ganea characterizations We start describing the Whitehead and Ganea characterizations for the sectional category of a continuous map f provided that our spaces are CW complexes. We will then translate them to the rational context in the category cdga. 3.1. The Whitehead characterization. Take i∶ A ↪ X a cofibration replacement for f . We define the m-th fat wedge of i as T m (i) ∶= {(x0 , . . . , xm ) ∈ X m+1 ∶ xj ∈ i(A) for at least one j} ⊂ X m+1 . Then, secat(f ) = secat(i) is the least m for which there exists the dashed map making the following a homotopy commutative diagram T m (i)

X

Δm+1

X m+1 .

Here Δm+1 denotes the diagonal map Δm+1 (x) = (x, x, . . . , x) 3.2. The Ganea characterization. Take p∶ E ↠ B a fibration replacement for f . We define the m-th Ganea fibration for f as m Gm f ∶ P (p) ↠ B,

where

⎧ ⎫ m ⎪ ⎪ ⎪m ⎪ P m (p) ∶= ⎨ ∑ tj ej ∶ p(e0 ) = p(ej ), tj ≥ 0 and ∑ tj = 1⎬ , ⎪ ⎪ ⎪ ⎪ j=0 ⎩j=0 ⎭ m for which summands of the form 0e are dropped, and Gf (t0 e0 + t1 e1 + ⋯ + tm em ) ∶= p(e0 ). See [Hus94, Pg. 54] for a detailed description of the topology of P m (p). Then, secat(f ) = secat(p) is the smallest m for which Gm f admits a section.

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32

3.3. Whitehead vs. Ganea. The way to prove the Ganea characterization is to glue the local sections of p into a section of Gm p by means of a partition of the unity on B. Then, to prove the Whitehead characterization it is sufficient to see that there is a homotopy pullback[Mat76, Fas02] P m (p) (3.1)

T m (i)

Gm p

X

Δm+1

X m+1 .

3.4. First algebraic characterizations. Let ϕ∶ (A, d) ↠ (B, b) be a surjective cdga model for a cofibration replacement i∶ A ↪ X of a continuous map f . Then by [FT09, Thm. 1] the inclusion of the m-fat wedge T m (i) ↪ X m+1 is modeled by the quotient A⊗m+1 , d) π∶ (A⊗m+1 , d) ↠ ( (ker ϕ)⊗m+1 while Δm+1 ∶ X ↪ X m+1 is modeled by the multiplication μ∶ (A⊗m+1 , d) → (A, d). Now choose a relative Sullivan model for π, jm ∶ (A⊗m+1 , d) ↪ (A⊗m+1 ⊗ ΛW, D), then Diagram 3.1 is modeled by the following pushout (A⊗m+1 , d)

jm

μ

(A, d)

(A⊗m+1 ⊗ ΛW, D) μ

im

(A ⊗ ΛW, D).

Here, im is a relative Sullivan model for Gm f . We have that secat(f0 ) is the smallest m for which one of the following equivalent conditions hold ● Whitehead: There exists a cdga morphism r∶ (A⊗m+1 ⊗ ΛW, D) → A such that r ○ jm = μ. ● Ganea: There exists a cdga retraction for im . 4. Rational approximations of sectional category One can now impose less restrictive conditions to the existence of morphisms in the characterizations of Section 3.4 to get algebraic lower bounds for sectional category. Let f be a continuous map and im ∶ (A, d) ↪ (A ⊗ ΛW, d) be a relative Sullivan model for Gm f (for example, the one constructed in Section 3.4). If r∶ (A⊗ΛW, D) → (A, d) is a retraction of im (that is r(a⊗1) = a, for a ∈ A) then we have the following implications: r is a cdga morphism ⇒ r is an (A, d)-module morphism ⇒ H(i) is injective. Definition 4.1. With previous notation, ● the module sectional category of f , msecat(f ), is the least m for which im admits an (A, d)-module retraction.

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33

● the homology sectional category of f , Hsecat(f ), is the least m for which H(im ) = H∗ (Gm f , Q) is injective. In previous definition one can replace im by any model for Gm p and asking the retractions to be homotopy retractions. We can now deduce Proposition 4.2. If f is a continuous map, then nil ker H∗ (f, Q) ≤ Hsecat(f ) ≤ msecat(f ) ≤ secat(f0 ) ≤ secat(f ). Observe that, if we take f as the inclusion of the base point into a based space X, we get nil H+ (X, Q) ≤ e(X) ≤ mcat(X) ≤ cat(X0 ) ≤ cat(X). In contrast with Theorem 2.6, there are maps f for which msecat(f ) < secat(f0 ): Example 4.3 ([Sta00]). Consider the cdga (Λ(a2 , b2 , x3 ), dx = a2 + b2 ), then the inclusion ϕ∶ (Λ(a), 0) ↪ (Λ(a, b, x), dx = a2 + b2 ) admits a retraction as (Λ(a), 0)-module but not as cdga. In fact, if r is a cdga retraction then it must verify r(a) = a, r(b) = αa, r(x) = 0, but then r cannot commute with differentials since r(dx) = r(a2 + b2 ) = (1 + α2 )a2 ≠ 0. Let us now define an (A, d)-module retraction as r(bi x) = 0 and r(bi ) = 0 for i odd and r(bi ) = (−1)i/2 ai for i even. Since we are particularly interested in topological complexity, by taking f as the diagonal map, ΔnX , we introduce Definition 4.4. For a space X, ● the module topological complexity of X is mTCn (X) ∶= msecat(ΔnX ), ● the homology topological complexity is HTCn (X) ∶= Hsecat(ΔnX ). Since (ΔnX )0 = ΔnX0 , we have nil ker H∗ (ΔnX , Q) ≤ HTCn (X) ≤ mTCn (X) ≤ TCn (X0 ) ≤ TCn (X). 4.1. Module sectional category. Most of this section is based on [FGKV06, CKV16, CPV16]. Let (A, d) be a cdga. An (A, d)-module is a chain complex (M, d) together with an action of A, A ⊗ M → M verifying d(am) = d(a)m + (−1)∣a∣ ad(m). A semi-free extension of an (A, d)-module (M, d) is an (A, d) module of the form (M ⊕ (A ⊗ X), d), where X = ⊕i≥0 Xi , d(X0 ) ⊂ M , d(Xk ) ⊂ M ⊕ (A ⊗ (⊕k−1 i=0 Xi ) and the inclusion (M, d) → (M ⊕ (A ⊗ X), d) is an (A, d)-module morphism. Observe that relative Sullivan algebras (A ⊗ ΛV, D) are semi-free extensions of (A, d)[FHT01, Lemma 14.1]. In fact, we have that the category of (A, d)-modules is a closed proper model category [FGKV06, Theorem 4.1]. Any cdga morphism ϕ∶ (A, d) → (B, d) can be seen as an (A, d)-module morphism by taking in (B, d) the (A, d)-module structure ab = ϕ(a)b. A semi-free ≃ model for ϕ is an (A, d)-module quasi-isomorphism (A ⊕ (A ⊗ X), d) → (B, d) from a (A, d)-semi-free extension of (A, d). A semi-free model for a continuous map f is just a semi-free model for any cdga model of f .

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34

Theorem 4.5 ([FGKV06]). If (A, d) ↪ (A ⊕ (A ⊗ X), d) is a semi-free model for a map f then a semi-free model for the m-th Ganea fibration Gm f is jm ∶ (A, d) ↪ (A ⊕ (A ⊗ s−m X ⊗m+1 ), d) Here d = d0 + d+ (in A ⊕ A ⊗ s−m X ⊗m+1 ) is given by m

d(s

−m

+

∑ (k∣xm−k ∣+k−1)

x0 ⊗ ⋯ ⊗ xm ) = (−1)

k=1

m

(∣aiji ∣+1)(∣x0 ∣+⋯+∣xi−1 ∣+m)

∑ ∑(−1)

d0 x0 ⋅ ⋯ ⋅ d0 xm

aiji ⊗ s−m x0 ⊗ ⋯ ⊗ xiji ⊗ ⋯ ⊗ xm ,

i=0 ji

for x0 ,..., xm ∈ X and d+ xi = ∑ aiji ⊗ xiji with aiji ∈ A and xiji ∈ X. ji

Corollary 4.6. If f is a map, then msecat(f ) is the smallest m for which jm admits an (A, d)-module retraction and Hsecat(f ) is the smallest m for which H(jm ) is injective. 4.2. Poincar´ e Duality. A finite dimensional commutative graded algebra H is said to be a Poincar´e duality algebra with formal dimension n when H 0 = Q, H = ⊕ni=o H i and there exists an element Ω ∈ H n such that the map of degree −n Φ

H

hom(H, Q) b ↦ Ω# (ab)

a

is an isomorphism, where Ω# denotes the dual of Ω. Theorem 4.7 ([CKV16]). Let ϕ∶ (A, d) → (B, d) be a cdga morphism with H(A, d) a Poincar´e duality algebra, then H(ϕ) is injective if and only if ϕ admits a homotopy retraction as a morphism of (A, d)-modules. Corollary 4.8. If f ∶ X → Y is a map with H∗ (Y, Q) a Poincar´e duality algebra, then msecat(f ) = Hsecat(f ). In particular, if X is a Poincar´e duality complex, then mTCn (X) = HTCn (X). 5. Characterization ` a la F´ elix-Halperin Let f be a continuous map and ϕ∶ (A, d) ↠ (B, d) a surjective cdga model for f . Recall the notation from Section 3.4. Then, since μ((ker ϕ)⊗m+1 ) ⊂ (ker ϕ)m+1 , we have a diagram ⊗m+1

A ( (ker , d) ϕ)⊗m+1 π

(A⊗m+1 , d)

j

≃

(A⊗m+1 ⊗ ΛW, D)

μ

(A, d)

μ

i

(A ⊗ ΛW, D) τ A ( (ker ϕ) m+1 , d) . ρm

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35

Then taking (A ⊗ ΛZ, D) a relative Sullivan model for ρm , the lifting lemma gives a diagram (A ⊗ ΛZ, D)

A i

(A ⊗ ΛW, D)

τ

≃ τ

A (ker ϕ)m+1

which implies Proposition 5.1 ([JMP12,Car15]). If ρm admits a cdga homotopy retraction then secat(f0 ) ≤ m. If ρm admits an (A, d)-module homotopy retraction, then msecat(f ) ≤ m. If H(ρm ) is injective, then Hsecat(f ) ≤ m. Observe that the opposite implication need not hold (see Example 2.2) and this remains true even for topological complexity: Example 5.2. Let us compute the topological complexity of S03 . The path fibration π2 is modeled by the multiplication morphism μ2 ∶ (Λ(a1 , a2 ), 0) → (Λa, 0) with ∣a1 ∣ = ∣a2 ∣ = ∣a∣ = 3. We remark that ker H(μ2 ) = ([a1 ] − [a2 ]), therefore nil ker H(μ2 ) = 1 ≤ TC(S03 ). Now, since ker μ2 = (a1 −a2 ), we have that (ker μ2 )2 = 0 and ρ1 = Id admits a (homotopy) retraction. This proves, by Proposition 5.1, that TC(S03 ) = 1. Another way to show this is through TC(S03 ) ≤ TC(S 3 ) = 1. Now consider another cdga model for S 3 , namely the cdga (A, d) ∶= (Λ(a, b, c), d) with ∣a∣ = ∣b∣ = ∣c∣ = 1, with d(a) = bc, d(b) = ac and d(c) = ab. Observe that A is a free cga but that (A, d) is not a Sullivan algebra. Observe also that (A, d) is a cdga ≃ model for S 3 as there is a quasi-isomorphism (Λv3 , 0) → (A, d) defined as v ↦ abc. We have then that another cdga model for the path fibration π2 is the multiplication on (A, d), μ2 ∶ (Λ(a1 , b1 , c1 , a2 , b2 , c2 ), d) = (Λ(a, b, c), d). We now see that ρ2 ∶ (Λ(a1 , b1 , c1 , a2 , b2 , c2 ), d) → (

Λ(a1 , b1 , c1 , a2 , b2 , c2 ) , d) (ker μ2 )3

is not homology injective. In fact, the element ω ∶= (a1 − a2 )(b1 − b2 )(c1 − c2 ) ∈ (ker μ2 )3 can be written as ω = a1 b1 c1 − a2 b2 c2 − d(a1 a2 − b1 b2 + c1 c2 ). This means that [ω] ≠ 0 and that H(ρ2 )([ω]) = 0. If I is an ideal of A, define the homology nilpotency of I, Hnil I, to be the smallest m such that I m+1 is contained in an acyclic ideal of A, that is, a differential ideal J, with H(J) = 0. Using previous proposition one can deduce that, if ϕ is a surjective cdga model for f , then secat(f0 ) ≤ Hnil(ker ϕ) ≤ nil ker ϕ.

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36

In fact, if (ker ϕ)m+1 ⊂ J with J an acyclic ideal of (A, d). Then, by the five lemma, there is a diagram (A, d) ρm

≃

(A , d) J

A ( (ker ϕ) m+1 , d)

which can be used, together with the lifting lemma, to give a homotopy retraction for ρm . As a consequence, if f is a formal map such that H∗ (f, Q) is surjective, then secat(f0 ) = nil ker H∗ (f, Q). The hypothesis of H∗ (f, Q) being surjective is necessary: 2 ,b2 ) , 0) defined Example 5.3. Consider the cdga morphism ϕ∶ (Λa2 , 0) → ( Λ(a (a2 +b2 ) by ϕ(a) = a. Let f be a continuous map whose model is ϕ (the spatial realization of ϕ, for instance), then H∗ (f, Q) = ϕ. Obviously nil ker ϕ = 0 but ϕ does not admit a homotopy retraction. In fact, a relative Sullivan model for ϕ is given by

ϕ

(Λ(a), 0)

Λ(a,b) ( (a 2 +b2 ) , 0) ≃ θ

j

(Λ(a) ⊗ Λ(b, v1 , v2 , . . .), D), with D(b) = 0, D(v1 ) = a2 + b2 , θ(b) = b, θ(v1 ) = 0, and ∣vi ∣ ≥ 4 for i ≥ 2. But, as in Example 4.3, j cannot have a cdga retraction. This means that secat(f0 ) ≥ 1. 5.1. When f admits a homotopy retraction. Now suppose that our continuous map f admits a homotopy retraction. Observe that this is the case of the path fibrations πn ∶ X I ↠ X n . We now construct a special type of model for such maps which we will call s-models. We can suppose that f ∶ X ↪ Y is a cofibration and that it admits a strict retraction r∶ Y → X, so that r ○ f = IdX . Now, take a surjective quasi-isomorphism θ∶ (ΛV, d) → APL (X) and a relative Sullivan model APL (X)

APL (r)

APL (Y )

θ ≃

≃ ξ

(ΛV, d)

(ΛV ⊗ ΛW, D).

Now the lifting lemma Id

(ΛV, d) (ΛV ⊗ ΛW, D)

(ΛV, d)

ϕ′

ξ

APL (Y )

≃ θ APL (f )

APL (X),

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37

gives a cdga morphism, ϕ′ , which is a model for f . Now suppose (A, d) is any cdga model for X. Then there is a pushout diagram (ΛV, d)

(ΛV ⊗ ΛW, D) ≃

ψ ≃

(A, d)

ψ○ϕ′

(A ⊗ ΛW, D) ϕ

(A, d),

IdA

which gives a model ϕ for f which is a retraction for the inclusion (A, d) ↪ (A ⊗ ΛW, D). Such a model is said to be an s-model for f . We can now state a generalization of the F´elix-Halperin theorem which lets us compute the topological complexity of rational spaces. Theorem 5.4 ([CV17]). Let f be a continuous map and ϕ∶ (A ⊗ ΛW, D) → (A, d) an s-model for f . Then secat(f0 ) is the smallest m for which the cdga projection A ⊗ ΛW ρm ∶ (A ⊗ ΛW, D) → ( , D) (ker ϕ)m+1 admits a homotopy retraction. Also, ● msecat(f ) is the smallest m for which ρm admits a homotopy retraction as (A ⊗ ΛW, D)-module, and ● Hsecat(f ) is the smallest m for which H(ρm ) is injective. Observe that the augmentation is an s-model for the base point inclusion. In this case, previous theorem is just the F´elix-Halperin Theorem 2.1. 5.2. Applications to topological complexity. Let (A, d) be any cdga model for a space X. We build an s-model for the path fibration πn ∶ X I → X n as follows: Take (ΛV, d) a Sullivan model for X (not necessarily minimal) and take ≃ a quasi-isomorphism θ∶ (ΛV, d) → (A, d). Then the cdga morphism μθn ∶= (IdA , θ, . . . , θ)∶ (A, d) ⊗ (ΛV, d)⊗n−1 → (A, d) is an s-model for πn . Moreover, as with the multiplication (μn = μId n ), one can see that ker μθn = (θ(v) − vi ) where, as usual, vi , i = 2, . . . , n stands for v in the i-th factor. As a corollary to Theorem 5.4, we get Corollary 5.5. Let X be a space, (ΛV, d) a Sullivan model for X, and ≃

θ∶ (ΛV, d) → (A, d) a quasi-isomorphism. Then TCn (X0 ) is the least m such that the projection ρm ∶ (A ⊗ (ΛV )⊗n−1 , d) → (

A ⊗ (ΛV )⊗n−1 , d) (ker μθn )m+1

´ CARRASQUEL JOSE

38

admits a homotopy retraction. Moreover, mTCn (X) is the least m such that ρm admits a retraction as (A ⊗(ΛV )⊗n−1 , d)-module and HTCn (X) is the least m such that H(ρm ) is injective. Example 5.6. Compare [FGKV06, Example 6.5] and [GLO15, Example 3.5]. Let X = Sa3 ∨Sb3 ∪e8 ∪e8 where the cells are attached through the iterated Whitehead products [Sa3 , [Sa3 , Sb3 ]] and [Sb3 , [Sa3 , Sb3 ]]. By [FHT01, p. 179], the minimal model of X is of the form (ΛV, d) with V ≤10 = Q⟨a, b, x, y⟩, ∣a∣ = ∣b∣ = 3, d(a) = d(b) = 0, d(x) = ab, d(y) = abx and H(ΛV, d) = Q⟨1, [a], [b], [ax], [bx]⟩. Therefore, writing (A, d) ∶= (

Λ(a, b, x) , d) , (abx)

the projection θ∶ (ΛV, d) → (A, d) is a quasi-isomorphism and an s-model for the diagonal map ΔX is μθ ∶ (

Λ(a, b, x) Λ(a1 , b1 , x1 ) ⊗ Λ (⟨a2 , b2 , x2 , y2 ⟩ ⊕ V ≥11 ) , d) → ( , d) . (abx) (abx)

Now consider the element ω ∶= (a1 −a2 )(b1 −b2 )(x1 −x2 ) ∈ (ker μθ )3 . A computation yields that ω = a1 x1 b2 − b1 a2 x2 + a1 b2 x2 − b1 x1 a2 − d(x1 x2 + y2 ). This means that [ω] ∈ ker H(ρ2 ) is non-zero. Therefore HTC(X) ≥ 3. On the other hand, nil ker μ = 3 where μ∶ A ⊗ A → A is the multiplication. This proves that 3 = TC(X0 ) ≤ TC(X) whilst nil ker H∗ (Δ2X , Q) = 2. Finally we have Theorem 5.7 ([CPV16]). If X and Y are topological spaces, then mTCn (X × Y ) = mTCn (X) + mTCn (Y ). 6. A mapping theorem for topological complexity In this section we will give a slight generalization of the mapping theorem for rational topological complexity of Grant-Lupton-Oprea[GLO15, Theorem 3.2] and give a proof using Sullivan models. Theorem 6.1. Let fi ∶ Yi → X be continuous maps, i = 1, . . . , n, between rational spaces such that π∗ (fi ) are injective. If im (π∗ (f1 )) ∩ im (π∗ (f2 )) = 0 then cat(Y1 ) + ⋯ + cat(Yn ) ≤ TCn (X). Proof. Since π∗ (fi ) is injective, there is a surjective model for fi , ϕi ∶ (ΛV, d) ↠ (ΛWi , d) between minimal Sullivan models. The condition im (π∗ (f1 )) ∩ im (π∗ (f2 )) = 0 implies that for each w1 ∈ W1 and w2 ∈ W2 , there exist v1 , v2 ∈ V such that ϕ1 (v1 ) = w1 , ϕ2 (v1 ) = 0, ϕ1 (v2 ) = 0 and ϕ2 (v2 ) = w2 . As in the proof of Theorem 2.9, we model the (homotopy) pullback XI

P q

Y1 × ⋯ × Yn

πn n

f1 ×⋯×fn

X ,

RATIONAL METHODS APPLIED TO SECTIONAL CATEGORY AND TC

39

through the pushout (ΛV1 ⊗ ⋯ ⊗ ΛVn , d)

ϕ1 ⊗⋯⊗ϕn

(ΛW1 ⊗ ⋯ ⊗ ΛWn , d) j

i

(ΛV1 ⊗ ⋯ ⊗ ΛVn ⊗ ΛVˆ , D)

(ΛW1 ⊗ ⋯ ⊗ ΛWn ⊗ ΛVˆ , D),

where i is the relative model for πn from Section 1.8, j is a model for q, Vˆ = Vˆ1 ⊕ ⋯ ⊕ Vˆn−1 , and D0 (ˆ vi ) = ϕi+1 (vi+1 ) − ϕi (vi ). We will prove that q is trivial. Observe that, by the properties of the ϕi ’s, D0 ∶ Vˆ → W1 ⊕ ⋯ ⊕ Wn is surjective. Now take Z ∶= ker D0 ⊂ Vˆ . Since D(Vˆ ) ⊂ Λ+ (W1 ⊕ ⋯ ⊕ Wn ) ⊗ ΛVˆ , the projection ξ∶ (ΛW1 ⊗ ⋯ ⊗ ΛWn ⊗ ΛVˆ , D) ↠ (ΛZ, 0) is well defined. By construction, H(Q(ξ))∶ H(W1 ⊕ ⋯Wn ⊕ Vˆ , D0 ) → H(Z, 0) is an isomorphism, therefore, by Proposition 1.1, ξ is a quasi isomorphism. Since ξ ○ j is trivial we have that q is trivial, thus cat(Y1 × ⋯ × Yn ) = secat(q) ≤ secat(πn ) = TCn (X). But by Theorem 2.7, cat(Y1 × ⋯ × Yn ) = cat(Y1 ) + ⋯ + cat(Yn ).



Acknowledgements The author thanks the organizers of Workshop on TC and Related Topics, held in Oberwolfach in March 2016 and acknowledges the Belgian Interuniversity Attraction Pole (IAP) for support within the framework “Dynamics, Geometry and Statistical Physics” (DYGEST). References A. K. Bousfield and V. K. A. M. Gugenheim, On PL de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94, DOI 10.1090/memo/0179. MR0425956 [Car15] J. G. Carrasquel-Vera, Computations in rational sectional category, Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 3, 455–469. MR3396996 [Car16a] J. G. Carrasquel-Vera, The Ganea conjecture for rational approximations of sectional category, J. Pure Appl. Algebra 220 (2016), no. 4, 1310–1315, DOI 10.1016/j.jpaa.2015.09.001. MR3423449 [CV17] J. G. Carrasquel-Vera, The rational sectional category of certain maps, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no. 2, 805–813. MR3700384 [CKV16] J. G. Carrasquel-Vera, T. Kahl, and L. Vandembroucq, Rational approximations of sectional category and Poincar´ e duality, Proc. Amer. Math. Soc. 144 (2016), no. 2, 909–915, DOI 10.1090/proc12722. MR3430865 [CPV16] J.G. Carrasquel-Vera, P.-E. Parent, and L. Vandembroucq, Module sectional category of products, J. Homotopy Relat. Struct. (2017). DOI:10.1007/s40062-017-0192-0 [Far03] M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211–221, DOI 10.1007/s00454-002-0760-9. MR1957228 [Fas02] A. Fass` o Velenik, Relative homotopy invariants of the type of the LusternikSchnirelmann category, Eingereichte Dissertation (Ph. D. Thesis), Freien Universit¨ at Berlin (2002). [FGKV06] L. Fern´ andez Su´ arez, P. Ghienne, T. Kahl, and L. Vandembroucq, Joins of DGA modules and sectional category, Algebr. Geom. Topol. 6 (2006), 119–144, DOI 10.2140/agt.2006.6.119. MR2199456 [FH82] Y. F´ elix and S. Halperin, Rational LS category and its applications, Trans. Amer. Math. Soc. 273 (1982), no. 1, 1–38, DOI 10.2307/1999190. MR664027 [BG76]

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[FHL98]

[FHT01] [FT88]

[FT09]

[GLO15]

[Hal83] [Hes91] [Hus94] [JMP12] [LS34] [LS81]

[Mat76] [Opr86] [Qui67] [Rud10] [Sch66] [Sta00] [Sul77] [Vig79]

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Y. F´ elix, S. Halperin, and J.-M. Lemaire, The rational LS category of products and of Poincar´ e duality complexes, Topology 37 (1998), no. 4, 749–756, DOI 10.1016/S00409383(97)00061-X. MR1607732 Y. F´ elix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR1802847 Y. F´ elix and D. Tanr´ e, Formalit´ e d’une application et suite spectrale d’EilenbergMoore (French), Algebraic topology—rational homotopy (Louvain-la-Neuve, 1986), Lecture Notes in Math., vol. 1318, Springer, Berlin, 1988, pp. 99–123, DOI 10.1007/BFb0077798. MR952575 Y. F´ elix and D. Tanr´ e, Rational homotopy of the polyhedral product functor, Proc. Amer. Math. Soc. 137 (2009), no. 3, 891–898, DOI 10.1090/S0002-9939-08-09591-9. MR2457428 M. Grant, G. Lupton, and J. Oprea, A mapping theorem for topological complexity, Algebr. Geom. Topol. 15 (2015), no. 3, 1643–1666, DOI 10.2140/agt.2015.15.1643. MR3361146 S. Halperin, Lectures on minimal models, M´ em. Soc. Math. France (N.S.) 9-10 (1983), 261. MR736299 K. P. Hess, A proof of Ganea’s conjecture for rational spaces, Topology 30 (1991), no. 2, 205–214, DOI 10.1016/0040-9383(91)90006-P. MR1098914 D. Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. MR1249482 B. Jessup, A. Murillo, and P.-E. Parent, Rational topological complexity, Algebr. Geom. Topol. 12 (2012), no. 3, 1789–1801, DOI 10.2140/agt.2012.12.1789. MR2979997 L. Lusternik and L. Schnirelmann, M´ ethodes topologiques dans les probl` emes variationnels, vol. 188, Hermann, Paris, 1934. J.-M. Lemaire and F. Sigrist, Sur les invariants d’homotopie rationnelle li´ es a ` la L. S. cat´ egorie (French), Comment. Math. Helv. 56 (1981), no. 1, 103–122, DOI 10.1007/BF02566201. MR615618 M. Mather, Pull-backs in homotopy theory, Canad. J. Math. 28 (1976), no. 2, 225–263, DOI 10.4153/CJM-1976-029-0. MR0402694 J. F. Oprea, DGA homology decompositions and a condition for formality, Illinois J. Math. 30 (1986), no. 1, 122–137. MR822387 D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, SpringerVerlag, Berlin-New York, 1967. MR0223432 Y. B. Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010), no. 5, 916–920, DOI 10.1016/j.topol.2009.12.007. MR2593704 A. Schwarz, The genus of a fiber space, A.M.S Transl. 55 (1966), 49–140. D. Stanley, The sectional category of spherical fibrations, Proc. Amer. Math. Soc. 128 (2000), no. 10, 3137–3143, DOI 10.1090/S0002-9939-00-05468-X. MR1691006 ´ D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Etudes Sci. Publ. Math. 47 (1977), 269–331 (1978). MR0646078 M. Vigu´ e-Poirrier, Formalit´ e d’une application continue (French, with English summary), C. R. Acad. Sci. Paris S´ er. A-B 289 (1979), no. 16, A809–A812. MR558804

Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umul´, Poland towska 87, 61-614 Poznan Email address: [email protected], [email protected]

Contemporary Mathematics Volume 702, 2018 http://dx.doi.org/10.1090/conm/702/14104

Topological complexity of classical configuration spaces and related objects Daniel C. Cohen Abstract. We survey results on the topological complexity of classical configuration spaces of distinct ordered points in orientable surfaces and related spaces, including certain orbit configuration spaces and Eilenberg-Mac Lane spaces associated to certain discrete groups.

Contents 1. Introduction 2. The plane and the sphere 2.1. Arrangements, I 2.2. Cohomology 2.3. Genus zero 3. Genus one 3.1. Fadell-Neuwirth theorem 3.2. Cohen-Taylor/Totaro spectral sequence 4. Higher genus 4.1. Gr¨obner bases 4.2. Zero divisors 5. Orbit configuration spaces 5.1. Generalized Fadell-Neuwirth theorem 5.2. Arrangements, II 6. Some discrete groups 6.1. Almost-direct products of free groups 6.2. Fiber-type arrangements 6.3. Subgroup conditions 7. Sins of omission 7.1. Graph configuration spaces 7.2. Unordered configuration spaces Acknowledgements References

2010 Mathematics Subject Classification. 20F36, 55M30, 55R80. The author was partially supported by NSF 1105439. c 2018 American Mathematical Society

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DANIEL C. COHEN

1. Introduction Investigation of the collision-free motion of n distinct ordered particles in a topological space X leads one to study the (classical) configuration space F (X, n) = {(x1 , . . . , xn ) ∈ X n | xi = xj if i = j} of n distinct ordered points in X, and the topological complexity of this space. For a path-connected topological space Y , I = [0, 1] the unit interval, and Y I the space of all continuous paths γ : I → Y (with the compact-open topology), the topological complexity of Y is the sectional category (or Schwarz genus) of the fibration η : Y I → Y × Y , γ → (γ(0), γ(1)), TC(Y ) = secat(η). This homotopy invariant, introduced by Farber [16], provides a topological approach to the motion planning problem from robotics. For any s ≥ 2, one may more generally consider the sectional category of the fibration ηs : Y I → Y s , where ηs sends a path to s points on the path, ηs (γ) = (γ(t1 ), . . . , γ(ts )), where tk = (k − 1)/(s − 1), 1 ≤ k ≤ s. This is the higher topological complexity of Y , TCs (Y ) = secat(ηs ) introduced by Rudyak [36], extending Farber’s notion above, as TC2 (Y ) = TC(Y ). We survey results on the topological complexity of configuration spaces F (X, n) in the case where X is an orientable surface, as well as related objects. The discussion is focused primarily on the “classical” topological complexity, TC = TC2 , and includes remarks on the higher topological complexity in those instances where results on this invariant are known. The general principle is as follows: The topological complexity is as large as possible, given natural constraints. For instance, as is well known (and discussed in Section 2 below), the configuration space of n distinct ordered points in the plane has the homotopy type of the product of a circle and a CW-complex of dimension n − 2. These constraints, together with the fact that TC(S 1 ) = 2, and the known bounds recorded next, yield the topological complexity of F (R2 , n), recorded in Theorem 2.1. If Y is a topological space with the homotopy type of a finite-dimensional CW-complex, let hdim(Y ) denote the homotopy dimension of Y . Throughout the discussion, we will make use of the following basic tools. For details and other relevant facts, see Farber’s survey [19]. • •

TC(Y ) ≤ 2 · hdim(Y ) + 1

•

TC(Y × Z) ≤ TC(Y ) + TC(Z) − 1  

∪ TC(Y ) > zcl H ∗ (Y ) = cup length ker H ∗ (Y ) ⊗ H ∗ (Y ) −−→ H ∗ (Y )

We call the first two of these the dimension and product inequalities, and use cohomology with C-coefficients (unless stated otherwise) in the context of the third, the zero divisor cup length. We use the unreduced notions of topological complexity and higher topological complexity. For instance, TC(Y ) = secat(η : Y I → Y × Y ) is equal to the smallest integer m such that there exists of cover of Y × Y by m open sets, on each of which the fibration η : Y I → Y × Y admits a continuous local section. In particular, for Y contractible, TC(Y ) = 1. 2. The plane and the sphere The topological complexity of the configuration space of ordered points in the plane X = R2 = C was determined by Farber-Yuzvinsky. Theorem 2.1 ([23]). TC(F (C, n)) = 2n − 2 for n ≥ 2.

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2.1. Arrangements, I. To discuss this result, we recall some relevant facts from the theory of complex hyperplane arrangements. Let V = C be a complex vector space of dimension  < ∞. A hyperplane in V is a codimension one affine subspace H. A hyperplane arrangement in V is a finite collection A = {H1 , . . . , Hm } of hyperplanes in V . If we fix coordinates x = (x1 , . . . , x ) on V , each hyperplane of A may be realized as j = {x ∈ C | fj (x) = 0}, where fj is a linear polynomial. The H m product Q(A) = j=1 fj is said to be a defining polynomial of the arrangement A. An arrangement A is said to be essential if there are  hyperplanes in A whose intersection is a point. If the intersection of all hyperplanes in A is nonempty, m H =  ∅, we can choose coordinates so that fj is a linear form and 0 ∈ Hj for j=1 j each j. In this situation, A is said to be central, and the defining polynomial Q(A) is homogeneous. Refer to Orlik-Terao [35] as a general reference on arrangements. A principal object

m of topological study of arrangements is the complement M = M (A) = V  j=1 Hj = V Q(A)−1 (0). The complement is an open, smooth manifold of real dimension 2, which has the homotopy type of a connected, finite CW-complex of dimension at most  (M is a Stein manifold). If A is essential, then the homotopy dimension of M is precisely . n Example 2.2. The  braid arrangement An in V = C is nthe central arrangement n consisting of the 2 diagonal hyperplanes Hi,j = {x ∈ C | xi − xj = 0}. Note that An is not essential, as the diagonal line {(x, x, . . . , x)} is contained in (the intersection of) all hyperplanes of An . The complement of the braid arrangement  M (An ) = Cn  Hi,j = {(x1 , . . . , xn ) ∈ Cn | xi = xj if i = j} = F (C, n) i b1,1 > · · · > a1,g > b1,g > a2,1 > b2,1 > · · · > a2,g > b2,g > · · · · · · > an,1 > bn,1 > · · · > an,g > bn,g . Proposition 4.3. The set ⎧ ai,r ai,s , ai,r bi,s , bi,r bi,s , ⎪ ⎪ ⎪ ⎪ a ⎪ i,k bi,k − ai,g bi,g , ⎪ ⎨ ai,1 bj,1 − bi,1 aj,1 − ai,g bi,g − aj,g bj,g , G= ai,p aj,q , ai,p bj,q , bi,p bj,q , ⎪ ⎪ ⎪ ⎪ a ⎪ i,1 aj,g bj,g + ai,g bi,g aj,1 , ⎪ ⎩ bi,1 aj,g bj,g + ai,g bi,g bj,1 ,

⎫ (1 ≤ i ≤ n, 1 ≤ r = s ≤ g), ⎪ ⎪ ⎪ (1 ≤ i ≤ n, 1 ≤ k ≤ g − 1), ⎪ ⎪ ⎪ ⎬ (1 ≤ i < j ≤ n), (1 ≤ i = j ≤ n, 2 ≤ p, q ≤ g),⎪ ⎪ ⎪ ⎪ (1 ≤ i < j ≤ n), ⎪ ⎪ ⎭ (1 ≤ i < j ≤ n)

is a Gr¨ obner basis for the ideal L = J + K in the exterior algebra E. Note that the elements of G are recorded with their initial terms first, and that we have used −Δi,j in place of Δi,j . Note also the presence of the cubic elements in G. The proposition may be established using [1, Corollary 1.5], by showing that all S- and T -polynomials involving elements of G reduce to zero with respect to G. This (lengthy) process may be inductively sped up, by successively considering the ideals Lk and sets G k involving the generators ai,p , bi,p of E with first index i ≥ k, for k = n − 1, n − 2, . . . . Remark 4.4. Gonz´ alez-Guti´errez [28] consider a further quotient of the algebra A. In the above notation, they work with the algebra E/(L + K  ), where K  is the ideal generated by the elements (ai,1 − a1,1 )(bj,1 − b1,1 ), 2 ≤ i = j ≤ n. This simplifies the zero divisor calculations carried out in [28], but complicates the above Gr¨ obner basis considerations. ∼ 4.2. Zero divisors. The algebras A = E/J and B = E/L = E/(J + K) = A/((J + K)/J) we consider are quotients of the exterior algebra E by ideals generated in degrees greater than or equal to two, so we identify the degree-one generators

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DANIEL C. COHEN

of all of these algebras and denote them by the same symbols ai,p , bi,p , 1 ≤ i ≤ n, 1 ≤ p ≤ g. Let ZA be the ideal in A ⊗ A generated by all degree-one zero divisors ZA = ¯ ai,p = 1 ⊗ ai,p − ai,p ⊗ 1, ¯bi,p = 1 ⊗ bi,p − bi,p ⊗ 1, (1 ≤ i ≤ n, 1 ≤ p ≤ g). Similarly, denote the ideal generated by degree-one zero divisors in B ⊗ B by ZB . 2n+2 = 0. To show that zcl(A) ≥ 2n + 2, it suffices to show that ZA 2n+2 is nonzero in A ⊗ A. Proposition 4.5. The ideal ZA

¯1,g ¯b1,g a ¯2,1¯b2,1 · · · a ¯n,1¯bn,1 . We assert that the image ζA of ζ is Let ζ = a ¯1,1¯b1,1 a 2n+2 nonzero in A ⊗ A. Clearly, this image is in ZA . For n = 1, this is immediate, ∗ ¯ ¯ ¯1,1 b1,1 a ¯1,g b1,g = 2a1,g b1,g ⊗ a1,g b1,g = 2ω1 ⊗ ω1 in this as A = H (Σg ) and ζ = a instance. For n ≥ 2, as the natural projection A ⊗ A  B ⊗ B takes the generators and powers of ZA to those of ZB , it is enough to show that the image ζB of ζ in B ⊗ B is nonzero. A calculation with the description of the ideal L = J + K defining B = E/L given in §4.1 reveals that, for n ≥ 2, the image of ζ in B ⊗ B is given by   n−1  n−1 (−1)n−k−1 a1,g b1,g Uk ⊗ a1,g b1,g Vk , ζB = 2 k k=0

where Uk = b2,1 · · · bk+1,1 ak+2,1 · · · an,1 and Vk = a2,1 · · · ak+1,1 bk+2,1 · · · bn,1 . In particular, U0 = Vn−1 = a2,1 · · · an,1 and V0 = Un−1 = b2,1 · · · bn,1 . The Gr¨ obner basis G recorded in Proposition 4.3 may be used to show that this element is nonzero in B ⊗ B. For instance, the leading term of this element (in E ⊗ E) is the tensor product a1,g b1.g a2,1 · · · an,1 ⊗ a1,g b1.g b2,1 · · · bn,1 of two monomials neither of which reduce to zero with respect to G. It follows that (the 2n+2 = 0, and leading term of) ζB is nonzero in B ⊗ B. Consequently, ζA = 0, ZA zcl(A) ≥ 2n + 2 as was required. 5. Orbit configuration spaces Let Γ be a group and X a Γ-space. The orbit configuration space FΓ (X, n) is the space of all ordered n-tuples of points in X which lie in distinct Γ-orbits, FΓ (X, n) = {(x1 , . . . , xn ) | Γ · xi ∩ Γ · xj = ∅ if i = j}. Orbit configuration spaces, introduced by Xicot´encatl [39], are natural generalizations of classical configuration spaces. If Γ = {1} is trivial, F{1} (X, n) = F (X, n) is the classical configuration space. 5.1. Generalized Fadell-Neuwirth theorem. We will focus on the case where X is a connected manifold without boundary of positive dimension, and Γ is a finite group acting freely on X. Let OnΓ denote the union of n distinct orbits, Γ · x1 , . . . , Γ · xn , in X. The Fadell-Neuwirth theorem recorded in Theorem 3.2 was generalized by Xicot´encatl to orbit configuration spaces as follows. Theorem 5.1 ([39]). For  ≤ n, the projection onto the first  coordinates, pΓ : FΓ (X, n) → FΓ (X, ), is a locally trivial bundle, with fiber FΓ (X  OnΓ , n − ). The proof of this result given in [39] is a modification of that of [14] for classical configuration spaces. In the special case  = n − 1 an alternative argument, which informs on the structure of these bundles, is given in [6].

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Assume that the order of the finite group Γ is r, and write Γ = {g1 , . . . , gr }. Define a map from the orbit configuration space to the classical configuration space by sending an n-tuple of points in X to their orbits. That is, define f : FΓ (X, n) → F (X, rn) by f (x1 , . . . , xn ) = (g1 x1 , . . . , gr x1 , g1 x2 , . . . , gr x2 , . . . . . . , g1 xn , . . . , gr xn ). Theorem 5.2 ([6]). The orbit configuration space bundle pΓ : FΓ (X, n + 1) → FΓ (X, n) is equivalent to the pullback of the classical configuration space bundle p : F (X, rn + 1) → F (X, rn) under the map f . Now specialize to the case where the finite cyclic group Γ = Z/rZ acts freely on the manifold X = C∗ by multiplication by the primitive r-th root of unity ζ = exp(2πi/r). The associated orbit configuration space is FΓ (C∗ , n) = {(x1 , . . . , xn ) ∈ (C∗ )n | xj = ζ k xi , i = j, 1 ≤ k ≤ r}, which may be realized as the complement in Cn of the hyperplane arrangement Ar,n k consisting of the hyperplanes Hi = ker(xi ), 1 ≤ i ≤ n, and Hi,j = ker(xi − ζ k xj ), 1 ≤ i < j ≤ n, 1 ≤ k ≤ r. The arrangement Ar,n consists of the reflecting hyperplanes of the full monomial group, the complex reflection group isomorphic to the wreath product of the symmetric group Sn and Γ = Z/rZ. For instance, when r = 2, this is the type B Coxeter group, and π1 (FZ/2Z (C∗ , n)) is the type B pure braid group. Discussions of reflection arrangements, including the full monomial arrangements Ar,n , may be found in references including [35, 41]. Theorem 5.3. TC(FZ/rZ (C∗ , n)) = 2n This result may be established using techniques from the theory of hyperplane arrangements as discussed in Section 2 and below, or by using the group theoretic methods presented in Section 6. 5.2. Arrangements, II. Beginning with work of Arnol’d and Brieskorn (see §2.2), the cohomology ring of the complement of a complex hyperplane arrangement is a well-studied object, facilitating analysis of the (higher) zero divisor cup length in this context. Let A = {H1 , . . . , Hm } be an arrangement of m hyperplanes in V = C . For convenience, we will assume that A is essential, and we will use cohomology with coefficients in C. The Orlik-Solomon theorem [34] shows that H ∗ (M (A); C) is isomorphic to the Orlik-Solomon algebra A(A), the quotient of the exterior algebra E(A) generated by one-dimensional classes ej , 1 ≤ i ≤ m, by a homogeneous ideal I(A). Detailed expositions may be found in [35,40]. Let [m] = {1, . . . , m}, refer to the hyperplanes of A by their subscripts, and order them accordingly. Given S ⊂ [m], denote the flat i∈S Hi by ∩S. If ∩S = ∅, call S independent if the codimension of ∩S in V is equal to |S|, and dependent if codim(∩S) < |S|. If S = (i1 , i2 , . . . , ip ) is an increasingly ordered subset of [m], recall that eS = ei1 ei2 · · · eip denotes the corresponding p k−1 standard monomial in the exterior algebra. Define ∂eS = eS\{ik } . k=1 (−1) The Orlik-Solomon ideal is generated by  {∂eS | S is dependent} {eS | ∩S = ∅}. A circuit is a minimally dependent subset T ⊆ [m], that is, T is dependent, but every nontrivial subset of T is independent. If S is dependent and T ⊂ S a circuit, then eS = ±eT eS\T , and ∂eS = ±∂eT · eS\T ± eT · ∂eS\T . Also, note that if

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∩T = ∅, then ∩S = ∅ for any S containing T . In light of these two observations, the generating set for the Orlik-Solomon ideal given above may be reduced as follows: The ideal I(A) is generated by  (5.1) G = {∂eT | T is a circuit} {eS | S is minimal such that ∩ S = ∅}. The ordering of the hyperplanes of A induces the degree-lexicographic order on the set of standard monomials eS in the exterior algebra E(A). Call a subset S of [m] a broken circuit if there exists k ∈ [m] so that k < i for all i ∈ S and (k, S) is a circuit. Broken circuits correspond to the initial monomials of the elements ∂eT appearing in (5.1). In [40, Theorem 2.8], Yuzvinsky shows that the initial monomials of elements of G generate in(I(A)), the ideal generated by the initial terms of elements of I(A), whence G is a Gr¨ obner basis for the Orlik-Solomon ideal I(A), see [1]. This yields a basis for the quotient A(A) = E(A)/I(A), the Orlik-Solomon algebra of the arrangement A. For S ⊂ [m], let aS denote the image of the standard monomial eS in the Orlik-Solomon algebra A(A). The nbc basis for A(A) consists of all elements aS corresponding to subsets S of [m] which contain no broken circuits [35, 40]. This basis has been used to study the (higher) zero divisor cup length of A(A) by a number of authors, including [23, 41]. Following these references, we restrict our attention to a central arrangement A. Recall that we assume A is essential in C . Consequently, a maximal independent set (resp., nbc set) has cardinality . ¯ = B ∪C. Let Π = (B, C) be an ordered pair of disjoint subsets of [m], and let Π ¯ The pair Π is said to be basic if B and C are nbc sets for some linear order on Π and B is a maximal independent set, |B| = . The central arrangement A is said to be large if there is a basic pair Π = (B, C) with |C| =  − 1. In [41], Yuzvinsky uses basic pairs to find lower bounds on the (higher) zero divisor cup length of the Orlik-Solomon algebra, and proves the following. Theorem 5.4 ([41]). Let A be an essential central arrangement in C with complement M (A), and let s ≥ 2 be a positive integer. If (B, C) is a basic pair, then TCs (M (A)) > (s − 1) + |C|. If A is large, then TCs (M (A)) = s. The arrangements Ar,n in Cn associated to the full monomial groups and arising in the context of cyclic group orbit configuration spaces are large. Recall that Ar,n k has hyperplanes Hi = ker(xi ) and Hi,j = ker(xi −ζ k xj ), where ζ = exp(2πi/r), and take B = {H1 , . . . , Hn } and C = {H1,2 , H1,3 , . . . , H1,n }. Thus, Theorem 5.3 follows from Theorem 5.4. More generally, Yuzvinsky establishes an analogous result for the reflection arrangement associated to any irreducible complex reflection group. A complex reflection in V = C is a finite order linear transformation τ : V → V whose fixed point set is a hyperplane Hτ . A reflection group is a finite subgroup of GL(V ) that is generated by reflections. A reflection group is irreducible if its tautological representation in GL(V ) is irreducible. The reflection arrangement AW associated to the reflection group W is the set of hyperplanes {Hτ | τ a reflection in W }. Theorem 5.5 ([41]). Let W be an irreducible reflection group of rank , and let s ≥ 2 be a positive integer. If AW is the associated reflection arrangement, then TCs (M (AW )) = s. Remark 5.6. If Γ is a simple graph with vertices {1, . . . , n}, the associated graphic arrangement AΓ consists of the hyperplanes ker(xi − xj ) in Cn corresponding to the edges {i, j} of Γ . For instance, if Γ = Kn is the complete graph, then

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AΓ = An is the braid arrangement introduced in Example 2.2. In [24], Fieldsteel uses Yuzvinsky’s result stated in Theorem 5.4 to find conditions on the graph Γ , related to the arboricity, which insure that the (higher) topological complexity of the complement M (AΓ ) of a graphic arrangement is as large as possible. 6. Some discrete groups Let X be an aspherical space, that is, a space whose higher homotopy groups vanish: πi (X) = 0 for i ≥ 2. Farber [19] poses the problem of computing the topological complexity of such a space in terms of algebraic properties of the fundamental group G = π1 (X). In other words, given a discrete group G, define the topological complexity of G to be TC(G) := TC(K(G, 1)), the topological complexity of an Eilenberg-Mac Lane space of type K(G, 1), and express TC(G) in terms of invariants such as the cohomological or geometric dimension of G if possible. Example 6.1. Associated to a simple graph Γ on n vertices is a right-angled Artin group GΓ with generators corresponding to the vertices of Γ , and commutator relators corresponding to the edges. For instance, if Γ = Kn is the complete graph, then GΓ = Zn is free abelian, while if Γ has no edges, then GΓ = Fn is free. For any right-angled Artin group, one has TC(GΓ ) = z(Γ ) + 1, where z(Γ ) is the maximal number of vertices of Γ covered by two (disjoint) cliques in Γ , see [9, 29, 31]. Many of the configuration spaces discussed previously are K(G, 1)-spaces, for surface pure braid groups, for pure braid groups associated to reflection groups, etc. For example, π1 (F (C, n)) = Pn is the Artin pure braid group. From the homotopy exact sequence of the Fadell-Neuwirth bundle F (C, m) → F (C, m − 1), with fiber C{m−1 points} and cross section, we see (inductively) that F (C, n) is a K(Pn , 1)space, and obtain a split, short exact sequence 1 → Fn−1 → Pn → Pn−1 → 1, where Fk is the free group on k generators. Thus, Pn = Fn−1  Pn−1 = Fn−1  (Fn−2  Pn−2 ) = · · · = Fn−1  (· · ·  (F3  (F2  F1 ))) is an iterated semidirect product of free groups. The iterated semidirect product structure of Pn is apparent in the classical presentation of this group. The pure braid group Pn has generators Ai,j , 1 ≤ i < j ≤ n, and relations ⎧ ⎪ if r < s < i < j, ⎪ ⎪Ai,j ⎪ −1 −1 ⎪ ⎪ if r = i < s < j, ⎨Ar,j As,j Ar,j As,j Ar,j −1 −1 (6.1) Ar,s Ai,j Ar,s = Ar,j As,j Ar,j if r < i = s < j, ⎪ ⎪ ⎪[Ar,j , As,j ]Ai,j [Ar,j , As,j ]−1 if r < i < s < j, ⎪ ⎪ ⎪ ⎩A if i < r < s < j, i,j where [u, v] = uvu−1 v −1 denotes the commutator, see, for instance Birman [4]. Observe that, for s < j as in the relations above, the action of Fs−1 = A1,s , . . . , As−1,s  on Fj−1 = A1,j , . . . , Aj−1,j  (via the Artin representation) is by conjugation. It follows that the induced action of Pn−1 on H∗ (Fn−1 , ; Z) is trivial. 6.1. Almost-direct products of free groups. An almost-direct product of free groups is an iterated semidirect product G = Fdn · · ·Fd1 of finitely generated free groups for which Fdi acts trivially on H∗ (Fdj ; Z) for i < j. Thus, Pn is an almost-direct product of free groups. The fundamental groups of the orbit

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configuration spaces FZ/rZ (C∗ , n) considered in the previous section provide another family of examples. Let Γ = Z/rZ, and ζ = exp(2πi/r). The pure braid group Pr,n = π1 (FΓ (C∗ , n)) associated to the full monomial group G(r, n) may also be realized as an almostdirect product of free groups. From Theorems 5.1 and 5.2, the map FΓ (C∗ , n) → FΓ (C∗ , n − 1) defined by forgetting the last coordinate is a bundle, with fiber C∗  {n − 1 orbits} = C  {r(n − 1) + 1 points}. A minor modification of these results is useful in revealing the almost-direct product structure of Pr,n . Given a configuration of m distinct ordered points (x1 , . . . , xm ) in C∗ , one obtains a configuration of m + 1 distinct ordered points (0, x1 , . . . , xm ) in C, yielding a homotopy equivalence F (C∗ , m)  F (C, m + 1). Using this observation, together with Theorem 5.2, one can check that the bundle FΓ (C∗ , n) → FΓ (C∗ , n − 1) may be realized as the pullback of the classical configuration space bundle F (C, N + 1) → F (C, N ) where N = r(n − 1) + 1, under the map g : FΓ (C∗ , n − 1) → F (C, N ) given by g(x1 , . . . , xn−1 ) = (0, ζx1 , . . . , ζ r x1 , ζx2 , . . . , ζ r x2 , . . . . . . , ζxn−1 , . . . , ζ r xn−1 ). It follows that the orbit configuration space bundle FΓ (C∗ , n) → FΓ (C∗ , n − 1) admits a section, and the fundamental group Pr,n−1 of the base acts trivially on the homology of the fiber. Hence, an inductive argument reveals that Pr,n is an almost-direct product of free groups. Under natural assumptions on the ranks of the constituent free groups, the topological complexity of an almost-direct product of free groups was determined by Cohen. Theorem 6.2 ([7]). If G = Fdn  · · ·  Fd1 is an almost-direct product of free groups with dj ≥ 2 for each j, and m is a nonnegative integer, then TC(G × Zm ) = 2n + m + 1. For an arbitrary iterated semidirect product of free groups G = nj=1 Fdj of cohomological dimension n, a K(G, 1)-complex of dimension n is constructed in [10]. Thus, for such groups, the dimensional upper bound on topological complexity may be stated in terms of the cohomological dimension as TC(G) ≤ 2 cd(G) + 1. The integral homology H∗ (G; Z) istorsion-free and the Poincar´e polynomial is  given by P (G, t) = nk=0 bk (G) · tk = nj=1 (1 + dj t), where bk (G) is the k-th Betti number of G, see [15]. A minimal, free ZG-resolution of Z, which we denote by  C• (G) −−→ Z, is constructed in [10]. Let N = b1 (G) = d1 +d2 +· · ·+dn . The abelianization map a : G → ZN induces a chain map a• : C• → K• , where C• = C• (G)⊗ZG ZZN and K• → Z is the standard ZZN -resolution of Z. The induced map a∗ : H ∗ (ZN ; Z) → H ∗ (G; Z) in integral ∼ cohomology is surjective, and is an isomorphism a∗ : H 1 (ZN ; Z) −−→ H 1 (G; Z) in dimension one, see [7, Theorem 2.1]. Let J be the ideal in the exterior algebra H ∗ (ZN ; Z) generated by the elements of the kernel of the surjection a2 : H 2 (ZN ; Z) → H 2 (G; Z), J = (ker(a2 )). An explicit Gr¨obner basis for J is exhibited in [7, §3] (in the degree-lexicographic order on a standard basis for the exterior algebra), and this is used to shown that the integral cohomology ring of G is given by H ∗ (G; Z) ∼ = H ∗ (ZN ; Z)/J. ∗ ∗ Passing to field coefficients, H (−) = H (−; C), if G = nj=1 Fdj is an almostdirect product of free groups with dj ≥ 2 for each j, one can exhibit pairs of generators of xi , yj ∈ H 1 (G) corresponding to distinct generators of the free groups

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Fdj , 1 ≤ j ≤ n. As shown in [7, Theorem 4.2], this yields 2n zero-divisors x ¯j , y¯j in H 1 (G) ⊗ H 1 (G) with nonzero product. These considerations yield TC(G) = 2n + 1 for G as in the statement of the theorem. The general case TC(G×Zm ) = 2n+m+1 may be obtained from this, the product inequality, and a straightforward analysis of the zero-divisor cup length of H ∗ (G × Zm ). 6.2. Fiber-type arrangements. The Artin pure braid group Pn associated to the symmetric group, and, more generally, the pure braid groups Pr,n associated to the full monomial groups may be realized as the fundamental groups of the hyperplane arrangements defined by the polynomials  (xri − xrj ), r ≥ 1. Q(Ar,n ) = x1 · · · xn · 1≤i dim W so that the mechanism is redundant. A standard approach to robot manipulation is the following: given a robot’s initial joint configuration c ∈ C and end-effector’s required final position w ∈ W, one first computes the initial position f (c) ∈ W, and finds a motion from f (c) to w represented by a path α : [0, 1] → W from α(0) = f (c) to α(1) = w. Then the path α is lifted to C starting from the initial point c, thus obtaining a path α & : [0, 1] → C. The lifting α & represents the motion of joints that steers the robot to the required position w, which is is reminiscent of the path-lifting problem in covering spaces or fibrations. However, we know that the kinematic map is not a fibre bundle in general, and so the path lifting must avoid singular points. From the computational viewpoint the most natural approach to path lifting is by solving differential equations but there are also other approaches. We will follow [1] and call any such lifting method a tracking algorithm. Clearly, every (smooth) inverse kinematic map determines a tracking algorithm but the converse is not true in general. In fact, a tracking algorithm determined by inverse kinematics is always cyclic in the sense that it lifts closed paths in W to closed paths in W. Baker and Wampler [2] proved that a tracking algorithm is equivalent to one determined by inverse functions if, and only if the tracking is cyclic. Therefore Theorem 3.1 implies that, notwithstanding the available redundant degrees of freedom, one cannot construct a cyclic tracking algorithm for pointing or orienting. This is not very surprising if we know that most tracking algorithms rely on solutions of differential equations and are thus of a local nature. In particular the most widely used Jacobian method with additional constraints (cf. extended Jacobian method in [25] or augmented Jacobian method in [5]) yields tracking that is only locally cyclic, in the sense that there is an open cover of W, such that closed paths contained in elements of the cover are tracked by closed paths in C. However, this does not really help, because Baker and Wampler [2] (see also [1, Theorem 2.3]) proved that if there is a tracking algorithm defined on an entire W = S 2 or W = SO(3), then there are arbitrarily short closed paths in the working space that are tracked by open paths in C. Therefore Theorem 3.4. The extended Jacobian method (or any other locally cyclic method) cannot be used to construct a tracking algorithm for pointing or orienting a mechanism with revolute joints. Let us also mention that an analogous result can be proved for positioning mechanisms where the working space is a 2- or 3-dimensional disk around the base

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of the mechanism and whose radius is sufficiently big. See [1, Theorem 2.4] and the subsequent Corollary for details. Our final result is again due to Gottlieb [17] and is related to the question of whether it is possible to restrict the angles of the joints to stay away from the singular set Sing(f ) of the Denavit-Hartenberg kinematic map f : (S 1 )n → SO(3). We obtain the following surprising restriction. Theorem 3.5. Let M be a closed smooth manifold. Then there does not exist a smooth map s : M → (C − Sing(f )) such that the map f ◦ s : M → SO(3) is a submersion (i.e. non-singular). The proof is based on a simple lemma which is also of independent interest. Lemma 3.6. The Denavit-Hartenberg map f : (S 1 )n → SO(3) can be factored up to a homotopy as (S 1 )n EE EE E m EEE "

f

S1

/ SO(3) y< y y y yy g yy

where m is the n-fold multiplication map in S 1 and g is the generator of π1 (SO(3)). Proof. First observe that a rotation of the z-axis in the Denavit-Hartenberg frame of any joint induces a homotopy between the resulting forward kinematic maps. Therefore, we may deform the robot arm until all z-axes are parallel (so that the arm is effectively planar). Then the rotation angle of the end-effector is simply the sum of the rotations around each axis. As the sum of angles correspond to the multiplication in S 1 , it follows that the Denavit-Hartenberg map factors as f = g ◦ m for some map g : S 1 → SO(3). Clearly, g must generate π1 (SO(3)) = Z2 because f induces an epimorphism of fundamental groups.  Proof. (of Theorem 3.5) Assume that there exists s : M → (C − Sing(f )) such that f ◦ s : M → SO(3) is a submersion. It is well-known that the projection of an orthogonal 3 × 3-matrix to its last column determines a map p : SO(3) → S 2 that is also a submersion. Then the composition M

s

/ C − Sing(f )

f

/ SO(3)

p

/ S2

is a submersion, and hence a fibre bundle by Ehresmann’s theorem [11]. It follows that the fibre of p ◦ f ◦ s is a closed submanifold of M . On the other side, p ◦ f ◦ s is homotopic to the constant, because by Lemma 3.6 f factors through S 1 and S 2 is simply-connected. This leads to a contradiction, as the fibre of the constant map M → S 2 is homotopy equivalent to M ×ΩS 2 , which cannot be homotopy equivalent  to a closed manifold, because ΩS 2 has infinite-dimensional homology. In particular, Theorem 3.5 implies that even if we add constraints that restrict the configuration space of the robotic device to some closed submanifold C  of the set of non-singular configurations of the joints, the restriction f : C  → SO(3) of the Denavit-Hartenberg kinematic map still has singular points. Clearly, the new singularities are not caused by the configuration of joints but are a consequence of the constraints that define C  .

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4. Overview of topological complexity The concept of topological complexity was introduced by M. Farber in [12] as a qualitative measure of the difficulty in constructing a robust motion plan for a robotic device. Roughly speaking, motion planning problem for some mechanical device requires to find a rule (’motion plan’) that yields a continuous trajectory from any given initial position to a desired final position of the device. A motion plan is robust if small variations in the input data results in small variations of the connecting trajectory. Toward a mathematical formulation of the motion planning problem one considers the space C of all positions (’configurations’) of the device, and the space C I of all continuous paths α : I → C. Let π : C I → C × C be the evaluation map given by π(α) = (α(0), α(1)). A motion plan is a rule that to each pair of points c, c ∈ C assigns a path α(c, c ) ∈ C I such that π(α(c, c )) = (c, c ). For practical reason we usually require robust motion plans, i.e. plans that are continuously dependent on c and c . Clearly, robust motion plans are precisely the continuous sections of π. Farber observed that a continuous global section of π exists if, and only if, C is contractible. For non-contractible spaces one may consider partial continuous sections, so he defined the topological complexity TC(C) as the minimal n for which C × C can be covered by n open sets each admitting a continuous section to π. Note that other authors prefer the ’normalized’ topological complexity (by one smaller that our TC) as it sometimes leads to simpler formulas. We list some basic properties of the topological complexity: (1) It is a homotopy invariant, i.e. C  C  implies TC(C) = TC(C  ). (2) There is a fundamental estimate cat(C) ≤ TC(C) ≤ cat(C × C), where cat(C) denotes the (Lusternik-Schnirelmann) category of C (see [7]). (3) Furthermore   TC(C) ≥ nil Ker Δ∗ : H ∗ (C × C) → H ∗ (C) . Here Ker Δ∗ is the kernel of the homomorphism between cohomology rings induced by the diagonal map Δ : C → C×C, and nil(Ker Δ∗ ) is the minimal n such that every product of n elements in KerΔ∗ is zero. There are many other results and explicit computations of topological complexity – see [14] for a fairly complete survey of the general theory. 5. Complexity of a map In [23] we extended Farber’s approach to study the more general problem of robot manipulation. Robots are usually manipulated by operating their joints in a way to achieve a desired pose of the robot or a part of it (usually called end-effector ), so we must take into account the kinematic map which relates the internal joints states with the position and orientation of the end-effector. To model this situation, we take a map f : C → W and consider the projection map πf : C I → C ×W, defined as   πf (α) := (1 × f )(π(α)) = α(0), f (α(1)) . Similarly to motion plans, a manipulation plan corresponds to a continuous sections of πf , so it would be natural to define the topological complexity TC(f ) as the minimal n such that C × W can be covered by n sets, each admitting a continuous

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section to πf . This is analogous to the definition of TC(C), but there are two important issues that we must discuss before giving a precise description of TC(f ). In the definition of TC(C) Farber considers continuous sections whose domains are open subsets of C × C. In most applications C is a nice space (e.g. manifold, semi-algebraic set,...), and in fact Farber [14] shows that for such spaces alternative definitions of topological complexity based on closed, locally compact or ENR domains yield the same result. This was further generalized by Srinivasan [26] who proved that if C is a metric ANR space, then every section over an arbitrary subset Q ⊂ C can be extended to some open neighbourhood of Q. Therefore, for a very general class of spaces (including metric ANRs) one can define topological complexity by counting sections of the evaluation map π : C I → C ×C over arbitrary subsets of the base. Another important fact is that the map π is a fibration, which implies that one can replace sections by homotopy sections in the definition of TC(C) and still get the same result. This relates topological complexity with the so called Schwarz genus [24], a well-established and extensively studied concept in homotopy theory. The genus g(h) of a map h : X → Y is the minimal n such that Y can be covered by n open subsets, each admitting a continuous homotopy section to h; the genus is infinite if there is no such n. Therefore, we have TC(C) = g(π), a result that puts topological complexity squarely within the realm of homotopy theory. The situation is less favourable when it comes to the complexity of a map. Firstly, πf : C I → C × W is a fibration if, and only if, f : C → W is a fibration, and that is an assumption that we do not wish to make in view of our intended applications (cf. Theorem 3.3). Every section is a homotopy section but not viceversa, and in fact, the minimal number of homotopy sections for a given map can be strictly smaller than the number of sections. For example, the map h : [0, 3] → [0, 2] given by ⎧ t ∈ [0, 1] ⎨ t 1 t ∈ [1, 2] h(t) := ⎩ t − 1 t ∈ [2, 3] (see Figure 10) admits a global homotopy section because its codomain is con-

Figure 10. Projection with genus 1 and sectional number 2. tractible, but clearly there does not exist a global section to h. Furthermore, the following example (which can be easily generalized) shows that the difference between the minimal number of sections and the minimal number of homotopy sections can be arbitrarily large. Actually, many results on topological complexity depend heavily on the fact that the evaluation map π : C I → C × C is a fibration, and so some direct generalizations of the results about TC(C) are harder to prove while other are simply false.

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Figure 11. Projection with genus 1 and sectional number 3. The second difficulty is related to the type of subsets of C ×W that are domains of sections to πf . While the spaces C or W are usually assumed to be nice (e.g. manifolds), the map f can have singularities which leads to the problem that we explain next. Given a subset Q of C × W and a point c ∈ C let Q|c be the subset of W defined as Q|c := {w ∈ W | (c, w) ∈ Q}. Assume that Q|c is non-empty and that there is a partial section α : Q → C I to πf . Then the map αc : Q|c → C

αc (w) := α(c, w)(1) .

satisfies f (αc (w)) = w, so αc is a partial section to f . Furthermore H : Q|c ×I → C, given by H(w, t) := α(c, w)(1 − t) deforms the image of the section αc (Q|c ) in C to the point c, while f ◦ H deforms Q|c in W to the point f (c). These observations have several important consequences. Assume that (c, w) is an interior point of some domain Q ⊂ C × W of a partial section to πf . Then w is an interior point of Q|c that admits a partial section to f . Therefore, if f is not locally sectionable around w (like the previously considered map h : [0, 3] → [0, 2] around the point 1), then it is impossible to find an open cover of C × W that admits partial sections to πf . A similar argument shows that we cannot use closed domains for a reasonable definition of the complexity of f . One way out would be to follow the approach by Srinivasan [26] and consider sections with arbitrary subsets as domains, but that causes problems elsewhere. After some balancing we believe that the following choice is best suited for applications. Let C and W be path-connected spaces, and let f : C → W be a surjective map. The topological complexity TC(f ) of f is defined as the minimal n for which there exists a filtration of C × W by closed sets ∅ = Q0 ⊆ Q1 ⊆ . . . ⊆ Qn = C × W, such that πf admits partial sections over Qi − Qi−1 for i = 1, 2, . . . , n. By taking complements we obtain an equivalent definition based on filtrations of C × W by open sets. If W is a metric ANR, then g(πf ) ≤ TC(f ), and the two coincide if f is a fibration. Suppose TC(f ) = 1, i.e. there exists a section α : C × W → C I to πf . Then (C × W)|c = W for every c ∈ C, and by the above considerations, f : C → W admits a global section that embeds W as a categorical subset of C. Even more, W can be deformed to a point within W, so W is contractible and TC(W) = cat(W) = 1.

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To get a more general statement, let us say that a partial section s : Q → C to f : C → W is categorical if s(Q) can be deformed to a point within C. Then we define csec(f ) to be the minimal n so that there is a filtration ∅ = A0 ⊆ A1 ⊆ . . . ⊆ An = W, by closed subsets, such that f admits a categorical section over Ai − Ai−1 for i = 1, . . . , n (and csec(f ) = ∞, if no such n exists). For W a metric ANR space we have csec(f ) ≥ cat(W) because A = f (s(A)) is contractible in W for every categorical section s : A → C. If furthermore f : C → W is a fibration, then csec(f ) = cat(W). Theorem 5.1. Let W be a metric ANR space and let f : C → W be any map. Then cat(W) ≤ csec(f ) ≤ TC(f ) < csec(f ) + cat(C) . Proof. If TC(f ) = n, then we have shown before that there exists a cover of W by n sets, each admitting a categorical section to f , therefore TC(f ) ≥ csec(f ). As a preparation for the proof of the upper estimate, assume that C ⊆ C admits a deformation H : C × I → C to a point c0 ∈ C, and that A ⊆ W admits a categorical section s : A → C with a deformation K : s(A) × I → C to a point c1 ∈ C. In addition, let γ : I → C be a path from c0 to c1 . Then we may define a partial section α : C × A → C I by the formula ⎧ 0 ≤ t ≤ 1/3 ⎨ H(c, 3t) γ(3t − 1) 1/3 ≤ t ≤ 2/3 α(c, w)(t) := ⎩ K(s(w), 2 − 3t) 2/3 ≤ t ≤ 1 By assumption, there is a filtration of C by closed sets ∅ = C0 ⊆ C1 ⊆ . . . ⊆ Ccat(C) = C, such that each difference Ci − Ci−1 deforms to a point in C, and there is also a filtration of W by closed sets ∅ = A0 ⊆ A1 ⊆ . . . ⊆ Acsec(f ) = W, such that each difference Ai − Ai−1 admits a categorical section to f . Then we can define closed sets  Ci × Aj Qk = i+j=k

that form a filtration ∅ = Q1 ⊆ Q2 ⊆ · · · ⊆ Qcat(C)+csec(f ) = C × W . Each difference Qk − Qk−1 =



(Ci − Ci−1 ) × (Aj − Aj−1 )

i+j=k

is a mutually separated union of sets that admit continuous partial sections, and so there exists a continuous partial section plan on each Qk − Qk−1 . We conclude that TC(f ) is less then or equal to cat(C) + csec(f ) − 1.  Topological complexity of a map can be used to model several important situations in topological robotics. In the rest of this section we describe some typical examples: Example 5.2. The identity map on X is a fibration, so if X is a metric ANR, then TC(X) = TC(IdX ).

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Example 5.3. In the motion planning of a device with several moving components one is often interested only in the position of a part of the system. This situation may be modelled by considering the projection p : C → C  of the configuration space of the entire system to the configuration space of the relevant part. Then TC(p) measures the complexity of robust motion planning in C but with the objective to arrive at a requested state in C  . A similar situation which often arises and can be modelled in this way is when the device can move and revolve in three dimensional space (so that its configuration space C is a subspace of R3 × SO(3)), but we are only interested in its final position (or orientation), so we consider the complexity of the projection p : C → R3 (or p : C → SO(3)). Example 5.4. Our main motivating example is the complexity of the forward kinematic map of a robot as introduced in [23]. A mechanical device consists of rigid parts connected by joints. As we explained in the first part of this paper (see also [27, Section 1.3]), although there are many types of joints only two of them are easily actuated by motors – revolute joints (denoted as R) and prismatic or sliding joints (denoted as P). Revolute joints allow rotational movement so their states can be described by points on the circle S 1 . Sliding joints allow linear movement with motion limits, so their states are described by points of the interval I. Other joints are usually passive and only restrict the motion of the device, so a typical configuration space C of a system with m revolute joints and n sliding joints is a subspace of the product (S 1 )m × I n . Motion of the joints results in the spatial displacement of the device, in particular of its end-effector. The pose of the endeffector is given by its spatial position and orientation, so the working space W of the device is a subspace of R3 × SO(3) (or a subspace of R2 × SO(2) if the motion of the device is planar). In the following examples, for simplicity, we will disregard the orientation of the end-effector. Given two revolute joints that are pinned together so that their axes of rotation are parallel, the configuration space is C = S 1 × S 1 and the working space is an annulus W = S 1 × I. The forward kinematic map f : C → W can be given explicitly in terms of polar coordinates. This configuration is depicted in Figure 12 with the mechanism and its working space overlapped, and the complexity of the corresponding kinematic map is 3, see [23, 4.2]. If instead we pin the joints so that the axes of rotation are orthogonal then we obtain the so-called universal or Cardan joint. The configuration spaces is a product of circles C = S 1 × S 1 , but the working space is the two dimensional sphere and the forward kinematic map may be expressed by geographical coordinates (see Figure 13). By the computation in [23, 4.3] the complexity of the kinematic map for the universal joint is either 3 or 4 (we do not know the exact value). One of the most commonly used joint configurations in robotics is SCARA (Selective Compliant Assembly Robot Arm), which is based on the (RRP) configuration as in Figure 14, and is sometimes complemented with a screw joint or even with a 3 degrees-of-freedom robot hand. The configuration space is C = S 1 × S 1 × I and the working space is W = S 1 ×I ×I. The forward kinematic map may be easily given in terms of cylindrical coordinates. Since the kinematic map is the product of the kinematic map for the planar two-arm mechanism and the identity map on the interval, its complexity is equal to 3.

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Figure 12. (RR) planar configuration. Position of the arm is completely described by the angles α and β, C = S 1 × S 1 , W = S 1 × [R1 − R2 , R1 + R2 ].

Figure 13. (RR) universal joint. C = S 1 × S 1 , W = S 2 . The last computation can be easily generalized to products of arbitrary maps and we omit the proof since it follows the standard lines as in [22, 3.8]. Proposition 5.5. The product of maps f : C → W and f  : C  → W  satisfies the relation max{TC(f ), TC(f  )} ≤ TC(f × f  ) < TC(f ) + TC(f  ) . Example 5.6. Robotic devices are normally employed to perform various functions and it often happens that different states of the device are functionally equivalent (say for grasping, welding, spraying or other purposes as in Figure 15). Functional equivalence is often described by the action of some group G on C and there

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Figure 14. (RRP) SCARA design. C = S 1 × S 1 × I, W = S 1 × I × I.

Figure 15. (RRR) design with functional equivalence for grasping. are several versions of equivariant topological complexity – see [6], [20], [10] or [3]. Some of them require motion plans to be equivariant maps defined on invariant subsets of C × C, while other consider arbitrary paths that are allowed to ’jump’ within the same orbit (see [3, Section 2.2] for an overview and comparison of different approaches). However, none of the mentioned papers give a convincing interpretation in terms of a motion planning problem for a mechanical system. We believe that the navigation planning for a device with a configuration space C in which different configurations can have the same functionality should be modelled in terms of the complexity of the quotient map q : C → C/ ∼ associated to an equivalence relation on C (or q : C → C/G if the equivalence by a group action). Then TC(q) can be interpreted as a measure of the difficulty in constructing a robust motion plan that steers a device from a given initial position to any of the final positions that have the required functionality. It would be interesting to relate this concept with the above mentioned versions of equivariant topological complexity. 6. Instability of robot manipulation Let us again consider the robot manipulation problem determined by a forward kinematic map f : C → W. A manipulation algorithm for the given device

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is a rule that to every initial datum (c, w) ∈ C × W assigns a path in C starting at c and ending at c ∈ f −1 (w). In other words, a manipulation algorithm is a (possibly discontinuous) section of πf : C I → C × W, patched from one or more robust manipulation plans. Let αi : Qi → C I be a collection of robust manipulation plans such that the Qi cover C × W. In general the domains Qi may overlap, so in order to define a manipulation algorithm for the device we must decide which manipulation plan to apply for a given input datum (c.w) ∈ C × W. We can avoid this additional step by partitioning C × W into disjoint domains, e.g. by defining Q1 := Q1 , Q2 := Q2 −Q1 , Q3 := Q3 −Q2 −Q1 , . . ., and restricting the respective manipulation plans accordingly. Since C and W are by assumption path-connected, if we partition C × W into several domains, then there exist arbitrarily close pairs of initial data (c, w) and (c , w ) that belong to different domains. This can cause instability of the robot device guided by such a manipulation algorithm in the sense that small perturbations of the input may lead to completely different behaviour of the device. The problem is of particular significance when the input data is determined up to some approximation or rounding because the instability may cause the algorithm to choose an inadequate manipulation plan. Also, a level of unpredictability significantly complicates coordination in a group of robotic devices, because one device cannot infer the action of a collaborator just by knowing it’s manipulation algorithm and by determining its position. Farber [13] observed that for any motion planning algorithm the number of different choices that are available around certain points in X increases with the topological complexity of X. He defined the order of instability of a motion planning algorithm for X at a point (x, x ) ∈ X ×X to be the number of motion plan domains that are intersected by every neighbourhood of (x, x ). Then he proved ([13, Theorem 6.1]) that for every motion planning algorithm on X there is at least one point in X × X whose order of instability is at least TC(X). We are going to state and prove a similar result for the topological complexity of a forward kinematic map. Our proof is based on the approach used by Fox [15] to tackle a similar question on Lusternik-Schnirelmann category. Theorem 6.1. Let f : C → W be any map and let C × W = Q1  . . .  Qn be a partition of C × W into disjoint subsets, each of them admitting a partial section αi : Qi → C I of πf . Then there exists a point (c, w) ∈ C × W such that every neighbourhood of (c, w) intersects at least TC(f ) different domains Qi . Proof. If every neighbourhood of (c, w) intersects Qi then (c, w) is in Qi , the closure of Qi . Therefore, we must prove that there exist TC(f ) different domains Qi such that their closures have non-empty intersection. To this end for each k = 1, 2, . . . , n we define Rk as the set of points in C × W that are contained in at least k sets Qi . Each Rk is a union of intersections of sets Qi , hence it is closed, and we obtain a filtration C × W = R1 ⊇ R2 ⊇ . . . Rm ⊇ ∅, where m is the biggest integer such that Rm is non-empty. For each k = 1, . . . , m the difference Rk − Rk+1 consists of points that are contained in exactly k sets Qi . To construct a manipulation plan over Rk − Rk+1 let us define sets SI for every subset of indices I ⊆ {1, . . . , n} as the set of points that are contained in Qi if i ∈ I and are not contained in Qi if i ∈ / I. It is easy to check that Rk − Rk+1 is the

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disjoint union of sets SI where I ranges over all k-element subsets of {1, . . . , n}. Even more, if I and J are different but of the same cardinality, then the closure of SI does not intersect SJ (i.e., SI and SJ are mutually separated ). In fact, there is an index i contained in I but not in J, and clearly S I ⊆ Qi while SJ ∩ Qi = ∅. Since for I of fixed cardinality k the sets SI are mutually separated we can patch a continuous section βk : Rk − Rk+1 → C I to πf by choosing i ∈ I for each I and defining βk |SI := αi |SI . By definition of TC(f ) we must have m ≥ TC(f ) so that RTC(f ) is non-empty and there exists a point in C × W that is contained in at least TC(f ) different sets Qi .  The order of instability of a manipulation algorithm with n robust manipulation plans clearly cannot exceed n, so there is always a cover of C × W by sets admitting section to πf , whose order of instability is exactly TC(f ). As a corollary we obtain the characterization of TC(f ): it is the minimal n for which every manipulation plan on C × W has order of instability at least n. In applications the robot manipulation problem is often solved numerically, using gradient flows or successive approximations. Again, one may identify domains of continuity as well as regions of instability. It would be an interesting project to compare different approaches with respect to their order of instability. References [1] Daniel R. Baker, Some topological problems in robotics, Math. Intelligencer 12 (1990), no. 1, 66–76, DOI 10.1007/BF03023989. MR1034876 [2] D. R. Baker, C. W. Wampler, On the inverse kinematics of redundant manipulators, International Journal of Robotics Research, 7 (1988), 3–21. [3] Z. Baszczyk, M. Kaluba, Yet another approach to equivariant topological complexity, arXiv: 1510.08724 (2015). [4] M. Brady (ed.), Robotics Science (MIT Press, 1989). [5] S. Chiaverini, G. Oriolo, I. D. Walker, Kinematically Redundant Manipulators, in B. Siciliano, O. Khatib (eds.), Springer Handbook of Robotics, Chapter 1, (Springer, Berlin, 2008). [6] Hellen Colman and Mark Grant, Equivariant topological complexity, Algebr. Geom. Topol. 12 (2012), no. 4, 2299–2316, DOI 10.2140/agt.2012.12.2299. MR3020208 [7] Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanr´ e, Lusternik-Schnirelmann category, Mathematical Surveys and Monographs, vol. 103, American Mathematical Society, Providence, RI, 2003. MR1990857 [8] J. Denavit and R. S. Hartenberg, A kinematic notation for lower-pair mechanisms based on matrices, J. Appl. Mech. 22 (1955), 215–221. MR0068936 [9] P. S. Donelan, Singularities of robot manipulators, Singularity theory, World Sci. Publ., Hackensack, NJ, 2007, pp. 189–217, DOI 10.1142/9789812707499 0006. MR2342912 [10] Alexander Dranishnikov, On topological complexity of twisted products, Topology Appl. 179 (2015), 74–80, DOI 10.1016/j.topol.2014.08.017. MR3270927 [11] Charles Ehresmann, Les connexions infinit´ esimales dans un espace fibr´ e diff´ erentiable (French), Colloque de topologie (espaces fibr´ es), Bruxelles, 1950, Georges Thone, Li` ege; Masson et Cie., Paris, 1951, pp. 29–55. MR0042768 [12] Michael Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211–221, DOI 10.1007/s00454-002-0760-9. MR1957228 [13] Michael Farber, Instabilities of robot motion, Topology Appl. 140 (2004), no. 2-3, 245–266, DOI 10.1016/j.topol.2003.07.011. MR2074919 [14] Michael Farber, Invitation to topological robotics, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Z¨ urich, 2008. MR2455573 [15] Ralph H. Fox, On the Lusternik-Schnirelmann category, Ann. of Math. (2) 42 (1941), 333– 370, DOI 10.2307/1968905. MR0004108

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[16] D. Gottlieb, Robots and topology, Proc. 1986 IEEE International Conference on Robotics and Automation, Vol 3., 1689–1691. [17] Daniel H. Gottlieb, Robots and fibre bundles, Bull. Soc. Math. Belg. S´ er. A 38 (1986), 219–223 (1987). MR885533 [18] Daniel H. Gottlieb, Topology and the robot arm, Acta Appl. Math. 11 (1988), no. 2, 117–121, DOI 10.1007/BF00047283. MR937313 [19] J.M. Hollerbach, Optimal kinematic design for a seven degree of freedom manipulator, 2nd International Symposium on Robotics Research, Kyoto, (Japan, 1984). [20] Wojciech Lubawski and Waclaw Marzantowicz, Invariant topological complexity, Bull. Lond. Math. Soc. 47 (2015), no. 1, 101–117, DOI 10.1112/blms/bdu090. MR3312969 [21] Richard N. Murray, Ze Xiang Li, and S. Shankar Sastry, A mathematical introduction to robotic manipulation, CRC Press, Boca Raton, FL, 1994. MR1300410 [22] P. Paveˇsi´ c, Formal aspects of topological complexity, in A.K.M. Libardi (ed.), Zbirnik prac Institutu matematiki NAN Ukraini ISSN 1815-2910, T. 6, (2013), 56–66. [23] P. Paveˇsi´ c, Complexity of the Forward Kinematic Map, to appear. [24] A.S. Schwarz, The genus of a fiber space, Amer. Math. Soc. Transl. (2) 55 (1966), 49–140. [25] B. Siciliano, Kinematic Control of Redundant Manipulator: A Tutorial, Journal of Intelligent and Robotic Systems 3 (1990), 201–212. [26] Tulsi Srinivasan, The Lusternik-Schnirelmann category of metric spaces, Topology Appl. 167 (2014), 87–95, DOI 10.1016/j.topol.2014.03.009. MR3193429 [27] K. Waldron, J. Schmiedeler, Kinematics, in B. Siciliano, O. Khatib (eds.), Springer Handbook of Robotics, Chapter 1, (Springer, Berlin, 2008). Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia Email address: [email protected]

Contemporary Mathematics Volume 702, 2018 http://dx.doi.org/10.1090/conm/702/14106

On the cohomology classes of planar polygon spaces Donald M. Davis Abstract. We obtain an explicit formula for the Poincar´ e duality isomorphism H n−3 (M (); Z2 ) → Z2 for the space of isometry classes of n-gons with specified side lengths, if  is monogenic in the sense of Hausmann-Rodriguez. This has potential application to topological complexity.

1. Main theorem If  = (1 , . . . , n ) is an n-tuple of positive real numbers, let M () denote the space of isometry classes of oriented n-gons in the plane with the prescribed side lengths. In [4], a complete description of H ∗ (M (); Z2 ) was given in terms of generators and a complicated set of relations. In [1], explicit calculations were made in H ∗ (M (); Z2 ) for length vectors  satisfying certain conditions, enabling us to prove that, for these , the topological complexity of M () satisfied (1.1)

2n − 6 ≤ TC(M ()) ≤ 2n − 5.

This is a result in topological robotics, as it specifies the number of motion planning rules required for a certain n-armed robot.([3]) However, our result only applied to a very restricted set of length vectors . The groups H k (M (); Z2 ) are spanned by monomials Rk−r Vj1 · · · Vjr for distinct positive subscripts j ≤ n − 1. Here R and Vj are elements of H 1 (M (); Z2 ). Since M () is an (n − 3)-manifold, there is a Poincar´e duality isomorphism (1.2)

φ : H n−3 (M (); Z2 ) → Z2 .

For the cases considered in [1], we obtained an explicit formula for φ(Rn−3−r Vj1 · · · Vjr ). The contribution of this paper is to extend that formula to a broader class of length vectors. Note that it tells for each monomial whether it is 0 or the nonzero class, hence the title. Our result here was subsequently applied in [2]. In order to describe these length vectors, we review the notion of genetic code introduced in [5]. Since permuting the length vectors does not affect the homeomor(), we may phism type of M assume that 1 ≤ · · · ≤ n . A subset S ⊂ {1, . . . , n} is called short if i∈S i < i ∈S i . We assume that  is generic, which says that   there are no subsets S for which i∈S i = i ∈S i . We define a partial order 2010 Mathematics Subject Classification. Primary 55M30, 58D29, 55R80. Key words and phrases. Topological complexity, planar polygon spaces, cohomology. c 2018 American Mathematical Society

85

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DONALD M. DAVIS

on sets of integers by {s1 , . . . , sk } ≤ T if there exist distinct t1 , . . . , tk in T with si ≤ ti . The genetic code of  is the set of maximal elements in the set of short subsets for  which contain n. An element in the genetic code is called a gene. One of the main theorems of [1] was that, with three exceptions, (1.1) holds if  has a single gene of size 4. In order to prove this, we needed and obtained the explicit formula for φ(Rn−3−r Vj1 · · · Vjr ) for such length vectors.([1, Thm 4.1]) In this paper, we extend this formula to all monogenic codes. We hope that this formula will enable us to study the cohomological implications for topological complexity of these spaces. The huge variety of genetic codes makes it seem unlikely that a formula such as ours might be extended to all genetic codes. In [1], we introduced the term gee to refer to a gene with the n omitted. This is sensible since, for all genes G, n ∈ G. Also, most of the formulas do not involve n. We say that a subgee is any set of positive integers which is less than or equal to a gee. Thus the subgees are all sets S for which S ∪ {n} is short. We write the elements of a gee in decreasing order. Only those {j1 , . . . , jr } which are subgees can have Rk−r Vj1 · · · Vjr = 0.([4, Cor 9.2]) Now we can state our theorem. It involves the following new definition. Definition 1.3. Let Sk denote the set of k-tuples of nonnegative integers such that, for all i, the sum of the first i components of the k-tuple is ≤ i.  For B = (b1 , . . . , bk ), let |B| = bi . Theorem 1.4. Suppose  has a single gee, G = {g1 , . . . , gk } with ai = gi − gi+1 > 0. (gk+1 = 0.) If J is a set of distinct positive integers ≤ g1 , let θ(J) = (θ1 , . . . , θk ), where θi is the number of elements j ∈ J satisfying gi+1 < j ≤ gi . Then, for φ as in ( 1.2), (1.5)

φ(R

n−3−r

V j1 · · · V jr ) =

k  

 ,

ai +bi −2 bi

B i=1

where B ranges over all (b1 , . . . , bk ) for which |B| = k − r and B + θ({j1 , . . . , jr }) ∈ Sk . Note that J = {j1 , . . . , jr } is a subgee if and only if θ(J) ∈ Sk . For example, if two j’s are greater than g2 , then θ(J) has first component greater than 1, so θ(J) is not in Sk , and J cannot be ≤ G since it has two elements greater than the second largest element of G. Thus (1.5) is only relevant when J is a subgee, but it yields 0 in other cases, anyway. An important special case of the theorem appears in the following corollary. Corollary 1.6. If J is a subgee with r = k, then φ(Rn−3−r Vj1 · · · Vjr ) = 1. The following elegant independent proof of this corollary was provided by the referee. Proof of Corollary 1.6. First note that Vj1 · · · Vjk is nonzero, since this is also true in H ∗ (M (); Z e duality, 2 )/R, which is an exterior face ring. By Poincar´ there must be an X = i Xi ∈ H n−3−k (M (); Z2 ) with X · Vj1 · · · Vjk = 1. If Xi contains as a factor any Vt with t ∈ {j1 , . . . , jk }, then Xi · Vj1 · · · Vjk = 0 since {t, j1 , . . . , jk } is not a subgee. Any factors Vt with t ∈ {j1 , . . . , jk } can be replaced by R, by the relation Vt2 = RVt . Thus each Xi with Xi · Vj1 · · · Vjk = 0 can be replaced by Rn−3−k , and the number of such Xi must be odd.

ON THE COHOMOLOGY CLASSES OF PLANAR POLYGON SPACES

87

In working with our formula, it is useful to denote by YT any term Rn−3−r Vj1 · · · Vjr for which θ({j1 , . . . , jr }) = T . The reader can verify that the case k = 3 of Theorem 1.4 agrees with [1, Thm 4.1], when the latter is expressed as in Example 1.7. For example, in φ(Y0,1,0 ), B = (0, 0, 2), (1, 0, 1), and (0, 1, 1) satisfy B + (0, 1, 0) ∈ S3 , but B = (1, 1, 0) does not, since the sum of the first two entries of (1, 2, 0) is greater than 2. (Our method of subscripting Y here differs from that in [1].) This φ(Y0,1,0 ) refers to φ(Rn−4 Vj ) for g3 < j ≤ g2 . Example 1.7. If  has a single gee {a1 + a2 + a3 , a2 + a3 , a3 }, then, writing ai for ai − 1, φ(YT ) φ(Y0,2,0 ) = φ(Y1,1,0 ) φ(Y1,0,1 ) φ(Y0,0,2 ) = φ(Y0,1,1 )

= = = =

φ(Y1,0,0 )

=

φ(Y0,1,0 )

=

φ(Y0,0,1 )

=

φ(Y0,0,0 )

=

1 if |T | = 3 and T ∈ S3 a3 a2 + a3 a + a + a a13  2   3 + a2 a3 a23  + a1 a3 + a2 a3 a22  a3  + + a1 a2 + a1 a3 + a2 a3 a23 +1 2a2   a3   + 2 a3 + 2 (a1 + a2 ) + a1 a2 a3 . 3

2. Proof In this section, we prove Theorem 1.4. As noted above, H n−3 (M (); Z2 ) is spanned by monomials Rn−3−r Vj1 · · · Vjr for which J = {j1 , . . . , jr } ≤ G, i.e., J is a subgee. Using [4, Cor 9.2] as interpreted in [1, Thm 2.1], a complete set of relations is given by relations RI for each subgee I except the empty set. This relation RI says (2.1)



Rn−3−|J|



Vj = 0,

j∈J

J ∩I


where the sum is taken over all subgees J disjoint from I. To prove our theorem, it suffices to show that our proposed φ sends each relation RI to 0, since it will then be the unique nonzero homomorphism (1.2). Similarly to [1, (4.2)], the number of subgees J disjoint from I and satisfying k  ai −mi  , if θ(I) = (m1 , . . . , mk ). Note that mi ≤ ai θ(J) = C = (c1 , . . . , ck ) is ci i=1

since mi is the size of a subset of ai integers. Since our φ applied to a term in (2.1) is determined by θ(J), φ(RI ) becomes (2.2)

k  

ai −mi ci

C i=1

k  ai +bi −2 , bi B i=1

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DONALD M. DAVIS

where |B| = k − |C| and B + C ∈ Sk . Letting T = B + C, this can be rewritten, with the outer sum taken over all T ∈ Sk satisfying |T | = k, as k  ai −mi ai +bi −2 ti −bi

=

k   

ai −mi ti −bi

k   

ai −mi ti −bi

=

k  

1−mi ti

1−ai  bi

mod 2

 .

i=1

T

a+b−2

bi

i=1 bi

T

(2.3)

ai +bi −2

i=1 bi

T

≡

bi

B≤T i=1

T

  = ± 1−a Here we use that b b , and the Vandermonde identity. The sum (2.3) can be considered as a sum Σ1 over all k-tuples T of nonnegative over those which are not in Sk . The sum integers summing  k minus the sum Σ   to 2 mi = 0, but that case has been excluded. Σ1 equals k− k mi , which is 0 unless (I = ∅.) If tj , . . . , tk are nonnegative integers for which tj + · · · + tk ≤ k − j, but tj  + · · · + tk > k − j  for all j  > j, let U (tj , . . . , tk ) denote the set of k-tuples T indexing the sum Σ2 which end with (tj , . . . , tk ). Since tj + · · · + tk ≤ k − j is equivalent to saying that the sum of the first j − 1 components is not ≤ j − 1, these sets U (tj , . . . , tk ) partition the set of T ’s which occur in Σ2 . We show that the sum Σ2 restricted to any such set U (tj , . . . , tk ) is 0, which will complete the proof. We have 

k  

1−mi ti



T ∈U(tj ,...,tk ) i=1

=

k  

1−mi ti

·



j−1 

1−mi  ti

|T  |=k−tj −···−tk i=1

i=j

=



k  

1−mi ti

 j−1−m1 −···−mj−1  · . k−tj −···−tk

i=j 

In the second line, T = (t1 , . . . , tj−1 ). Since (m1 , . . . , mk ) arises from a subgee, m1 +· · ·+mj−1 ≤ j −1. But k −tj −· · ·−tk ≥ j. Thus the final binomial coefficient consists of a nonnegative integer atop a larger integer, and hence is 0.

ON THE COHOMOLOGY CLASSES OF PLANAR POLYGON SPACES

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References [1] Donald M. Davis, Topological complexity of planar polygon spaces with small genetic code, Forum Math. 29 (2017), no. 2, 313–328, DOI 10.1515/forum-2016-0018. MR3619115 [2] Donald M. Davis, Topological complexity (within 1) of the space of isometry classes of planar n-gons for sufficiently large n, JP Jour of Geom and Topology 20 (2017), no. 1, 1–26. [3] Michael Farber, Invitation to topological robotics, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Z¨ urich, 2008. MR2455573 [4] J.-C. Hausmann and A. Knutson, The cohomology ring of polygon spaces (English, with English and French summaries), Ann. Inst. Fourier (Grenoble) 48 (1998), no. 1, 281–321. MR1614965 [5] Jean-Claude Hausmann and Eugenio Rodriguez, The space of clouds in Euclidean space, Experiment. Math. 13 (2004), no. 1, 31–47. MR2065566 Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015 Email address: [email protected]

Contemporary Mathematics Volume 702, 2018 http://dx.doi.org/10.1090/conm/702/14111

Sectional category of a class of maps Jean-Paul Doeraene, Mohammed El Haouari, and Carlos Ribeiro

Abstract. We propose a definition of ‘sectional category of a class of maps’. This combines the notions of ‘sectional category’ of James, and ‘category of a class of spaces’ of Clapp and Puppe.

The category cat X of a space X in the sense of Lusternik and Schnirelmann is the smallest number n such that there exists an open covering {U0 , . . . , Un } of X for which each inclusion Ui → X is nullhomotopic. In [1], M. Clapp and D. Puppe introduced the A-category of X, where A is a class of spaces, replacing ‘is nullhomotopic’ in the previous definition by ‘factors through some space of A’. On the other hand, the sectional category secat p of a fibration p : E → X, originally defined by Schwarz [12], is obtained by replacing ‘each inclusion Ui → X is nullhomotopic’ in the previous definition by ‘p has a local section on each of the open sets Ui ’. Here we gather these ideas by defining the sectional category of a class of maps with same target X. We propose the Ganea and the Whitehead versions of this definition, as well as the open covering approach. Sectional category earned its renown recently thanks to Farber’s notion of topological complexity of a space A ([6]), which measures the difficulty of solving the motion planing problem: the topological complexity of A is the sectional category of the diagonal Δ : A → A × A. Hence, particular attention is devoted to the sectional category of classes of maps with target A × A containing (or not) the diagonal. Throughout this paper T will be a category of topological spaces and maps. It can be just topological spaces and continuous maps, but also pointed topological spaces and maps, G-equivariant topological spaces and maps, or else filtered topological spaces and maps. To assure that everything goes well, T should be a J-category in the sense of [3]. In [11], it is shown that different notions of sectional category are obtained for different J-structures, but coincide under reasonable conditions.

2010 Mathematics Subject Classification. Primary 55M30. Key words and phrases. Sectional category, A-category, Ganea, Whitehead. c 2018 American Mathematical Society

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1. The Ganea point of view Definition 1.1. For any finite sequence S = (ι0 : A0 → X, . . . , ιn : An → X) of maps of T, the Ganea construction of S is the following sequence of homotopy commutative diagrams (0  i < n): Ai+1 GG ιi+1 {= GG { { GG { { αi+1 GG { # {{ Gi+1 gi+1 Fi D DD w; γi ww DD w w D ww βi DD gi ! ww Gi ηi

'/

7X

where the outside square is a homotopy pullback, the inside square is a homotopy pushout and the map gi+1 = (gi , ιi+1 ) : Gi+1 → X is the whisker map induced by this homotopy pushout. The iteration starts with g0 = ι0 : A0 → X. We can summarize all this by saying that gn is the iterated join over X of all maps in S. We denote Gn by G(S) and gn by g(S). We also write gn (ιX ) instead of g(S) when S = (ιX , . . . , ιX ). Definition 1.2. Let A be a class of maps of T with same target X. The sectional category of A is the least integer n such that there exists a sequence S of n + 1 maps in A, g(S) : G(S) → X having a homotopy section, i.e. a map σ : X → G(S) such that g(S) ◦ σ  idX . We denote the sectional category by secat (A). We write secat (ιX ) = secat (A) when A is reduced to the single map ιX : A → X. In this case, there is only one sequence of length n + 1 of maps in A which is (ιX , . . . , ιX ). If T is pointed with ∗ as zero object, we write cat (X) = secat (A) when A is reduced to the single map ∗ → X. The integer cat (X) is the ‘normalized’ version of the LusternikSchnirelmann category. We shall also write: infcat (A) = inf{secat (ι) | ι ∈ A}. Remark 1.3. Clearly, for any class A, secat (A)  infcat (A). Example 1.4. Let X be a fixed space in T, and let A be a class of spaces in ¯ where A¯ is the class consisting of T. Then A-cat(X) in the sense of [1] is secat (A) all maps from any space in A to X. Example 1.5. Let T be the category of stratified spaces and maps. Consider X a foliated manifold in T and let A be the class of all inclusions A → X where A is a transverse subspace of X, i.e. A ∩ F is at most countable for any leaf F of X. Then secat (A) is actually the transverse LS-category of X introduced by H. Colman [2] while infcat (A) is actually the open LS-category of X introduced by J.-P. Doeraene, E. Macias-Virg´os and D. Tanr´e [5]. In fact, it appears that here secat (A) = infcat (A). Indeed in the light of Theorem 3.3, secat (A)  n when there is a covering of X with open subspaces Ui (0  i  n) which are each deformable in X to a transverse subspace Ai of X, in n

a stratified way. Then each Ui is deformable in a stratified way in X to A = Ai which is also transverse. Hence infcat (A)  n.

i=0

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Proposition 1.6. Let f : X → Y be a map in T and assume that we have a sequence of homotopy commutative squares in T (0  i  n): ιi

A (†)

/X f

 B

τi

 / Y.

Then, for the corresponding sequences S = (ι0 , . . . , ιn ) and R = (τ0 , . . . , τn ) of maps in T, there is a homotopy commutative diagram g(S)

G(S) (‡)

 G(R)

/X f

g(R)

 / Y.

In particular, if for any map ι : A → X in a class A there exists a map τ : B → Y in a class B with a homotopy square (†), and if f has a homotopy section, then secat (B)  secat (A). On the other hand, if for any map τ : B → Y in a class B the square (†) is a homotopy pullback and the map ι is in a class A, then the diagram (‡) is also a homotopy pullback and in this case secat (A)  secat (B). Proof. We can see that there is a map ϕ : G(S) → G(R) such that g(R) ◦ ϕ  f ◦ g(S), using the Join Theorem ([4, Theorem 51]) recursively in the following diagram: / Ai+1 Fi (S) DD ι r s r s DD i+1 D! ysss xrrr /X / Gi+1 (S) Gi (S)  ϕi Fi (R) ss  ysss

Gi (R)

ϕi+i



ζi+1

/ Bi+1 f DDτi+1 r r D DD   xrrr ! / Gi+1 (R) /Y

beginning with ϕ0 = ζ0 and ending with ϕ = ϕn . Assume f has a homotopy section s. If g(S) has a homotopy section σ, then g(R) has a homotopy section ϕ ◦ σ ◦ s. Assume the starting squares (†) are homotopy pullbacks. Then so is the front rightmost one in the above diagram for any i < n, thus (‡) is a homotopy pullback. If g(R) has a homotopy section σ, then g(S) has a homotopy section which is the  induced map (σ ◦ f , idX ). Definition 1.7. There is a preorder on maps of T with same target X defined by: ι : A → X  τ : B → X if ι factors through τ up to homotopy, i.e. there is a map ζ : A → B such that τ ◦ ζ  ι. This preorder extends to classes of maps of T with same target X: we write A  B if each map of A factors through at least one map of B up to homotopy. We write A ≈ B if A  B and B  A.

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Remark 1.8. If Aˆ is a subclass of A, then Aˆ  A. With f = idX in Proposition 1.6 we get: Proposition 1.9. For any classes A and B of maps in T with same target X: B  A =⇒ secat (B)  secat (A). Corollary 1.10. Let T be pointed. For any class A of maps in T with same target X: secat (A)  cat (X). Corollary 1.11. For any class A of maps in T with same target X, and any ˆ If, moreover, each map subclass Aˆ of A, we have A  Aˆ and secat (A)  secat (A). ˆ then also Aˆ  A and of A factors up to homotopy through at least one map of A, ˆ secat (A) = secat (A). Remark 1.12. From this fact, we may often replace a class A by a smaller or a greater one to compute secat (A). In particular, we can keep only one representative for each homotopy class of maps of A. Conversely, we can always assume that all maps equivalent (for the relation ≈) to some map of A are also in A. Corollary 1.13. For any class A of maps in T with same target X, if A contains a map τ : B → X such that each map of A factors through τ , then secat (A) = infcat (A) = secat (τ ). Example 1.14. Let T be pointed and let A be the set of the two maps in1 : A → A ∨ B and in2 : B → A ∨ B. It is known that secat (in1 ) = cat (B) and secat (in2 ) = cat (A); hence infcat (A) = min{cat A, cat B}. But secat (A) = 1 (or 0 if A  ∗ or B  ∗). Indeed apply the ‘Whisker Maps inside a Cube’ Lemma ([4, Lemma 49]) to the following diagram to get the section of g(in1 , in2 ):

A

A

| || ~||

~~ ~~ ~

∗

 F0

/BI III uu u I$ u zu / A∨B A∨B σ

/B w HHH w HH w $  w {w / G1 / A ∨ B. g1

This shows that secat (A) can be strictly less than infcat (A). Example 1.15. Let A and B be the homotopy cofibres of two applications S 2 → S 2 of degrees relatively prime numbers p and q respectively, let X = A × B; let A be the set of the two maps in1 : A → A × B and in2 : B → A × B. It is known that secat (in1 ) = cat (B) and secat (in2 ) = cat (A). But A and B are suspensions, hence cat A = cat B = 1. Thus secat (A) = 1. Now consider the map τ = g(in1 , in2 ) which is a lower bound of A (for the preorder ) by construction. This is the inclusion A ∨ B → A × B. Then H2 (A) = unneth formula, H∗ (τ ) is an isomorphism, and, Zp and H2 (B) = Zq , hence by the K¨ by Whitehead’s theorem, τ is a homotopy equivalence. Thus secat (τ ) = 0. This shows that secat τ , where τ is the join of two minimal maps of A, can be strictly less than secat (A).

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Proposition 1.16. Let us denote by q the integer part of any rational number q. For any class A of maps with same target X, consider the class Ak (respectively: Ak ) of all maps g(S) where S is any sequence of k + 1 (respectively: at most k + 1) maps of A (not necessarily distinct). Then: secat (Ak ) = secat (Ak ) =

' secat (A) ( . k+1

Proof. Any sequence of n+1 maps of Ak is a sequence R = (g(S0 ), . . . , g(Sn )). By associativity of the join, g(R)  g(S0 + · · · + Sn ) where S0 + · · · + Sn is the concatenation of the sequences Si , which is a sequence of (n + 1)(k + 1) maps of A. But secat (Ak ) is the least integer n such that there exists a sequence R of n + 1 maps of Ak such that g(R) has a homotopy section. Thus, if secat (A) = m, then n m k m 1 − k+1  n < k+1 + k+1 , will be such n(k + 1) < m + 1  (n + 1)(k + 1), that is k+1 m  hence n =  k+1 . Finally secat (Ak ) = secat (Ak ) by Corollary 1.11. As a particular case, when A is made of only one map ιX , then Ak is made of the single map gk (ιX ). Then: (ιX ) . Corollary 1.17. For any map ιX : A → X, secat (gk (ιX )) =  secat k+1

2. The Whitehead point of view Definition 2.1. For any finite sequence T = (τ0 : B0 → X0 , . . . , τn : Bn → Xn ) of maps of T, the Whitehead construction of T is the following sequence of homotopy commutative diagrams (0  i < n): i  0 7 Xj × BLi+1 ( i0 idXj )×τi+1 p LLL ti ×idBi+1 ppp L LLL pp LLL ppp p p ) p % / i+1 Xj Ti × Bi+1 T i+1 8 ti+1 OOO 5 0 r r OOO r r r OO rrr idTi ×τi+1 OOOO ' rrr ti ×idXi+1 Ti × Xi+1 where the outside square is a homotopy pullback, the inside square is a homotopy  Xj is the whisker map induced by this pushout and the map ti+1 : Ti+1 → i+1 0 homotopy pushout. The induction starts with t0 = τ0 : B0 → X0 . We denote Tn by T (T ) and tn by t(T ). Remark 2.2. The product symbol × means here the homotopy pullback over the terminal object e; it is the true pullback when the objects are e-fibrant. In the category Top or Top∗ , all objects are e-fibrant, hence these are true pullbacks. Theorem 2.3. For 0  i  n, let / Bi

Ai ιi

 X

τi

fi

 / Xi

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be homotopy pullbacks in which T = (τ0 , . . . , τn ) are sequences of maps in T. Then denoting S = (ι0 , . . . , ιn ), the map g(S) : G(S) → X has a homotopy secn tion if and onlyif the induced map fˆ = (f0 , . . . , fn ) : X → 0 Xj factors through n t(T ) : T (T ) → 0 Xj up to homotopy. Keep in mind the important particular case in which fi = idX , so that ιi = τi and fˆ is the diagonal map Δ : X → X n+1 . Proof. It is a standard argument (following the lines of [7, Theorem 8]) to prove that there is a homotopy pullback: / T (T )

G(S) g(S)

 X

t(T )

fˆ

 / n X j 0



and the result follows. We extend the notion of ‘category of a map’ by the following definition:

Definition 2.4. Let T be pointed and let be a class of maps X with same source X. The category of X is the least integer n such that there exists a sequence ˆ : X → Xn of n + 1 maps in X such that f0 : X → X0 , . . . , fn nthe induced map f = n (f0 , . . . , fn ) : X → 0 Xj factors through t(T ) : T (T ) → 0 Xj up to homotopy, where T = (∗ → X0 , . . . , ∗ → Xn ). We denote this integer by cat X . As a particular case, when there is only one map f : X → X0 in X , we recover the usual definition of cat f , and when this map f is the identity on X (so that fˆ = Δ), we recover cat X. Observe that Theorem 2.3 shows that the category of a class is nothing but a particular case of sectional category of (another) class: Corollary 2.5. Let T be pointed and let be a class of maps X with same source. Then cat X = secat A where A is the class consisting of the homotopy fibers of the maps of X . Example 2.6. Consider any A  ∗ and B  ∗ in T and let X = {pr1 : A×B → A, pr2 : A × B → B} the set of the two projections. The set of homotopy fibers of X is A = {in2 : B → A × B, in1 : A → A × B}. By Corollary 2.5 cat X = secat A. Indeed, in this case g(in2 , in1 )  t(∗ → A, ∗ → B) : A ∨ B → A × B and fˆ = (pr1 , pr2 )  idA×B . Example 2.7. Consider any A  ∗ and B  ∗ in T and let X = {pr1 : A ∨ B → A, pr2 : A ∨ B → B} the set of the two projections. Consider the set of homotopy fibers of X : A = {ι1 : F1 → A ∨ B, ι2 : F2 → A ∨ B}. Hence by Corollary 2.5, cat X = secat A. In this case t(∗ → A, ∗ → B)  fˆ = (pr1 , pr2 ) : A ∨ B → A × B and of course fˆ factors through t(∗ → A, ∗ → B) up to homotopy. Hence, cat X = 1.

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Example 2.8. Let A be a connected, CW H-space, and let D : A × A → A the map such that pr1 ·D  pr2 . The diagonal map Δ : A → A×A is the homotopy fibre of D ([10, Proposition 3.7]). Thus secat (Δ) = cat (D) and secat ({in1 , in2 , Δ}) = cat ({pr2 , pr1 , D}). Note that in this case secat (Δ) = cat (A) by Proposition 1.6. 3. The open covering point of view In this section, we work in the category Top∗ , even if some things can be done in a wider context of a category T. Proposition 3.1. Let be any sequence T = (τ0 : B0 → X0 , . . . , τn : Bn → Xn ) of closed cofibrations in Top∗ . Then: T (T ) = {(x0 , . . . , xn ) ∈ and T (T ) →

n j=0

n 

Xj | xk ∈ Bk for some k}

j=0

Xj is a closed cofibration.

Proof. We have the following commutative diagram where all squares are pullbacks, and, since the projections are fibrations, homotopy pullbacks as well: / i Xj × Bi+1 / / Bi+1 Ti × Bi+1 / 0     Ti × Xi+1 /

 / i Xj × Xi+1 0

 / / Xi+1

 Ti /

 / i X j 0

 / / ∗.

Since Bi+1 → Xi+1 is a closed cofibration and Ti × Xi+1 → Xi+1 is a fibration, Ti × Bi+1 → Ti × Xi+1 is also a closed cofibration, by [13, Theorem 12]. And  similarly, assuming that Ti → i0 Xj is a closed cofibration by induction hypothesis, i Ti × Bi+1 → 0 Xj × Bi+1 is also a closed cofibration. But then, the homotopy i+1 pushout Ti+1 is the true pushout. Moreover, the map Ti+1 → 0 Xj is closed, by [9, Proposition 2.46], and it is a cofibration by [13, Theorem 6].  Definition 3.2. Let τ : B → Y and f : X → Y be maps in Top∗ . A subspace U of X is said (τ, f )-categorical if there is a map s : U → B so that the restriction of f to U is homotopic to τ ◦ s. If the context makes it clear what τ and f are, we say also that U is B-categorical. Saying that τ : B → Y is a closed cofibration means that τ is an embedding and (Y, B) is a NDR-pair; in particular there is an open subset N of Y such that B ⊂ N ⊂ Y and N is (τ, idY )-categorical. We have the following characterization of secat A or cat X in terms of open categorical covering: Theorem 3.3. Let A be a class of maps with the same target X, a well-pointed normal space. Then secat A is the least integer n, such that there exists a sequence S = (ι0 : A0 → X, . . . , ιn : An → X) of n + 1 maps of A and there is an open covering (Ui )0in of X, each Ui being Ai -categorical.

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Theorem 3.4. Let X be a class of maps with the same source X, a well-pointed normal space, and whose targets are path-connected spaces. Then cat X is the least integer n, such that there exists a sequence (f0 , . . . , fn ) of n + 1 maps of X and there is an open covering (Ui )0in of X, each fi |Ui being nullhomotopic. These theorems are consequences of the following proposition: Proposition 3.5. Let T = (τ0 : B0 → X0 , . . . , τn : Bn → Xn ) be any sequence of closed cofibrations and let f0 : X → X0 , . . . , fn : X → Xn be a sequence of maps in which X is a normalspace. Then the induced map fˆ = (f0 , . . . , fn ) factors n through t(T ) : T (T ) → 0 Xj up to homotopy if and only if there is an open covering (Ui )0in of X, each Ui being Bi -categorical. Proof. (⇐.) By hypothesis, there is a covering (U0 , . . . , Un ) of X by open sets, and deformations Hi : Ui × I → Xi of fi |Ui into a map with values in Bi , for 0  i  n. As X is normal, there exists a covering of X by open sets, (W0 , . . . , Wn ), such that W i ⊂ Ui , for 0  i  n. For any i, we choose a Urysohn function ϕi : X → I such that ϕi (x) = 1 if x ∈ W i and ϕi (x) = 0 if x ∈ Ui . We define now ˆ i : X × I → Xi by: a continuous map H  Hi (x, ϕi (x)t) if x ∈ Ui , ˆ i (x, t) = H otherwise. fi (x) n We collect these maps in a continuous map H : X ×I → 0 Xj defined by H(x, t) = ˆ n (x, t)). Observe that H(x, 0) = (f0 (x), . . . , fn (x)) = fˆ(x). ˆ 0 (x, t) . . . , H (H Set r(x) = H(x, 1). Since the Wi s are a covering of X, for any point x ∈ ˆk, H ˆ k (x, 1) = Hk (x, 1) ∈ X, there is a Wk with x ∈ Wk . By definition of H n Bk . As the maps ιi are closed cofibrations, T (T ) = {(x0 , . . . , xn ) ∈ 0 Xj |xk ∈ Bk for some k}, and we deduce r(X) ⊂ T (T ) and r is a lifting up to homotopy (by the homotopy H) of fˆ. (⇒.) By hypothesis, there is a map r : X → T (T ) and a homotopy H : X ×I → n ˆ X i between f and the composite t(T ) ◦ r. 0 For any 0  i  n, as (Xi , Bi ) is a NDR-pair, there exists also an open set Ni , deformation Gi : Ni × I → Xi of Ni → Xi into a map with Bi ⊂ Ni ⊂ Xi , and a  n values in Bi . Let pi : 0 Xj → Xi be the i-th projection. We set hi = pi ◦ t(T ) ◦ r

n

n −1 and Ui = hi (Ni ). Then, since r(X) ⊂ T (T ) = i=0 p−1 i (Bi ), X = i=0 Ui . Hence the Ui s are a covering of X. Define Hi : Ui × I → Xi by:  pi H(u, 2t) if 0  t  1/2, Hi (u, t) = Gi (hi (u), 2t − 1) if 1/2  t  1. This is well defined since pi H(u, 1) = hi (u) an Hi is a homotopy between fi |Ui and a map with values in Bi .  Proof of Theorem 3.3. We can use Theorem 2.3 (where fi = idX ) and Proposition 3.5 directly if the maps in A are closed cofibrations. If they are not, we can replace them as follows: Ai {} }{{ ιi X fi |= i CC C | | ∼ !!  /X X ιi

id

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decomposing first ιi in A into a closed cofibration ιi followed by a fibration which is a homotopy equivalence (by [14, Proposition 2]), and then choosing a section fi of it; this section exists since X is well-pointed and thus cofibrant. For an open set Ui of X, being (ιi , fi )-categorical is equivalent to be (ιi , idX )-categorical, and we can apply Proposition 3.5.  Proof of Theorem 3.4. As in the previous proof, we make the following change of targets: ∗ τi xx| x |x Xi fi > CC } C }} ∼ !!  / Xi . X fi

For an open set Ui of X, fi |Ui nullhomotopic ⇐⇒ fi factors through ∗ → Xi (since Xi is path-connected) ⇐⇒ Ui is (τi , fi )-categorical, and we can apply Proposition 3.5.  Remark 3.6. Following the lines of [8, 1.3, 7.1 and 8.3], we can obtain lower bounds for secat and cat of a class of maps from cohomology. Consider the singular cohomology theory H ∗ , with any coefficient ring, and the corresponding reduced ˜ ∗ . Let nilR denote the nilpotency index of the ring R (this is the least theory H integer n such that Rn = 0). If A is a finite set of maps ιi : Ai → X with same target: secat A + 1  nil (∩ιi ∈A ker ι∗i ) ˜ ∗ (X) → H ˜ ∗ (Ai ) denotes the induced homomorphism. where ι∗i : H If X is a finite set of maps fi : X → Xi with same source: cat X + 1  nil (∩fi ∈X im fi∗ ) ˜ ∗ (Xi ) → H ˜ ∗ (X) denotes the induced homomorphism. where fi∗ : H Example 3.7. Let A and B be two spaces, and consider the inclusions in1 : A → A×B and in2 : B → A×B. Assume A and B are ‘reasonable’ spaces, so that the inclusions are closed cofibrations and A × B is normal. Then secat (A) = infcat (A) = min{cat A, cat B}. Indeed, first recall that secat in1 = cat B and secat in2 = cat A; hence by Remark 1.3, secat A  min{cat A, cat B}. Conversely, assume that secat A = p + q − 1 and that we have a covering of A × B formed by p open sets Ui (i = 1, 2, . . . , p) with deformations of Ui into A and q open sets Vj (j = 1, . . . , q) with deformations of Vj into B. If p = 0, then fix any point b0 ∈ B. The sets pr1 (A × {b0 } ∩ Vj ) (pr1 being the first projection) are contractible in A and cover A. Hence cat A  q − 1. We can do the same reasoning if q = 0 and conclude that cat B  p − 1. We now suppose that p = 0 and q = 0. As above fix a point b0 ∈ B. and consider the sets A × {b0 } ∩ Vj and their projections Oj = pr1 (A × {b0 } ∩ Vj ) which are contractible in A. If they cover A, we are done: cat A  q − 1. But in general there may be points of A which are not in these projections. These points should lie in the projections pr1 (A × {b0 } ∩ Ui ).

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If the projections Oi do not cover A, then for each integer i (1  i  p), there is a point bi and an integer ji (1  ji  q) such that pr1 (A × {b0 } ∩ Ui ) ⊆ pr1 (A × {bi } ∩ Vji ). For if this is not the case then there is a0 ∈ A with (a0 , b0 ) ∈ Ui for some i such that {a0 } × B ∩ Vj is empty for all j. It follows that {a0 } × B can be covered by the open sets Ui and the projections pr2 ({a0 } × B ∩ Ui ) form a covering of B by p contractible open sets. Hence cat B  p − 1, and since secat A  cat B, we should have q = 0. From this fact, we conclude that if p = 0 and q = 0, then the parts of A which are not possibly covered by the projections Oj (1  j  q), are covered by the projections pr1 (A × {bi } ∩ Vji ) (1  ji  q). Thus we obtain a covering of A by p + q contractible sets. Hence cat A  p + q − 1 = secat A. The same holds for B; hence secat {in1 , in2 } = min{cat A, cat B}. Note that incidentally we proved that cat A = cat B whenever we need p = 0 and q = 0 in order to realize the sectional category with respect to the inclusions in1 and in2 . 4. Topological complexity The topological complexity of a space A, as defined in [6], is the sectional category of the diagonal map Δ : A → A × A, i.e. TC (A) = secat (Δ). It is known that cat (A)  TC (A)  cat (A × A). The natural question we may ask in relation with the previous sections is: what about secat (A) where A is a class of maps with target A × A that contains the diagonal Δ : A → A × A? Of course secat (A)  TC (A); but, for instance, what is secat (A) when A is the set of all maps from A to A × A? Remark 4.1. For any class A of maps with target A × A, if A contains either in1 or in2 : A → A × A, then secat (A)  cat (A). Indeed secat (A)  infcat (A)  secat (ini ) = cat (A) (i = 1 or 2). Proposition 4.2. For any A in T, consider the maps in1 , in2 , Δ : A → A × A. Then: secat ({Δ, in1 }) = secat ({Δ, in2 }) = secat ({in1 , in2 }) = cat (A). Proof. Consider the following homotopy commutative diagram: ∗

/A

 A

 / A×A

/∗

in2

h

pr1

 / A.

The right square is a (homotopy) pullback. If h is either in1 or Δ, which are both sections of pr1 , then the outer rectangle is a (homotopy) pullback as well; hence the left one is also a homotopy pullback. By Proposition 1.6, with f = in2 , we get secat ({Δ, in1 })  secat ({∗ → A}) = cat (A), and with f = Δ, we get secat ({in1 , in2 })  secat ({∗ → A}) = cat (A). Use Remark 4.1 to get equalities. 

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Remark 4.3. If A is a surface, or any space with cat (A)  2, then also secat ({in1 , in2 , Δ}) = cat (A). This is clear for cat A  1. If the sectional category was strictly less than cat A = 2, we would have a homotopy section for the join of only 2 of the three maps in1 , in2 , Δ, and this is in contradiction with Proposition 4.2. References [1] M´ onica Clapp and Dieter Puppe, Invariants of the Lusternik-Schnirelmann type and the topology of critical sets, Trans. Amer. Math. Soc. 298 (1986), no. 2, 603–620, DOI 10.2307/2000638. MR860382 [2] Hellen Colman and Enrique Macias-Virg´ os, Transverse Lusternik-Schnirelmann category of foliated manifolds, Topology 40 (2001), no. 2, 419–430, DOI 10.1016/S0040-9383(99)00067-1. MR1808226 [3] Jean-Paul Doeraene, L.S.-category in a model category, J. Pure Appl. Algebra 84 (1993), no. 3, 215–261, DOI 10.1016/0022-4049(93)90001-A. MR1201256 [4] Jean-Paul Doeraene and Mohammed El Haouari, Up-to-one approximations of sectional category and topological complexity, Topology Appl. 160 (2013), no. 5, 766–783, DOI 10.1016/j.topol.2013.02.001. MR3022743 [5] Jean-Paul Doeraene, Enrique Macias-Virg´ os, and Daniel Tanr´ e, Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations, Topology Appl. 157 (2010), no. 9, 1680–1689, DOI 10.1016/j.topol.2010.03.007. MR2639834 [6] Michael Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211–221, DOI 10.1007/s00454-002-0760-9. MR1957228 [7] J. Garcia-Calcines and L. Vandembroucq, Weak sectional category, J. Lond. Math. Soc. (2) 82 (2010), no. 3, 621–642, DOI 10.1112/jlms/jdq048. MR2739059 [8] I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology 17 (1978), no. 4, 331–348, DOI 10.1016/0040-9383(78)90002-2. MR516214 [9] I. M. James, General topology and homotopy theory, Springer-Verlag, New York, 1984. MR762634 [10] Gregory Lupton and J´ erˆ ome Scherer, Topological complexity of H-spaces, Proc. Amer. Math. Soc. 141 (2013), no. 5, 1827–1838, DOI 10.1090/S0002-9939-2012-11454-6. MR3020869 [11] Marco Moraschini and Aniceto Murillo, Abstract sectional category in model structures on topological spaces, Topology Appl. 199 (2016), 23–31, DOI 10.1016/j.topol.2015.12.002. MR3442591 [12] Albert S. Schwarz. The genus of a fiber space. Amer. Math. Soc. Transl., 55:49–140, 1966. [13] Arne Strøm, Note on cofibrations. II, Math. Scand. 22 (1968), 130–142 (1969), DOI 10.7146/math.scand.a-10877. MR0243525 [14] Arne Strøm, The homotopy category is a homotopy category, Arch. Math. (Basel) 23 (1972), 435–441, DOI 10.1007/BF01304912. MR0321082 ´matiques, Universit´ D´ epartement de Mathe e de Lille 1, 59655 Villeneuve d’Ascq Cedex, France Email address: [email protected] ´matiques, Universite ´ de Lille 1, 59655 Villeneuve d’Ascq D´ epartement de Mathe Cedex, France Email address: [email protected] D´ epartement de Math´ ematiques, Universit´ e de Lille 1, 59655 Villeneuve d’Ascq Cedex, France Email address: [email protected]

Contemporary Mathematics Volume 702, 2018 http://dx.doi.org/10.1090/conm/702/14110

Q-topological complexity Luc´ıa Fern´ andez Su´arez and Lucile Vandembroucq Abstract. By analogy with the invariant Q-category defined by Scheerer, Stanley and Tanr´e, we introduce the notions of Q-sectional category and Qtopological complexity. We establish several properties of these invariants. We also obtain a formula for the behaviour of the sectional category with respect to a fibration which generalizes the classical formulas for Lusternik-Schnirelmann category and topological complexity.

1. Introduction The Q-category of a topological space X, denoted by QcatX, is a lower bound for the Lusternik-Schnirelmann category of X, catX, which has been introduced by H. Scheerer, D. Stanley, and D. Tanr´e in [17]. This invariant, defined using a fibrewise extension of a functor Qk equivalent to Ωk Σk , has been in particular used in the study of critical points (see [2, Chap. 7], [15]) and in the study of the Ganea conjecture. More precisely, after N. Iwase [13] showed that the Ganea conjecture, which asserted that cat(X × S n ) = catX + 1 for any n ≥ 1, was not true in general, although it was known to be true for many classes of spaces (e.g. [19], [14], [12], [16], [22]), one could ask for a complete characterization of the spaces X satisfying the equality above. In [17], H. Scheerer, D. Stanley, and D. Tanr´e conjectured that a finite CW-complex X satisfies the Ganea conjecture, that is the equality cat(X × S n ) = catX + 1 holds for any n ≥ 1, if and only if QcatX = catX. One direction of this equivalence has been proved in [23] but the complete answer is still unknown. In this paper we introduce the analogue of Q-category for Farber’s topological complexity [5] and establish some properties of this invariant. Since L.-S. category and topological complexity are both special cases of the notion of sectional category, introduced by A. Schwarz in [18], we naturally consider and study a notion of Qsectional category. Our definition, given in Section 2.3, is based on a generalized notion of Ganea fibrations and on a fibrewise extension of the functor Qk that we respectively recall in Sections 2.1 and 2.2. We next, in Section 3, establish various formulas for Q-sectional category and Q-topological complexity. It is worth noting that our study of the behaviour of the Q-sectional category in a fibration 2010 Mathematics Subject Classification. Primary 55M30. Key words and phrases. Sectional category, topological complexity. This research has been partially supported by Portuguese Funds from the “Funda¸c˜ ao para a Ciˆ encia e a Tecnologia”, through the Project UID/MAT/0013/2013. c 2018 American Mathematical Society

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led us to establish a new formula for the sectional category (see Theorem 3.8), which generalizes both the classical formula for the LS-category in a fibration and the formula established by M. Farber and M. Grant for topological complexity [6]. Finally, we use our results to study some examples. This permits us in particular to observe that the analogue of the Scheerer-Stanley-Tanr´e conjecture for topological complexity is not true. We also include a small observation about the original Scheerer-Stanley-Tanr´e conjecture. Throughout this text we work in the category of compactly generated Hausdorff spaces having the homotopy type of a CW-complex. 2. Q-sectional category and Q-topological complexity 2.1. Sectional category and Ganea fibrations. Let f : A → X be a map where X is a well-pointed path-connected space with base point ∗ ∈ X. The sectional category of f , secat(f ), is the least integer n (or ∞) for which there exists an open cover U0 , . . . , Un of X such that, for any 0 ≤ i ≤ n, f admits a local homotopy section on Ui (that is, a continuous map si : Ui → A such that f ◦ si is homotopic to the inclusion Ui → X). When f is a fibration, we can, equivalently, require local strict sections instead of local homotopy sections. As special cases of sectional category, Lusternik-Schnirelmann category and Farber’s topological complexity are respectively given by • cat(X) = secat(ev1 : P X → X) = secat (∗ → X) where P X ⊂ X I is the space of paths beginning at the base point ∗ and ev1 is the evaluation map at the end of the path, • TC(X) = secat (ev0,1 : X I → X × X) = secat(Δ : X → X × X) where ev0,1 evaluates a path at its extremities and Δ is the diagonal map. As is well-known, if f : A → X and g : B → Y are two maps with homotopy ∼ ∼ equivalences A → B and X → Y making the obvious diagram commutative then secat(f ) = secat(g). Also secat(f ) can be characterized through the existence of a global section for a certain join map which can be, for instance, explicitly constructed through an iterated fibrewise join of f when f is a fibration ([18]). Here we will assume (without loss of generality) that f : A → X is a (closed) pointed cofibration and consider the following constructions which give us a natural and explicit fibration, the Ganea fibration of f , equivalent to the join map characterizing secat(f ): • the fatwedge of f ([7], [9]): T n (f ) = {(x0 , . . . , xn ) ∈ X n+1 | ∃j, xj ∈ f (A)} which generalizes the classical fat-wedge T n (X) = {(x0 , . . . , xn ) ∈ X n+1 | ∃j, xj = ∗}, ) *  n+1 • the space Γn X = (γ0 , . . . , γn ) ∈ X I | γ0 (0) = · · · = γn (0) together with the fibration Γn X → X,

(γ0 , . . . , γn ) → γ0 (0) = · · · = γn (0)

which is a homotopy equivalence,

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• the nth Ganea fibration of f , gn (f ) : Gn (f ) → X, which is obtained by pull-back along the diagonal map Δn+1 : X → X n+1 of the fibration associated to the inclusion T n (f ) → X n+1 and explicitly given by: Gn (f ) = {(γ0 , . . . , γn ) ∈ Γn (X) | ∃j, γj (1) ∈ f (A)} → X (γ0 , . . . , γn ) → γ0 (0) = · · · = γn (0). All these spaces are considered with the obvious base points (∗, . . . , ∗) ∈ T n (f ) and (ˆ ∗, . . . , ˆ ∗) ∈ Gn (f ) ⊂ Γn (X) (where ˆ ∗ denotes the constant path). By construction, there exists a commutative diagram in which the square is a (homotopy) pull-back.

gn (f )

 X

∼

/•o

Gn (f )

Δn+1

 / X n+1

T n (f ) l v v L vv v v zv v

By [7], we know that secat(f ) is the least integer n such that the diagonal map Δn+1 : X → X n+1 lifts up to homotopy in the fatwedge of f , T n (f ). By the homotopy pull-back diagram above, this is equivalent to say that secat(f ) is the least integer n such that the fibration gn (f ) admits a (homotopy) section. By [9, Th. 8], the fibration gn (f ) is equivalent to the (fibrewise) join of n + 1 copies of f or of any map weakly equivalent to f . The fibre of gn (f ), denoted by Fn (f ), is homotopically equivalent to the (usual) join of n + 1 copies of the homotopy fibre F of f . When f is the inclusion ∗ → X (which is a cofibration since X is well-pointed) we recover a possible description of the classical Ganea fibration of X and we will use, in that case, the classical notation gn (X) : Gn (X) = {(γ0 , . . . , γn ) ∈ Γn (X) : ∃j, γj (1) = ∗} → X. We have cat(X) ≤ n

⇔

gn (X) : Gn (X) → X admits a (homotopy) section.

When X is a CW-complex, f = Δ : X → X × X is a cofibration and we have TC (X) ≤ n

⇔

gn (Δ) : Gn (Δ) → X × X admits a (homotopy) section.

We finally note that, if we have a commutative diagram where f and g are cofibrations ϕ /B A f

 X

ψ

 /Y

g

then we have a commutative diagram Gn (ψ,ϕ)

/ Gn (g)

Gn (f ) gn (f )

 X

gn (g)

ψ

 / Y.

Moreover, we have (a) if ϕ and ψ are homotopy equivalences, then so is Gn (ψ, ϕ); (b) if the first diagram is a homotopy pull-back, then so is the second diagram.

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Note that (b) follows (for instance) from the equivalence between gn (f ) and the fibrewise join of n + 1 copies of f together with the Join Theorem ([3]). 2.2. Fibrewise Q construction. The notion of Q-category defined in [17] is based on the Dror-Frajoun fibrewise extension [4] of a functor Qk equivalent to Ωk Σk . Instead of using the Dror-Farjoun construction, we will here use the (equivalent) explicit fibrewise extension of Qk given in [23]. We describe the case k = 1, recall more quickly the general case and refer to [17],[2, Sections 4.5, 4.6, 4.7] and [23] for further details. We denote by Σ1 Z the unreduced suspension of a space Z, i.e. Σ1 Z = Z ×I/ ∼ with (z, 0) ∼ (z  , 0) and (z, 1) ∼ (z  , 1) for z, z  ∈ Z. Denoting by [z, t] ∈ Σ1 Z the 1 : {0, 1} = ∂I → Σ1 Z given class of (z, t) ∈ Z × I, we have a canonical map α1 = αZ 1 1 by α (0) = [z, 0] and α (1) = [z, 1] where z ∈ Z. The functor Q1 is defined by 1 Q1 (Z) = {ω : I → Σ1 Z|ω|∂I = αZ } 1 and is given with a co-augmentation η 1 = ηZ : Z → Q1 Z which takes z ∈ Z to the & denotes the reduced suspension of Z, then path z → [z, t]. If Z is pointed and ΣZ 1 & there is a natural map Q (Z) → ΩΣZ, which is a homotopy equivalence induced by ∼ & the identification map Σ1 Z → ΣZ and which makes compatible the coaugmention & of Q1 with the usual coaugmentation η˜ : Z → ΩΣZ.

We now describe a fibrewise extension of Q1 . Let p : E → B be a fibration (over a path-connected space) with fibre F . The fibrewise suspension of p, Σ1B E → B, is defined by the push-out: E × {0, 1} / p×id

 B × {0, 1} /

/ E×I  / Σ1 E B E EE pˆ EE EE E"  , B.

The resulting map pˆ : Σ1B E → B is a fibration whose fibre over b is the (unreduced) suspension Σ1 Fb of the fibre of p : E → B over b. By construction, we have a canonical map μ1 : B × {0, 1} → Σ1B E 1 : {0, 1} → Σ1 Fb . which is a fibrewise extension of α1 : for any b ∈ B, μ1 (b, −) = αF b We define

Q1B (E) = {ω : I → Σ1B E | ∃b ∈ B, pˆω = b and ω|∂I = μ1 (b, −)} together with the map q 1 (p) : Q1B (E) → B given by ω → pˆω(0). This is a fibration whose fibre over b is Q1 (Fb ) ([23, Lemma 8]). We also have a fibrewise coaugmenta1 : E → Q1B (E) which extends η 1 : F → Q1 (F ) and we have a commutative tion ηB

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diagram E

1 ηB

/ Q1 (E) B

/ Q1 (E)

q 1 (p)

p

 B

 B

∼

/ ΩΣE &

∼

 & / ΩΣB

 ΩΣp

Q1 (p)

η1

 / Q1 (B)

where the map Q1B (E) → Q1 (E) is induced by the identification map Σ1B E → Σ1 E. In general, for any k ≥ 1, we consider: • The k fold unreduced suspension of a space Z, Σk Z, which can be described as (Z × I k )/ ∼ where the relation is given by (z, t1 , . . . , tk ) ∼ (z  , t1 , . . . , tk ) if, for some i, ti = ti ∈ {0, 1} and tj = tj for all j > i. We write [z, t1 , . . . , tk ] for the class of an element. • The kth fibrewise suspension of p : E → B, pˆk : ΣkB E → B, whose fibre over b is Σk Fb . As before, ΣkB E can be described as (E × I k )/ ∼ where the relation is given by (e, t1 , . . . , tk ) ∼ (e , t1 , . . . , tk ) if p(e) = p(e ) and, for some i, ti = ti ∈ {0, 1} and tj = tj for all j > i. We write [e, t1 , . . . , tk ] for the class of an element. • The canonical map αk : ∂I k → Σk Z given by αk (t1 , . . . , tk ) = [z, t1 , . . . , tk ] (where z ∈ Z is any element) and its fibrewise extension μk : B × ∂I k → ΣkB E satisfying μk (b, −) = αk : ∂I k → Σk Fb for any b ∈ B. • The fibration q k (p) : QkB (E) → B where QkB (E) is the (closed) subspace k of (ΣkB E)I given by QkB (E) = {ω : I k → ΣkB E | ∃b ∈ B, pˆk ω = b and ω|∂I k = μk (b, −)} and q k (p)(ω) = pˆk ω(0). The fibre is ∼

& kF Qk (F ) = {ω : I k → Σk F |ω|∂I k = αk } → Ωk Σ and is given with an obvious coaugmentation η k : F → Qk (F ), equivalent & k F . We denote by η k : to the classical augmentation η˜k : F → Ωk Σ B k k E → QB (E) the fibrewise extension of η . When it is relevant Qk (F ) and QkB (E) are considered with the base point given by the map u → [∗, u] where u ∈ I k and ∗ is the base point of F and E. We have, for any k ≥ 1, a commutative diagram: (1)

E p

 B

k ηB

/ Qk (E)

/ Qk (E)

B

q k (p)

 B

∼

/ Ωk Σ & kE

∼

 & k B. / Ωk Σ

kp Ωk Σ

Qk (p)

η

k

 / Qk (B)

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Setting Q0 = Q0B = id we also have a commutative diagram of the following form: (2)

E = Q0B E p

 B

/ Q1 (E) B

/ ···

/ Qk (E) B

···

 B

q 1 (p)

q k (p)

 B

bk B

/ Qk+1 (E) B

/ ···

q k+1 (p)

 B

···

where the map bkB is a fibrewise extension of the map bk : Qk (F ) → Qk+1 (F ) given by bk (ω)(t1 , . . . , tk+1 ) = [ω(t1 , . . . , tk ), tk+1 ]. Observe that the map bk is equivalent k &k k k+1 & k+1 & &k Σ ηΣ F . We can interpret to the map Ωk (˜  k F ) : Ω (Σ F ) → Ω (ΩΣΣ F ) = Ω the sequence (2) as a fibrewise stabilization of the fibration p since, for any integers & k F ). k and i, we have πi (Qk F ) = πi+k (Σ For any k ≥ 0, the functors Qk and QkB preserve homotopy equivalences. We also note that the Qk− construction is natural in the sense that, if we have a commutative diagram E

f

p

(†)

 B

/E p

g

 /B

where p and p are fibrations (over path-connected spaces), then, for any k, we obtain a commutative diagram / Qk (E) B

QkB  (E  ) (‡)

q k (p )

 B

q k (p)

g

 / B.

Moreover we have: Proposition 2.1. With the notations above, if Diagram ( †) is a homotopy pull-back, then so is Diagram ( ‡). Proof. Since the functors Qk and QkB preserve homotopy equivalences, it is sufficient to establish the statement when Diagram (†) is a strict pull-back. In this case, the whisker map ψ : E  → E ×B B  is a homeomorphism and its inverse φ satisfies p φ(e, b ) = b for (e, b ) ∈ E ×B B  . We can then check that the map φ¯ : ΣkB E ×B B  → ΣkB  E  induced by ((e, u), b ) → [φ(e, b ), u] for ((e, u), b ) ∈ (E × I k ) ×B B  is an inverse of the whisker map ΣkB  E  → ΣkB E ×B B  . In other words, the diagram / Σk E Σk  E  B

p

B

k

 B

pˆk

g

 /B

is a strict pull-back (and homotopy pull-back since pˆk is a fibration). We can next check that the map QkB (E) ×B B  → QkB  (E  ) that takes (ω, b ) ∈ QkB (E) ×B B  to ¯ b ) is an inverse of the the element I k → ΣkB  E  of QkB  (E  ) given by u → φ(ω(u),

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whisker map QkB  (E  ) → QkB (E) ×B B  . This means that Diagram (‡) is a strict  pull-back and therefore a homotopy pull-back since q k (p) is a fibration. Remark 2.2. Assuming that the fibres F and F  of p and p are path-connected spaces (having the homotopy type of a CW-complex), we can give the following more conceptual proof of Proposition 2.1. In this case, the spaces QkB (E) and QkB  (E  ) are path-connected spaces having the homotopy type of a CW-complex, since B and & k F and Qk (F  )  Ωk Σ & k F so are (see [20]). Since Diagram ( †) is Qk (F )  Ωk Σ a homotopy pull-back, the map e : F  → F induced by this diagram is a homotopy equivalence and so is Qk (e). Therefore Diagram ( ‡) is a homotopy pull-back since the whisker map from QkB  (E  ) to QkB (E) ×B B  induces Qk (e) between the fibres and is hence a homotopy equivalence. Finally, we also note the following constructions which will be, as in [17], useful in our study of products. If p : E → B and p : E  → B  are two fibrations then, there exist, for any k ≥ 0, commutative diagrams of the following form / Qk  (E × E  ) QkB (E) × E  B×B QQQ QQQ QQQ q k (p×p ) q k (p)×p QQQ(  B × B

/ Qk  (E × E  ) E × QkB  (E  ) B×B QQQ QQQ QQQ q k (p×p ) Q p×q k (p ) QQQ(  B × B

which are induced by the obvious fibrewise extensions of the maps Σk Z × Z  → Σk (Z × Z  ) ([z, t1 , . . . , tk ], z  ) → [(z, z  ), t1 , . . . , tk ] and

Z × Σk Z  → Σk (Z × Z  ) (z, [z , t1 , . . . , tk ]) → [(z, z  ), t1 , . . . , tk ]. By considering the fibrewise extensions of the evaluation 

ev : Σk Qk Z → Σk Z [ω, t1 , . . . , tk ] → ω(t1 , . . . , tk ) and of the map Qk ◦ Qk (Z) → ω : I → Σk Qk Z → we also get a commutative diagram k

Qk (Z) ev ◦ ω

/ Qk (E) QkB ◦ QkB (E) B NNN NNN NNN NNN  & B. This permits us to establish the following result: Proposition 2.3. Let p : E → B and p : E  → B  be two fibrations. For any k ≥ 0, there exists a commutative diagram / Qk  (E × E  ) QkB (E) × QkB  (E  ) B×B PPP oo PPP o o o PPP oook  q k (p)×q k (p ) PPP( wooo q (p×p )  B×B .

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110

Proof. Using the constructions above we obtain: QkB (E) × QkB  (E  ) q k (p)×q k (p )

 B × B

/ Qk  (E × Qk  (E  )) B×B B

/ Qk  ◦ Qk  (E × E  ) B×B B×B

/ Qk  (E × E  ) B×B

 B × B

 B × B

 B × B.

q k (p×p )

 2.3. Definition of Qsecat and QTC. Let f : A → X a cofibration where X is a well-pointed path-connected space and let k ≥ 0. By applying the QkX construction to the Ganea fibrations gn (f ) we define: Definition 2.4. Qk secat(f ) is the least integer n (or ∞) such that the fibration q k (gn (f )) : QkX (Gn (f )) → X admits a (homotopy) section. By Diagram (2), we have: · · · ≤ Qk+1 secat(f ) ≤ Qk secat(f ) ≤ · · · ≤ Q1 secat(f ) ≤ Q0 secat (f ) = secat (f ) As a limit invariant, we set: Qsecat (f ) := lim Qk secat(f ). If f is the inclusion ∗ → X, we recover the notion of Qk cat(X) introduced by H. Scheerer, D. Stanley and D. Tanr´e. If X is a CW-complex and f is the diagonal map Δ : X → X × X, we naturally use the notation Qk TC(X) and QTC (X). Remark 2.5. The notion of Qk secat can be extended to any map g by applying the Q construction to any fibration equivalent to the join map characterizing secat(g) or, equivalently, by setting Qk secat(g) := Qk secat(f ) where f is any cofibration weakly equivalent to g. 3. Some properties of Qsecat and QTC In all the statements in this section, we consider cofibrations whose target is a well-pointed path-connected space. 3.1. Basic properties. We start with the following properties which permit us to generalize to QTC and Qcat the well-known relationships [5] between TC and cat (see Corollary 3.2): Proposition 3.1. Let f : A → X and g : B → Y be two cofibrations together with a commutative diagram /B A ϕ

g

f

 X

ψ

 / Y.

(a) If ψ is a homotopy equivalence then, for any k ≥ 0, Qk secat (f ) ≥ Qk secat(g). (b) If the diagram is a homotopy pull-back then, for any k ≥ 0, we have Qk secat(f ) ≤ Qk secat(g).

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Proof. By considering the Ganea fibrations and applying the QkX construction, we obtain the following commutative diagram / Qk (Gn (g)) Y

QkX (Gn (f )) q k (gn (f ))

q k (gn (g))

 X

ψ

 / Y.

If ψ is a homotopy equivalence, we deduce from a section of the left-hand fibration a homotopy section of the right-hand fibration, which proves the statement (a). For (b), the hypothesis implies that the diagram is a homotopy pull-back (see Proposition 2.1) which permits us to obtain a section of the left hand fibration from a section of the right hand one.  Corollary 3.2. Let X be a path-connected CW-complex. For any k ≥ 0, we have Qk cat(X) ≤ Qk TC(X) ≤ Qk cat(X × X). Proof. It suffices to apply the two items of the propostion above to the following two diagrams, respectively. The right hand diagram, where i2 is the inclusion on the second factor, is a homotopy pull-back: ∗

/X

 X ×X

 X ×X

Δ

∗

/X

 X

 / X × X.

Δ

i2

 3.2. Cohomological lower bound. Let f : A → X be a cofibration, and let f ∗ : H ∗ (X; k) → H ∗ (A; k) be the morphism induced by f in cohomology with coefficients in a field k. As is well-known ([18]), if we consider the index of nilpotency, nil, of the ideal ker f ∗ , that is, the least integer n such that any (n + 1)-fold cup product in ker f ∗ is trivial, then we have nil ker f ∗ ≤ secat(f ). Actually the proof of [8, Thm 5.2] permits us to see that nil ker f ∗ ≤ Hsecat(f ) ≤ secat(f ) where Hsecat (f ) is the least integer n such that the morphism induced in cohomology by the nth Ganea fibration of f , H ∗ (X; k) → H ∗ (Gn (f ); k), is injective. Here we prove: Theorem 3.3. For any k ≥ 0, nil ker f ∗ ≤ Hsecat (f ) ≤ Qk secat(f ). Proof. Suppose that Qk secat(f ) ≤ n. By applying Diagram (1) to the nth Ganea fibration of f , we obtain the following commutative diagram: Gn (f ) gn (f )

 X

/ Qk (Gn (f )) X

/ Qk (Gn (f ))

q k (gn (f ))

 X

η

∼

/ Ωk Σ & k Gn (f )

∼

 & k X. / Ωk Σ

 k gn (f ) Ωk Σ

Qk (gn )

k

 / Qk (X)

112

´ ´ L. FERNANDEZ SUAREZ AND L. VANDEMBROUCQ

∼ & k X, which is the usual coaugmentation, The composite η˜k : X → Qk (X) → Ωk Σ & k X through the k-fold adjunction. can be identified to the adjoint of the identity of Σ k & k Gn (f ) such From Q secat (f ) ≤ n we know that there exists a map ψ : X → Ωk Σ k &k k that Ω Σ (gn (f ))ψ  η˜ . Through the k fold adjunction, we get a homotopy & k (gn (f )) which implies that H ∗ (gn (f )) is injective.  section of Σ

When f = Δ : X → X × X, nil ker Δ∗ coincides with Farber’s zero-divisor cuplength, that is, the nilpotency of the kernel of the cup-product ∪X : H ∗ (X; k) ⊗ H ∗ (X; k) → H ∗ (X; k) and we have: Corollary 3.4. Let X be a path-connected CW-complex. For any k ≥ 0, we have nil ker ∪X ≤ Qk TC(X). 3.3. Products and fibrations. We here study the behaviour of Qsecat and QTC with respect to products and fibrations and establish generalizations of wellknown properties of LS-category and topological complexity. Classically these properties are proven using the open cover definition of these invariants. In order to be able to obtain generalizations to Qsecat, QTC and Qcat, we first need a proof of the classical property based on the Ganea fibrations. Theorem 3.5. Let f : A → X and g : B → Y be two cofibrations. Then, for any k ≥ 0, Qk secat(f × g) ≤ Qk secat (f ) + Qk secat(g). Proof. From the diagram constructed in [11, Remark 7.1] (from the product of fibrewise joins of two fibrations to the fibrewise join of the fibration product) we can deduce the existence of a commutative diagram of the following form: / Gm+n (f × g) Gn (f ) × Gm (g) OOO pp OOO ppp OOO p p p gn (f )×gm (g) OO ' wppp gn+m (f ×g) X × Y. By applying the QkX×Y construction and using the map of Proposition 2.3 we obtain: / Qk / Qk QkX (Gn (f )) × QkY (Gm (g)) X×Y (Gn (f ) × Gm (g)) X×Y (Gm+n (f × g)) VVVV UUUU VVVV U U U VVVV UUUU q k (gn+m (f ×g)) VV U q k (gn (f ))×q k (gm (g))VVVVV q k (gn (f )×gm (g)) UUUUU  V* * X ×Y X × Y.

From this diagram, we can establish that, if Qk secat (f ) ≤ n and Qk secat(g) ≤ m  then Qk secat(f × g) ≤ n + m. Corollary 3.6. Let X and Y be path-connected CW-complexes. For any k ≥ 0, we have Qk TC(X × Y ) ≤ Qk TC(X) + Qk TC(Y ). Proof. Since the diagonal map ΔX×Y : X × Y → X × Y × X × Y coincides, up to the homeomorphism switching the two middle factors, with the product ΔX × ΔY , we have Qk TC(X × Y ) = Qk secat(ΔX × ΔY ) and the result follows from the theorem above.  We now turn to the study of fibrations. We first establish the following result.

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Proposition 3.7. Suppose that there exists a commutative diagram /C

A

g

f

 X

 /Y

ι

in which f , g and ι are cofibrations. Then, for any k ≥ 0, we have Qk secat (g) ≤ (Qk secat(ι) + 1)(secat(f ) + 1) − 1. Proof. Suppose that secat(f ) ≤ p. Then the fibration gp (f ) admits a section. Using the commutative diagram at the bottom of the third page of this note, we obtain from this section a map λ : X → Gp (g) such that gp (g)◦λ = ι. Explicitly, for x ∈ X, λ(x) is an element (γ0 , · · · , γp ) ∈ Γp (Y ) such that γ0 (0) = · · · = γp (0) = ι(x) and at least one path γj satisfies γj (1) ∈ ιf (A) ⊂ g(C). Since ι is a cofibration and the fibration Γp (Y ) → Y , (γ0 , · · · , γp ) → γ0 (0) is a homotopy equivalence, the relative lifing lemma ([21, Th. 9]) in the diagram / Γp (Y )