Symplectic Manifolds with no Kaehler structure (Lecture Notes in Mathematics, 1661) 3540631054, 9783540631057

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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen

1661

Springer

Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

Aleksy Tralle John Oprea

Symplectic Manifolds with no Kahler Structure

Springer

Authors Aleksy Tralle Instytut Matematyczny Polskiej Academii Nauk ul. Sniadeckich 8, 00-950 Warszawa, Poland and Instytut Matematyczny Uniwersytet Wroclawski ul. Plac Grunwaldzki 2/4, 50-384 Wroclaw, Poland e-mail: [email protected] John Oprea Department of Mathematics Cleveland State University Cleveland,Ohio 44115, USA e-mail: [email protected] Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Oprea, John: Symplectic manifolds with no Kahler structure / John Oprea ; Aleksy Tralle. - Berlin; Heidelberg; New York; Barcelona; BUdapest; Hong Kong; London; Milan; Paris; Santa Clara ; Singapore; Tokyo : Springer, 1997 (Lecture Dotes in mathematics: 1661) ISBN 3-540-63105-4

Mathematics Subject Classification (1991): 55P62, 53C15 ISSN 0075-8434 ISBN 3-540-63105-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready AMS-TEX output by the authors SPIN: 10553259 46/3142-543210 - Printed on acid-free paper

Dedicated to Irena, Jan and Kathy

INTRODUCTION In recent years, methods of rational homotopy theory (especially Sullivan models) have been applied successfully to symplectic geometry. In particular, these techniques have proved useful in attacking the Thurston- Weinstein problem of constructing symplectic manifolds with no Kahlerian structure (d. [AT2, BG1, BG2, CFG, DFGM, FG1, FG2, FLS, Has, 01, Trl, Tr2]), certain cases of Arnold's conjecture (cf. [02, MO]) and other geometric problems. Concomitantly, some new and stimulating conjectures and problems have been formulated in the last several years due to this influx of homotopical ideas. Examples include the Lupton-Oprea conjecture [L01] and the Benson-Uordoii conjecture [BG2], both of which are in the spirit of some older and still unsolved problems (e.g. Tliurston's conjecture and Sullivan's pToblem). These results, problems and conjectures are scattered in various research articles. In this work, we intend to present them in a unified way, stressing qeometric techniques flavored with the spice of homotopy theory. Before starting our presentation, we emphasize some particular features of this work. Here, we collect a majority of known results on the problem of constructing symplectic manifolds with no Kahlerian structure. With this in mind, nilmanifolds, solvmanifolds, fiber bundles and surgery techniques are discussed. We also present some relevant homotopy theory, e.g. the Dolbeault raiionol hom.otopy theoru. We give many examples with the aim of claryfying methods of rational homotopy theory to geometers and attracting the attention of "rationalists" to some interesting geometric problems. As an example of the latter, we mention the existence theorems for symplectic fat bundles [ANT, TrK]. This book is meant to be a kind of "bridge" for mathematicians working in two different research areas, so we give proofs (especially geometric ones) of background material where we can while simply providing motivation (and references) where detailed proofs would bring the narrative to a halt. Our explicit aim is to clarify the interrelations between certain aspects of symplectic geometry and homotopy theory, so we try to present as much of the geometry "hidden" behind algebraic calculations as possible. Acknowledgement. The authors express their deep gratitude to Greg Lupton for valuable discussions and advices. They are also indebted to the VolkswagenStiftung "Research in Pairs" grant at Mathernatisches Forschungsinstitut (Oberwolfach) in June, 1996. The first author is grateful to the Polish Research Commitee (KBN) for a financial support. Also, we would like to express our sincere thanks to the referee for valuable suggestions and comments which improved the contents of this work.

CONTENTS

Introduction Chapter 1. The Starting Point: Homotopy Properties of Kahler Manifolds 1.1 1.2 1.3 1.4 1.5 1.6

Differential Graded Algebras and Minimal Models DC A- Homotopy and Invariance of Minimal Models Formality and Kahler Manifolds Rational Homotopy of Fibrations An Illustrative Geometric Example Higher Order Massey Products

VI

1 1 8

18 27 33

40

Chapter 2. Nilmanifolds

45

2.1 2.2 2.3

45 54 ,58

Nilmanifolds The Benson-Gordon-Hasegawa Theorem Symplectic Structures on Nilmanifolds and Related Miscellany

Chapter 3. Solvmanifolds

70

3.1 3.2 3.3 3.4 3.5

71 77

Solvmanifolds and their Mostow Bundles Cohomology of Solvmanifolds: Hattori's Theorem Rational Models of Solvmanifolds with Kahler Structures The Benson-Gordon Conjecture Higher Dimensional Examples and Twisted Tensor Products

87 96 100

Chapter 4. The Examples of McDuff

120

4.1 4.2 4.3 4.4

120

Classical Blow-Ups The Symplectic Blow-Up The Main Result Remarks

122

129 134

CONTENTS

viii

Chapter 5. Symplectic Structures in Total Spaces of Bundles

137

5.1 5.2 5.3 5.4 5.5

137 141 146 151

Preliminaries on Homogeneous Spaces Compact Homogeneous Symplectic Manifolds The Weinstein Problem for Fiber Bundles Koszul Complexes and Minimal Models of Homogeneous Spaces Symplectic Fat Bundles and The Formalizing Tendency of Symplectic Structures

161

Chapter 6. Survey

173

6.1 6.2 6.3 6.4 6.5

173 181 189 193 198

Brylinski's Conjecture and 'Conjectural' Symplectic Invariants Applications to The Original Arnol'd Conjecture Dolbeault Homotopy Theory Miscellaneous Examples Discussion of Problems and Conjectures

References

200

Index

206

CHAPTER 1

THE STARTING POINT: HOMOTOPY PROPERTIES OF KAHLER MANIFOLDS 1. Differential Graded Algebras and Mimimal Models The aim of this chapter is not to present a full-scale exposition of rational homotopy theory, but to provide the geometrically-minded reader with enough background in the subject to understand the applications which follow in later chapters. The reader is referred to [GM, HI, Su, T] for many of the most technical of details. While we shall follow Sullivan's theory of minimal models [Su], we recommend [T] to those who wish to learn the approaches of Quillen and Chen. The work of De Rham [DR] in the 1920's has been fundamental to our understanding of the bond between the geometry and topology of smooth manifolds. Recall that De Rham showed that the cohomology given by closed modulo exact differential forms is precisely the same as that given by the then-new singular cohomology theory (with real coefficients). Thus, differential forms hold within them a crucial piece of homotopical information about the manifold. In the 1950's, Thom and Whitney [Wh] (among others) seemed to believe more; that forms hold all torsionfree homotopical information about the manifold. In the 1950's, however, homotopy theory was just coming to grips with the notion of localization, so it was difficult to make this idea of torsionfree topology precise enough to be linked to forms. The 1960's and 70's saw the development of a huge theory of localization in topology whose goal was to allow spaces to be analyzed 'one prime at a time' (see [HMR] for example). In particular, the idea of doing torsionfree homotopy theory gained the precise meaning of localizing a space X to a space Xo via a map X -+ X o which induces isomorphisms on homology H.(X) ® Q -+ H.(Xo;;Z) and homotopy 1l".(X) @ Q -+ 1l".(Xo), The first major advance in understanding this torsionfree theory was the result of Quillen [Q] that homotopy theory over Q is entirely algebraic; that is, any aspect of the rational homotopy type of a space could be understood within an entirely algebraic category (e.g. differential graded Lie algebras). This led to Sullivan's choice [Su] of a particular algebraic category in which to work, a category in which the De Rham theory may be imitated to produce a convenient (i.e. minimal) model describing the rational type of a space. It is to this theory and some of its refinements, extensions and ramifications that we now turn. Consider the category k-DGA of commutative graded differential algebras over a field k of characteristic zero. Thus, if (A, dA ) is such a differential graded algebra, then A = EllA i is a graded vector space with multiplications AP @ Aq -+ Ap+q which are associative and commutative (in the graded sense); a· b

= (_I)lallblb. a

2

HOMOTOPY PROPERTIES OF KAHLER MANIFOLDS

where laf and fbi denote the degrees of a and b respectively in the underlying graded vector space. (Where convenient, we shall also denote the degree of a homogeneous element a E Ai by deg(a) = i.) The DGA A also possesses a differential d: AP -+ AP+l which is a graded derivation

d(a. b) = (da)· b + (_l)la l a . (db)

satisfying d 2 = 0 analogous to the exterior derivative for differential forms. If (A, d A ) E k - DCA, we denote its cohomology algebra by H*(A, d A ) . A morphism in the category k - DCA is simply an algebra homomorphism which commutes with the differentials. The DGA (A, dA) is augmented if there is a DGA homomorphism f : A ---+ k such that Ker f contains all elements of positive degree in A (denoted A " ). Generally, our DG A's will be augmented in a natural way. The De Rham algebra offorms on a smooth (connected) manifold M, SlOR(M), is augmented by the map on forms

SlOR(M)

-+

lR

=: SlOR(pt)

induced by the inclusion of a point into M, pt -+ M. The fact that i- lR, but HgR(M) == lR elicits the following definition. The DGA (A, dA ) is said to be connected if AO = k and c­connected if HO(A, d A ) = k. The De Rham algebra of a connected space is not connected, but is c-connected. By definition, a differential graded algebra (M A , d) is called a model for (A, d A ) if there exists a DGA-morphism PA :

(M A , d)

-+

(A, d A )

inducing an isomorphism on cohomology. If (M A , d) is freely generated in the sense that M A = A V for a graded vector space V = ED Vi, then it is called a free model for (A, d A). Recall that the notation M A = A V means that, as a graded algebra, M A is a polynomial algebra on even elements veven and an exterior algebra on odd elements VO d d. (Note that a freely generated DGA is not 'free' in the usual sense in the category k - DCA since the requirement that a DGAmorphism commute with differentials precludes the immediate extension of an arbitrary map from the generating set V to the whole DGA.) A morphism between DGA's inducing isomorphisms in cohomology is called a quasi­isomorpliism and the algebras themselves are then said to be quasi­isomorphic.

Definition 1.1. A DGA (MA, d) is called a minimal model of (A, d A), if: (i) (M A , d)

= (AV, d) is a free

model for (A, dA ) ;

(ii) d is indecomposable in the following sense: there exists an ordering in the set V of all free generators of MA, say, V = {x"" C\; E I} such that x (3 < x", ==? deg( x(3) < deg(x",) and the expression for d( x",) contains only generators x(3 < x",. For degree reasons, then, d(x",) is a polynomial in generators xf3 with no linear part. The decomposability criterion above has an equivalent formulation which is apparent in the proof of

DIFFERENTIAL GRADED ALGEBRAS AND MIMIMAL MODELS

3

Theorem 1.2. [GM, LeI] For any c-connected (A, d A ) E k-DGA, there exists a minimal model (MA, d).

Proof. We only give the proof in the 'simply connected case' H1(A, d A ) = a so that the basic ideas are not obfuscated. Because we wish to mimic the cohomology of A, we first define

M(O) = M(1) = rQ;

M(2)

= M(I) ® A(V 2)

where V 2 = H 2(A), d(V 2) = 0 and P2: M(2) A is defined by taking vector HO(A) and 0'2: V 2 = H 2(A) z2(A) C A 2 to the nat­ space splittings 0'0: rQ

­=.

ural (surjective) projection of cocycles onto cohomology ZO HO(A) = rQ and Z2(A) H 2(A) respectively and then freely extending. Then, M(2) is clearly minimal and the map P2 commutes with differentials (since d(V 2) = 0), induces cohomology isomorphisms through degree 2 and an injection on cohomology in degree 3 (trivially, since M(2) has no elements of degree 3). These properties form the basis of an inductive stage­by­stage construction of the minimal model for A. Take as an inductive hypothesis that Pn : M(n) A has been constructed to satisfy

(1) M(n) is minimal (2) Pn induces cohomology isomorphisms through degree n (3) Pn induces an injection in cohomology in degree n + 1. Let zn+l(A) Hn+1(A) and zn+2(M(n)) H n+2(M(n)) be projections of cocycles to cohomology and let An+I Image(d) C An+2 be the surjection of A n +1 onto degree n + 2 coboundaries (i.e. its image under the differential). Take vector space splittings for each of these maps and denote them, respectively, by

Image(d)

I'n+2 ­­­­­­t

A n +l

.

= Hn+l(M(n)) H n+1(A)) and W2 = Hn+2(M(n))) H n+2(A). To continue the induction, we need to extend Pn to A such that is an isomorphism through a DGA map Pn+l : M(n + 1) degree n + 1 and is injective in degree n + 2. But W 1 and W 2 have been defined to precisely achieve these requirements. Namely, define V n+I = WI EtJ W 2 and Now, let WI

M(n

+ 1) = M(n) C9 A(Vn + 1 )

with differential

d(v) and Pn+1 : M(n

+ 1)

=

{

dM(n)(V)

for v E M(n)

0

for v E W1

f'n+2 (v)

for v E W 2

A defined by

4

HOMOTOPY PROPERTIES OF KAHLER MANIFOLDS for v E M(n) for v E WI for v E W 2

.

It is easy to see that Pn+1 satisfies the inductive hypothesis. (In particular, note that the third part of the definition of Pn+1 makes sense because v E W 2 means that [v] = 0 E H n + 2(A), so v is a coboundary.) Thus, we can create n-models for A, M(n), for all n. Upon taking the union MA UM(n), the minimal model for A is then constructed. 0

=

Example 1.3.

(1) Consider an even sphere s2n and its De Rham algebra fl DR(S 2n ). To create the minimal model for fl DR(S 2n ) (which, remember, induces an isomorphism on cohomology), we need a cocycle in degree 2n which maps to the cocycle in fl DR(S 2n) representing the fundamental class of Call this generator x and take the DGA freely generated by it, (A(X2n), d = 0). Now (A(X2n), d = 0) itself maps to fl DR(S 2n) because x freely generates it and, since x maps to a cocycle, the requirement that the map commute with differentials is automatically satisfied. We do not yet have a model for s2n since all higher powers of x represent nontrivial cohomology classes in A(x) - classes which do not exist in H*(s2n). Thus these extra powers ofx must be killed by the addition of a generator y in degree 4n-l with dy = x 2 . It is easy to see that defining don y in this way kills all extra cohomology. Moreover, we can map y to fl DR(S 2n ) by taking it to zero. The minimal model of s2n is then

s»,

The definition of d on y is precisely the definition of the corresponding k-invariant in the rational Postnikov tower for s2n and the existence of generators only in degrees 2n and 4n - 1 reflects Serre's theorem that even spheres have finite homotopy groups except in those degrees. We shall make this precise below. (2) Similarly, an odd dimensional sphere s2n-1 has a minimal model given by (A(X2n-I), d = 0). (3) Complex projective space has a minimal model which closely mimics the cohomology generated by the symplectic cohomology class. Namely,

A(cpn)

dy = x n+I). In fact, the minimal model of cpn may be constructed from the cohomology algebra alone - an important property in rational homotopy. (4) The minimal model of a torus is A(Tn ) = (A(XI, ... ,x n ) , d = 0) where each generator has degree IXil = l.

= (A(X2, Y2n+I),

At the end of Example 1.3 (1), we hinted that the minimal model contains more information in it than simply the cohomology of a space. In order to

DIFFERENTIAL GRADED ALGEBRAS AND MIMIMAL MODELS

5

relate minimal models to homotopy theory (and more specifically, to localization) directly, we need a DGA over (ll to replace the De Rham algebra. In [Su], Sullivan created exactly such a DGA called the DGA of rational polynomial forms A;'L' Let n be a standard simplex in IPI. n +1 ,

= {(to, ...

, tn)

n

10::; t;

::; 1, Lti

= I},

i=O

and let (] denote the boundary operator. Consider the restriction to differential forms in IPI. n +1 of the form L

n

of all

cPi,ikdti, /\ ... /\ dti k

where cPi,.i k are polynomials of to, . . . ,t n over the rationals. Denote the set of all such forms by A * and note that there are two relations in this algebra: n

Lti

=1

n

and

i=O

Ldti

= O.

i=O

Let X be a simplicial complex (with a constituent simplex denoted by o ). Define A;'dX)

= {(w" )"EX I w" E A*(o),

and w E ----> B, 7I"l(B) acts on H.(F). This action arises from the homotopy lifting property of fibrations as follows. Consider the diagram DB x F x 0

---->

1 DB x F x I

E

1 w ---+

B

where DB denotes the space of based loops on B, the top row embeds F in E and w(iT, i, t) = iT(t). The homotopy lifting property then says that the homotopy W may be lifted to a homotopy w: DB x F x I ----> E with Wo the embedding of the top row. Also, since W1 (iT, f) = bo (i.e. the basepoint), commutativity of the diagram implies that W1: DB x F ----> F with adjoint W1: DB ----> FF. Taking path components (using 71"0), we obtain a map

where [F, F] denotes the free homotopy classes of self-maps of F. This process gives the action of 71"1 (B) on H. (F) by taking induced homomorphisms of self-maps in the image of ¢. Note that we do not usually obtain an action of 71"1 (B) on 71". (F) because free homotopy classes may have several based classes which project to them. A fibration for which 71"1 (B) acts nilpotently on H.( F) is called a quasi-nilpotent fibration. Exam.pIe 1.10.

(1) Any simply connected space is trivially nilpotent. (2) Any K(7I",1) with 71" nilpotent is a nilpotent space. (Recall that a space X is a K(7I", n) if 7I"n(X) = 71" and 7I"j(X) = 0 for all i :{:. n.) Thus, in this case, the nilpotency condition is fulfilled because first, nilpotency is assumed for 71" and secondly, there are no higher homotopy groups for 71" to act upon. In Chapter 2, we shall see that nilmanifolds are special cases of this example. Solvmanifolds, however, are not nilpotent spaces and this is a major complication in analyzing their homotopical properties.

8

HOMOTOPY PROPERTIES OF KAHLER MANIFOLDS

Theorem 1.7 is then saying that, for a nilpotent space, the minimal model is a perfect reflection of the rational homotopy type of the space - that is, of the localization X o. For instance, for i > 1, Vi '== Hom( 7ri (X), Q), where 7ri (X) is the i t h homotopy group of X. The minimal model (A V, d) is therefore an algebraic version of the Q-localization of X, X o. Now, it also must be noted that the categorical equivalence of Theorem 1.7 implies that a map of spaces I: Y --+ Z induces (in the usual fashion) a map of forms 1*: A*(Z) --+ A*(Y) and a map of minimal models (AVz, d z) --+ (AVy, dy) unique up to DGA-homotopy. We shall explore this in more detail in §2.

2. DGA-Homotopy and Invariance of Minimal Models In this section, we will examine some properties of minimal models including their uniqueness up to isomorphism. In order to understand this, we first look at what means for two DGA maps to be 'homotopic'. Of course, we should always keep Theorem 1.7 in mind. What we are actually doing is understanding how the notion of homotopy translates from spaces to DGA's, thereby becoming an essential ingredient in the categorical equivalence between homotopy types and minimal models. Let M = (AX, d) be a freely generated DGA which has the property that d(xoJ E AX O. Hence, if maps f and g are homotopic by H, then

HA1(X) g(x) g(x) - f(x)

= HAo(x) + H(dx) = f(x) + dH(x) = dH(x).

Thus, on cocycles, homotopic maps differ by coboundaries. In other words, homotopic maps induce the same homomorphism on cohomology - just as they should. Further, although we shall not prove the following result, it provides the key to understanding rational homotopy theory algebraically.

10

HOMOTOPY PROPERTIES OF KAHLER MANIFOLDS

Theorem 2.2. [HI, Property 15.12] Homotopic maps of spaces induce homotopic DCA maps of minimal models. Indeed, by the general theory (see Theorem 1.7), for localizations at zero X o and Yo with respective minimal models Mx and My, there is a bijection of sets of homotopy classes

[Xo, Yo] == [My,Mx]. Note that the correspondence is a contravariant one just as for forms and cohomology. Instead of giving a proof of this result, we shall show how induced maps of minimal models arise. First, let's simplify our notation a bit by writing

O(Xi)

(idt

= '" LJ ­(Xi) n! 00

n=l

so that Al(X;) = Xi + dXi + O(Xi)' It is sometimes convenient to write x; for AI(Xi) as well. Also, since a DGA homotopy is defined on M, it is often more convenient to define the homotopy on each end, Xi and x;, to begin with as opposed to Xi, xi and dXi. With this in mind, note the following Lemma 2.3. (A(Xi, Xi, dXi), d) = (A(Xi, x;, Xi), d). Proof We need only show that dx may be obtained from generators {Xi, x;, X;}. If xi is in the first stage of the construction, then O(Xi) = 0 and dXi = x; ­ Xi. Assume inductively that dXj may be obtained for all Xi through stage k, say. Then, for Xi in stage k + 1, dXi = x; ­ Xi ­ O(Xi)' But, by the inductive assumption, O(x;) may be obtained from the new generating set (since it is 0 decomposable). Hence, so can dXi.

Now, in order to see how spatial maps translate into maps of models, we will develop an obstruction theory for models (see [Mor] and [GM]). This theory plays a roughly analogous role to that of ordinary obstruction theory in homotopy theory. As a warm­up, consider Proposition 2.4. Let f: M ­+ A be a DCA map and suppose that M is extended to a DCA Nt = M (9 AV with d(V) C M for some vector space in degree n V = (Xl, ... ,Xk). Then the obstructions to extending f to a DCA map 1: Nt ­+ A are cohomology classes [j(dXi)] E H n +1(A), i = 1, ... , k. Proof First, if j exists, then f(dxi) = dj(x;) , so j(Xi) is a coboundary and [f(dxi)] = O. Secondly, suppose that [j(dXi)] = O. Then there exists a E A with da = f(dx;). Define j(Xi) = a so that the extension j is obtained with dj = fd.

o

Remark 2.5. An extension Nt as above is sometimes called an elementary extension or a Hirsch extension, the latter in honor of Guy Hirsch who pioneered DGA methods in fibration theory around 1950.

INVARIANCE OF MINIMAL MODELS

11

Consider now a more complicated situation corresponding to the homotopy lifting property for fibrations. Namely, consider the DGA homotopy commutative diagram f

M

--t

il

1¢>

M®AV where M ®AV is the following

A

f --t

B

= M as above and ¢>f c::!i via a homotopy H:

Lifting Problem 2.6. When does there exist an extension that ¢>1 c:: ! (where the homotopy fI extends H as well)?

M

1

-+

1: M

B. There

-+

A such

In order to formulate an answer to this question, recall that to any cochain map of cochain complexes (e.g. a DGA map) ¢>: A -+ B, there is a cochain complex C(¢» = {C n(¢», d} called the algebraic mapping cone of ¢> with

and differential d(a, b) = (-da, db + ¢>(a)) (where the differentials inside the parenthesis are those of A and B). It is easy to check that d 2 = 0 and that there is a long exact sequence in cohomology

where j: B" -+ C n+l(¢» is the obvious inclusion and p: C n+l(¢» obvious projection. With this device, we can now prove Theorem 2.7. The obstructions to the existence of 1: B in the lifting problem are cohomology classes

M -+ A

-+

An+l is the

and

it . M1 -+

[-f(dx;), /(x;) - H(D(x;))] E Hn+l(C(¢»). Proof. We first show that (- fdx;, !x; - HDx;) is a cocycle. For this, recall that

)'1(x;)

= x; + dx; + D(x;) = AO(Xi)

so that dD(xi)

= dAl(X;)

- dAo(x;)

+ dXi + D(x;)

= (Al -

AO)(dxi)' Now, by the definition of

12

HOMOTOPY PROPERTIES OF KAHLER MANIFOLDS

DGA homotopy, H >"0

= rjJf and

d( - fdxi,!Xi - Ho'Xi)

Now suppose that! and

H >"1

so

= (dfdxi' dfxi - dHo'Xi - rjJfdxi) = (Jd2xi, fdxi - H do'Xi - rjJfdxi) = (0, fdxi - H(>"1 - >"O)(dXi) - rjJfdxi) = (0, fdXi - fdxi + rjJfdxi - rjJfdxi) = (0,0).

H exist.

d(jXi' HXi)

= fi,

Then we have

= (-d!Xi, dHxi + rjJ!x;)

= (- !dXi, H dXi + rjJ!Xi) = (-fdxi,!Xi - rjJ!Xi - HnXi + rjJXi) since dXi is decomposable and! restricts to f on M. Then

d(jXi, HXi)

= (-fdxi,!Xi -

Ho'Xi)

since o'Xi is decomposable and H restricts to H on M. Therefore, if ! and H exist, then the obstruction cocycles (- fdxi, f Xi - Ho'Xi) are coboundaries and so zero elements in Hn+l(C(rjJ)). Conversely, suppose that [- fdxi, fXi - Ho'Xi] = in Hn+l(C(rjJ)). Then there exists (a, b) such that d( a, b) = (-da, db + rjJa) = (- [ti», ,!Xi - Ho'Xi). Hence, da = [d», and db+rjJa = fXi - Ho'Xi. Now define !Xi = a and note that d!Xi = da = fdxi. Also, using Lemma 2.3, define H by

°

HIM = H HXi = rjJXi

{

I! >"1 Xi = fXi

HXi = b and note that dH>"IXi

sits; = db = fXi

= dfxi = fdxi = H)Ildxi. - Ho'Xi - rjJ!Xi

Finally, then,

= H(>I1Xi -

>"OXi - o'Xi)

= it a«;

Thus, the vanishing of the obstruction class allows the extensions! and defined.

H to

be D

INVARIANCE OF MINIMAL MODELS

13

Corollary 2.8.

(1) If 1; is a quasi-isomorphism, then 1 and if exist. (2) If 1; is a surjective quasi-isomorphism and 1;f = f, then so that Ii = f and 1;1 = f.

1 may be chosen

Proof For (1), simply note that the hypothesis implies that H*(C(¢)) = 0 using the long exact cohomology sequence associated to the algebraic mapping cone. Hence, all obstructions to lifting vanish. For (2), because ¢f = fi, we may choose the original homotopy H to satisfy H y = ¢ fy, H >'1 Y = fy and H f) = 0 for all generators y in M. In particular, the last equality says that HD(y) = 0 for all y as well, so the obstruction classes (which vanish by (1)) are [- fdxi, jx;]. Now, since [-fdxi,JXi] = 0 and ¢ is assumed to be a quasi-isomorphism, there exists (a, b) with d( a, b) = (-da, db + ¢a) = (- fdxi ,JXi)' Hence, da = fdxi and db + ¢a = j Xi. Because ¢ is assumed to be surjective, there is an a' E A with ¢a' = b. Then fXi = ¢(a + da') and we may define

l xi=a+da'. Thus, djXi = da = [d», and ¢ lxi tativity in the diagram.

= j Xi, so the lifting j

induces strict commu0

The results above allow us to model maps as well as spaces. Indeed, in the following theorem, the map F is called a model for the DGA map f: A -+ B. The construction of F follows from Corollary 2.8 taking ¢ = PB and using the fact that the minimal model MA is built in stages by Hirsch extensions. Theorem 2.9. [LeI] Any DCA morphism f : (A, dA ) -+ (B, dB) can be lifted to a DCA morphism F completing the following homotopy commutative diagram

f

---+

Example 2.10.

(1) Let 53 -+ 52 be the Hopf map. Because is a fibration, the induced map on forms (either polynomial or De Rham), C: A*(5 2 ) -+ A*(53), is injective. Of course, 53 has no degree 2 cohomology, so the image of 5 2's volume form must be exact, C(V2) = d(3, and this should be reflected in the model. Indeed, it is easy to see that C V2 1\ (3 is a non-exact closed form in A*(53) and so generates H 3(5 3) ==' 22:. In fact, 53 is a contact manifold and (3 is a contact form on 53 and C V2 1\ (3 is a true volume form on 53. Also, it is well known that is a generator for 11"3(5 2 ) ==' 22: and this too should be visible in the model. Recall that the minimal models for 52 and 53 are M S 2 = A(X2, Y3) with dx = 0, dy = x 2 and M S 3 = A(Z3) with dz = O. Here, subscripts

14

HOMOTOPY PROPERTIES OF KAHLER MANIFOLDS refer to the degrees of the corresponding generators. Since M 53 has no elements in degree 2, there is no choice for the model map E but that :=:(x) = 0. Now, if :=:(y) = too, then the general correspondence is between minimal models and rational homotopy would say that homotopically trivial - which is false. Thus, :=:(y) is some nonzero multiple of z. Because the map on generators of the minimal models is supposed to reflect the induced map on rational homotopy groups and represents the generator of 7T3(5'2), we must have :=:(y) = z. In order to check all this, we note several facts: (1) For the minimal model map P52, we have P52(X) = V2 and P52(Y) = 0 - the latter since 5'2 has no forms above degree 2. (2) For P53, we may define P53(Z) = -CV2/\(3 since we only need to induce an isomorphism on cohomology. We now have two composite mappings f,g: M52 --+ A*(5'3) with

°

f(x)

= CV2 ,

f(y)

= 0,

To define a homotopy H:

M 152

g(x) --+

= 0,

g(y)

= -C V 2 /\ (3.

A*(5'3), We first write

" ,x,y . -) = A(x,y,x,y = A(x,y,x,y,dx,dy)

with differential most easily defined on the latter set of generators

dx

and d2 = y' = eBy,

= 0,

dy = x 2 , d(x)

° of course. dx = x' - x

= dx,

d(f}) = df}

Then, using the definitions of x'

and

df} = y' - y - xx - xx'

where the reader can verify that O(y) = 2xx + xdx = xx + xx'. Now, H may be defined on each 'end' by

H(x) H(x')

.::, V2,

0,

H(y) H(y')

but what should the definitions of H(x) and H(f}) be? The only restriction is that H must be a DGA map. That is, we require dH(x) = H(dx) and dH(f}) = H(df}). In fact, this is always the crucial point in the construction of any DCA homotopy. In order to see what these relations mean, we compute

INVARIANCE OF MINIMAL MODELS

15

H(dx)

= H(x') -

H(dy)

= H(y') - H(y) - H(x)H(x) - H(x)H(x') = -c 1/2 1\ (3 - °- H (X )c 1/2 - ° = -c 1/2 1\ (3 - H (X )c 1/2 .

H(x)

Clearly, if we define H(x) = -(3, then we obtain dH(x) if we define H(y) = 0, we obtain from above,

= H(dx).

Then,

by graded commutativity since C 1/2 has even degree. Plainly this is dH(y) also and H has been constructed. Therefore, on minimal models, the model S of the Hopf map is given by S(x) 0, S(y) z.

=

=

(2) Hopf's Theorem states that the set of homotopy classes [X, sn] of maps from a complex of dimension n to an n-sphere is in bijection with Hn(X; Z;:). The usual proof is by obstruction theory. If n is odd, then the minimal model of S" is A(zn) with dz = 0. A map X o -+ So then corresponds to a DGA map A(z) ........ AV where Mx = (A V, d). But, since dz = 0, z can go to any n-cocycle in Mx and the map is determined by this cocycle. Hence, there is a surjection zn(Mx) -+ [Xo, So], Furthermore, if f and 9 are two maps with f(z) 0: and g(z) (3, then

=

Msn

= A(z, z', z)

with

dz

= z' -

=

z

and the existence of a homotopy H with H(dz) = dH(Z) is equivalent to the cocycles 0: and (3 differing by an exact form (i.e. dH(z) = (3- 0:). Thus, the homotopy relation corresponds to exactness in zn(M x) and, so, Hn(X; Q) == [Xo , So]. We leave it to the reader to handle the case of an even sphere. We have given the examples above in great detail only to illustrate the idea of DGA homotopy. Of course, it is too much to hope that more complicated examples can be analyzed so explicitly. Nevertheless, the general existence results are quite powerful as we shall see. One point that we have neglected, but which is essential, is the uniqueness of the minimal model. How can we hope to model maps, for instance, if we don't have some sort of 'fixed' models for our DGA maps to transform? In order to understand uniqueness properties of the minimal model, we need the following result (which we prove along the lines of [GM]). Theorem 2.11. If ¢ : M ¢; is a DGA isomorphism.

-+

N is a quasi-isomorphism of minimal DGA's, then

To prove this 'minimality' result, we first require the following

16

HOMOTOPY PROPERTIES OF KAHLER MANIFOLDS

Lemma 2.12. Let M == (AV, d) and M(n) == (A(EBi H*(AV,d) by

'ljJ(c) { 'ljJ( n)

= [c] =0

for c E C i for n E N i

Extending freely to (AV), we see that 'ljJ is injective on AC and zero on Ideal(N), Clearly, 'ljJ dc = 0 = d'ljJc since de = 0 in (AV, d) and d = 0 constantly in H*(AV, d), In order for 'ljJ to be a DGA homomorphism, however, we must still show that 'ljJ dn = d'ljJn as well. Again, d = 0 in H * (AV, d), so the righthand side is zero, Therefore we must show 'ljJ dn = 0 too, Let dn = r + s where r E AC and s E Ideal(N), Then 0 = d 2 n = dr + ds = 0+ ds since r E AC and we see that ds = 0, But s E Ideal(N), so the property above implies that s = dt for some t. But then r = d(n - t), so r is exact with 'ljJ(1') = [1'] = 0 and 1/;(dn)

= 1/;(1') + 1/;(s) =0+0 =0

FORMALITY AND KAHLER MANIFOLDS

19

since 1.j; is zero on Ideal(N). Thus 1.j; commutes with the differentials and, therefore, is a DGA homomorphism. Because the cohomology is generated by C, 1.j; induces the identity on cohomology. Now suppose that (A V, d) is formal; that is, there exists 1.j;: (AV,d)

--+

H*(AV,d)

with 1.j;* the identity. First, let's focus on the minimal model MH of the DGA (H* (AV, d), d = 0). Recall from the construction of the minimal model in Theorem 1.2 that, for the (n + 1)­st stage, we define

MH(n

+ 1) = MH(n) !Xl A(V n + l )

where V n +1 = WI EB Wz with differential d zero on WI and injective on Wz (designed to kill excessive cohomology in degree n+ 2). Further, pn+1 : MH (n+ 1) --+ (H*(AV, d), d = 0) maps WI injectively into cocycles to create a cohomology isomorphism and maps Wz into elements which will be coboundaries in degree n+2. But, for (H*(AV, d), d = 0), the vanishing of the differential implies that there are no coboundaries and that every element is a cocycle. Hence, Pn+1 (W z ) = 0 and Pn+1 is surjective. Inductively, we have a surjection p: MH --+ H* (A V, d) which satisfies p(WD = 0 for all i. In the notation of the theorem then,

Moreover, if ex E Ideal(N), then clearly p(ex) = O. If do = 0, then this means that p*([ex]) = 0 as well. But p" is an isomorphism, so it must be the case that [ex] = 0 E H*(M H ) . This of course means that ex is a coboundary ex = d{3 for some {3. Hence the criterion of the theorem is satisfied for MH· Now, because p is a surjective quasi­isomorphism, by Corollary 2.8 (2), there is a strictly commutative diagram

MH

1p

(AV, d)

H*(AV,d)

where cf; is an isomorphism by Theorem 2.11. Let ut = cf;-I(W{) and = Then, since cf;# is an isomorphism, we see that AV = AU where U i = ut EB Now, it is plain that, in AU,

Let ex E Ideal(N) with dex = O. Then ¢(ex) E Ideal(NH) and d¢(ex) = 0 (since cf; commutes with differentials). Since MH satisfies the criterion of the theorem, there exists a {3 with d{3 = cf;( o ). But ¢ is an isomorphism, so there is a , E AU with cf;(J) = {3. Consequently, d, = (}' and AU = AV satisfies the criterion as well. Remark 3.2. The criterion of Theorem 3.1 due to Deligne­Griffiths­MorganSullivan is a way of saying that all Massey products vanish simultaneously. By

20

HOMOTOPY PROPERTIES OF KAHLER MANIFOLDS

this we mean that choices can be made for Massey representatives such that all Massey products are zero (see Section 6 of this chapter). Example 3.3.

(1) Spheres are formal; odd spheres trivially so and even spheres by a simple application of the criterion with C = (x), N = (y). It is also easy to construct a quasi-isomorphism 1/J: Msn --+ H* (sn) directly. (2) A similar argument shows that all complex projective spaces are formal. (3) A theorem of Halperin (see [GHV, H2J) says that all homogeneous spaces of maximal rank are formal spaces. This theorem will be proved in Chapter 5 (see 5.4.16). (4) If a space X has cohomology freely generated (i.e. polynomial on evens and exterior on odds), then there is an embedding H*(X) H*(A, 6)j([a], [cD

where ([a], [cD denotes the ideal generated by [a] and [c]. Now, we want to introduce quadruple Massey products. Note that as one should expect, the quadruple product is defined for cohomology classes

[a]' [b], [c]' [d], if the triple Massey products ([a], [b], [cD and ([b], [c], [d]) vanish. However, a stronger condition must hold. Namely, the triple Massey products given above must vanish simultaneouslsi in the sense of the definition below. This simultaneous vanishing will be essential in constructing symplectic manifolds without Kahler structure, especially in the solvmanifold case (Chapter 3). In Chapter 3 the reader can find examples of symplectic solvmanifolds which have simultaneously vanishing triple Massey products and non-vanishing quadruple Massey products. Since ([a], [b], [cD and ([b], [c]' [dJ) are well-defined,

[a] . [b]

= [b] . [c] = [c] . [d] = 0

42

HOMOTOPY PROPERTIES OF KAHLER MANIFOLDS

or

bX12 = a· b,

bX23

= b· c,

34

= c' d Consider the representatives of the triple for some cochains x , x and X Massey products as cohomology classes in H*(A), say 12

y

bx

3 4.

23

= a· x 23 + x 12 . c,

Z

= b· x 3 4 + x 23 . d

Definition 6.2. We say that the triple Massey products ([aJ, [bJ, [c])

and

([bJ, [cJ, [d))

vanish simultaneously if both cohomology representatives y and z can be chosen

as coboundaries (4) Remark 6.3. Recall that the indeterminancy of the choice of y and z lies in ideals ([aJ, [cD and ([bJ, [d)) respectively. Therefore the vanishing of the triple Massey products does not imply condition (4). An example of vanishing but not simultaneously vanishing triple Massey products is given in [O'N].

Definition 6.4. Let [aJ, [bJ, [c] and [d] be cohomology classes in H*(A, b) of degrees p, q, rand s respectively. Assume that

[a] . [b] = [b] . [c]

= [c] . [dJ = 0

and ([aJ, [bJ, [c]) and ([bJ, [c],[d]) vanish simultaneously. Select representative cocycles a, b, c, d and cochains x 12 , x 23 , 34 X such that

= c· d. Making use of the previous assumptions we know that there exist cochains x 13 and x 24 such that bX13

= n . b,

bx 23

= b· C,

= a . X23 + X 1 2 . C,

bX24

bX 12

bx 3 4

= b· x 3 4 + x 23 . d.

We can show by a strightforward calculation that the cochain w

= a·

X

24

+ X 12 . x 3 4 + X 13 . d

is actually a cocycle of degree p+ q + r+ s -1. Define a quadruple Massey product

[w] = ([aJ, [bJ, [c], [d)) as a collection of all cohomology classes [w] E HP+q+r+s-l(A, b) that we can obtain by the above procedure. Continuing this process one can define n-tuple Massey products for all n > 4. We have already mentioned that Massey products provide obstructions to formality. An exact meaning of this is given by the result below.

HIGHER ORDER MASSEY PRODUCTS

43

Theorem 6.5. If a minimal graded differential algebra (M, d) = (A V, d) is formal, one may make uniform choices so that the cochains representing all Massey products and higher order Massey products are exact. Proof. We make use of the Formality Criterion (Theorem 3.1). We know that

where the complement N satisfies the property that, for each

U

E (N) with

du = 0, we have u = dv. Consider, first, a triple Massey product, say ([aJ, [bl, [c)) represented by a cocycle

y=7i'X 23+X 12 · C .

Note that x 12 and x 23 are not cocycles and therefore they cannot be represented by expressions consisting only of generators belonging to C. Moreover, without loss of generality we can assume that they lie in (N). This means that y E (N) and y is a coboundary by Theorem 3.1. Using the expression

we get W E (N) (by the same argument as before). The process of constructing higher order Massey products shows that all corresponding representing cocycles can be chosen belonging to (N) and, therefore, must be coboundaries. 0 Examples of calculations will be given in Chapters 3 and 5. Here we restrict ourselves to an example of a simply connected minimal differential graded algebra which admits non­vanishing Massey products. Examples 6.6.

(1) Consider the minimal differential graded algebra

Ixl = Iyl = 2, 10:11 = 10:21 = 10:31 = 3 d(x)

= d(y) = 0,

d(o:d

= x 2,

d(0:2)

= xy,

d(0:3) = y2

One can check that

([x], [x], [y)) is a well­defined and non­vanishing triple Massey product. This example often occurs in situations when it is necessary to construct a non­formal differential graded algebra, therefore, it is often called a generic example of a non-formal differential graded algebra.

(2) For instance, by modifying this generic example, we can construct an elliptic simply connected and cohomologically symplectic, yet non­formal, minimal differential algebra. Recall that a minimal DGA (A V, d) is said to be elliptic ifits cohomology H*(AV, d) and generating vector space V are both finite dimensional. Namely, consider

44

HOMOTOPY PROPERTIES OF KAHLER MANIFOLDS

= (A(x,y,z,al,a2,a3,,6,,),d) Ixl = Iyl = Izi = 2, lall = la21 = la31 = 1,61 = 3, 1,1 = 9 d(x) = d(y) = d(z) = 0, d(aI) = x 2, d(a2) = xy, d(a3) = y2 d(,B) = yz, db) = z5 + (alY - x(2)(a3z - y,6) (AV,d)

This differential graded algebra is non-formal. One can guess that it is non-formal since it is a modification of the generic example, however, here the proof of the non-formality is better seen directly from the Formality Criterion (Theorem 3.1). If (A V, d) were formal, there would exist a decomposition V = C EB N with the properties given by the Formality Criterion. However, we see that V 2 C C and V 3 n C = {O} (where V 2 and V 3 denote, respectively, the subspaces of generators of degree 2 and 3). This means that

Since there are no elements of degree 4 which are not cocycles in this algebra, aly-xa2 is not a coboundary. However, it is a cocycle, which is a contradiction. It can be checked that z is a cohomologically symplectic element (this is straightforward, but not trivial). Remark 6.7. Example 6.6 (2) was constructed in [LOI] in order to give an answer to the question of whether or not a simply connected elliptic compact manifold with a cohomologically symplectic minimal model is formal. Although the DGA above is known to be the minimal model of a smooth manifold, it is not known if the manifold is actually symplectic. The formality of simply connected symplectic manifolds is still one of the most interesting questions in symplectic homotopy.

CHAPTER 2

NILMANIFOLDS The problem of constructing symplectic manifolds with no Kahlerian structure seems to be very difficult in general since its solution involves many subtle interactions between algebraic topology, differential and algebraic geometry and analysis. For example, the existence of a symplectic structure w on a manifold M implies that M admits some compatible almost complex structure J. If this almost complex structure were integrable, then M would be Kahlerian, However, if the almost complex structure is non­integrable, then this does not imply any definite result since it is necessary to consider a moduli space of compatible almost complex structures for various possible symplectic structures. Fortunately, the existence of a Kahlerian structure on M implies strong consequences (e.g. formality) concerning its minimal model (see Chapter 1). Thus, a possible way of constructing symplectic manifolds with no Kahlerian structure is to find a symplectic manifold whzch does not satisfy these conditions (e.g. is non-formal). In 1984 [McDl]' McDuff provided examples (which we shall talk more oflater) of closed simply connected symplectic non­Kahler manifolds. More recently, Gompf has given a general 'surgery' technique for the construction of such examples. The main result of this chapter solves the problem completely for the class of manifolds known as nilmanifolds.

1. Nilmanifolds Let's begin with the definition of the central objects of study in this chapter. Definition 1.1. A nilmanifold is a compact homogeneous space of the form N /r, where N is a simply connected nilpoten t Lie group and r is a discrete co-compact subgroup in N (i.e. a lattice). The general theory of nilmanifolds is contained in [Aus, VGS]. We restrict ourselves to certain relevant properties of nilmanifolds and to presenting some examples which reveal those properties. Examples 1.2. (1) An n­dimensional torus T" = /71 n is obviously a nilmanifold. (2) (Malcev) Let n be a nilpotent Lie algebra with the property that there exists a basis in n, el, e2, ... , en, such that the structural constants cfj arising in brackets [ei,ej]

= I>fjek k

NILMANIFOLDS

46

are rational numbers for all i, j, k. Then Malcev [VGS] shows that there is a simply connected nilpotent Lie group N corresponding to n which admits a lattice (i.e. a discrete cocompact subgroup) f so that N /f is a compact nilmanifold. (3) The 'universal' example of a nilpotent Lie group is the group of upper triangular matrices having 1's along the diagonal, 1IJ n (IPI. ). X12

X13

1

X23

o Let 1IJ n C 1IJ n (lPI.) denote the set of matrices having integral entries. Then, 1lJ n (2::) is a lattice and the quotient

is a nilmanifold. The group 1IJ 3(lPI.) is called the Heisenberg group and the resulting nilmanifold is called the Heisenberg nilmanifold. This leads to (4) The Generalized Heisenberq Manifolds (see [CFG]). Let H(I,p) be the group of (p+ 2) x (p+ 2) square matrices of dimension 2p+ I of the form A 1

o where E p denotes the p x p identity matrix, A and Care p x l-matrices and b is a real number. These are the generalized Heisenberg groups. Because H(l,p) is contained in the nilpotent group 1IJ p +2(lPI. ), it is a nilpotent Lie group. Clearly the integral matrices of H(1,p) form a discrete cocompact subgroup I' and H(l, p)/f is a nilmanifold. From Theorem 2.2, H(1, p)/f cannot carry any Kahler structure since it is a non-toral nilmanifold. Also note that, if La: H(I,p) ----+ H(1,p) denotes left translation by a E H (1, p), then

x

0

La

= X + A,

yo La

= y + b,

Z

0

La

= Z + Ay + C

where we have chosen a (global) system of coordinates {X, y, Z} satisfying, for : A(r) cI>(x)

1

= "2x,

cI>(y)

= y,

---+

A(U3(J:,)) by cI>(z)

= z.

Clearly, cI> commutes with differentials and, since it is an isomorphism on vector spaces of generators and minimal algebras are freely generated, cI> is an isomorphism at the DGA level as well. Using the fact that the homotopy type of an Eilenberg-Mac Lane space is determined by its fundamental group, we have proved that I' and U3(J:,) have the same rationalizations. In fact, note that the very same argument holds for any group of the form I'

= (x,y,z:

[x,y]

= zn, [x,z] = e = [y,z]).

Hence, all of these finitely generated torsionfree nilpotent groups rationalize to the same group. There is one other important feature to notice about minimal models of nilmanifolds. Namely, there are precisely n degree 1 generators for an n-dimensional nilmanifold. Because the minimal model is an exterior algebra (so that the square of each generator vanishes), there is only one element which can represent the top class in cohomology. That element is the product of all generators, v = Xl ... x n . Now let's tie in the symplectic geometry of nilmanifolds with the minimal model. First, it is important to realize that symplectic nilmanifolds form a very special class of symplectic manifolds. In particular, it is not the case that

NlLMANlFOLDS

53

any manifold having a degree two cohomology class which multiplies to a top class is symplectic. For instance, Cp2#Cp2 has several such classes, but it cannot be symplectic since it cannot even have an almost complex structure [Au2] (which is automatically provided by a symplectic structure). On the other hand, Nomizu's theorem guarantee's the existence of a cochain in (An*, 8) representing the cohomology class in which multiplies to a top class. Since the cohomology product cannot be zero, then the cochain (i.e. wedge) product of the cochain cannot be zero. But this then defines a left invariant volume form on the nilpotent Lie group. Both the cochain (as a left invariant form) and the volume form then descent to the nilmanifold to provide a symplectic form wedging to a volume form. Therefore, for nilmanifolds, the proper symplectic cohomology behavior is sufficient to guarantee the existence of a true symplectic structure. Now, if a nilmanifold M 2k is symplectic, then there must be a degree 2 element of the minimal model which multiplies up to the top element 1/ = Xl ... X2k. We can write this degree 2 element as

Since w k

=

1/, the sum in the expression for w must contain all the degree 1 generators. Of course not any '2-form' may be chosen. The symplectic 'form' w

must be closed. Example 1.7. What must a 4-dimensional nilmanifold look like rationally? By what we have said, the minimal model must have four generators and, for nonredundancy reasons, two of those must be cocycles. Thus, there are three rational homotopy types to deal with (up to isomorphism). (1) The Torus T 4 . The minimal model is (A(X1,X2,X3,X4),d symplectic 'form' is w = XlX2 + X3X4.

= 0)

and a

(2) The Kodaira-Thurston manifold KT. For a geometric description of this manifold, see §2 below. This manifold is obtained by taking the product of the Heisenberg manifold M = U3(lW.)/U3 (.::z ) and the circle 51. The minimal model is given by

(A(Xl, x2, x3, X4), d)

with

dx = 0, dy = 0, dz = xy, du = 0

where the first three generators (and differential definitions) come from M and u comes from the circle. A symplectic 'form' is then given by w = xu + yz. Note that the degree 1 cohomology of KT is generated by the classes of x, y and u. Hence, the first Betti number is three and, by the Hard Lefschetz property (see Example 1.3.6), KT cannot be Kahler.

This was the how it was first demonstrated that compact symplectic non- Kahler manifolds exist. (3) Take the minimal model (A(XI,X2,X3,X4),d) with dx = 0, dy = 0, dz xy and du xz. Recall that the corresponding finitely generated torsionfree nilpotent group may be realized as a nilmanifold. Then a symplectic 'form' is given by w = xu + yz. See Example 1.6 above.

=

=

54

NILMANIFOLDS

So, all 4-dimensional nilmanifolds admit symplectic structures. By contrast, the reader may calculate the following Example 1.8. The 6-dimensional nilmanifold U4(IP!.)/U4(!Z) is not symplectic because no closed 'form' w in its minimal model can be found whose cube does not vanish.

2. The Benson-Gordon-Hasegawa Theorem Nilmanifolds form one of the best test classes in geometry and topology. On the one hand, their nonabelian natures introduce a complexity not found in tori, while on the other hand, their homotopy theory is still amenable to study. In particular, they have minimal models. So, it is no surprise that nilmanifolds are often the first non-toral cases to be considered in geometric or topological problems (e.g. the construction of isospectral non-isometric manifolds). Of course, even dimensional tori carry Kahler structures, so the natural question to ask is whether something similar can be said about nilmanifolds. We have already seen in Example 1.8 that not every even dimensional nilmanifold is symplectic, so first, we must restrict ourselves to that subclass. Then we have the manifold mentioned in Example 1.7. Example 2.1: The Kodaira-Thurston Manifold KT. [Th] The manifold KT is the quotient IP!.4/r, where r is the discrete affine group generated by unit translations along the XI, X2 and X3 axes together with the transformation (XI, X2, X3, X4) f---+ (XI +X2, X2, X3, X4+ 1). KT is a T 2-bundle over T 2 and its symplectic form lifts to the standard form dXI 1\ dX2 + dX3 1\ dX4 on IP!.4 The Kodaira- Thurston manifold is a symplectic nilmanifold with first Betti number equal to 3. Because Kahler manifolds have even odd Betti numbers (see Example 1.3.6), KT cannot be Kiihler. The minimal model for KT is given in Example 1.7 (2). We shall give another description of KT below as a special case of a more general collection of symplectic non-Kahler nilmanifolds. The Kodaira-Thurston example is, in fact, characteristic of nilmanifold behavior. That is, no nilmanifolds can be Kahler except for ton. This result, Theorem 2.2 below, is the main theorem of this section, solving the Weinstezn- Thurston problem in the class of nilmanifolds. Theorem 2.2. [BG 1, Has] Let M = N /f be a compact nilmanifold. If M admits a Kiihlerian structure, it is diffeomorphic to a torus. Proof. We have seen in Theorem 1.4 (and the development leading to it) that the minimal model of M is an exterior algebra MM = A(x I, ... ,x n ) with top class represented by the cocycle lJ XI" ·X n . Now, except when dx, 0 for all i = 1, ... ,n, no matter how we write the generators of the model, lJ is always an element of the ideal Ideal(N) generated by the complement to the closed degree 1 generators. Since lJ is a closed non-exact cocycle in Ideal(N), Theorem 1.3.1

=

=

NILMANIFOLDS

55

says that MM is not formal. By Theorem 1.3.7, M cannot be Kahler. In case all dXi = 0, then M has the rational homotopy type of a torus. Now, since the fundamental group Jrl M of a nilmanifold is torsionfree and ({Jl-Iocalization has torsion kernel, Jrl M must inject into EB({Jl. Hence, since Jrl M is finitely generated and injects into an abelian torsionfree group, it must be the case that JrIM EB!.Z and M has the homotopy type of a torus. But M is a true nilmanifold, so its fundamental group classifies its diffeomorphism type ([Mos]). 0 Therefore, M is diffeomorphic to a torus. Remark 2.8. Theorem 2.2 was also proved by Hano in [Hano] and was evidently known to Koszul. Hano's proof is quite geometric and relies on a description of the covariant derivative under the Kahler condition. In fact, however, the hypothesis of Theorem 2.2 is too strong. Recall from Definition 1.3.12 that a (cohomologically) symplectic manifold M has Lefscheiz type if

is an isomorphism.

Theorem 2.4. ([BG1, McD2, L01, L02] A symplectic nilmanifold M of Lefschetz type is diffeomorphic to a torus. Proof [L01]. We have seen that the minimal model for M is (A(XI, ... ,X2n), d) with symplectic 'form' given by W

=

L

aijXiXj

i,j T M such that J2 = ­ iT M. Because J behaves as the complex number i, it may be used to give the tangent vector spaces complex structures at each point of M. Of the spheres, only 52 = CP1 and 56 have almost complex structures. Complex manifolds certainly have almost complex structures, but the existence of an almost complex structure does not necessarily mean that the underlying manifold is a complex manifold however. Also, almost complex structures provide an equivalent definition of Kahler manifold for the more geometricly minded reader. Namely, a complex manifold is Kahler if the associated almost complex structure is preserved by parallel translation. Proposition 3.4. A symplectic manifold has an almost complex structure. Sketch of Proof. Given a Riemannian metric (­, ­), the nondegeneracy of w allows us to define an isomorphism A (on each tangent space) by w(X, Y) = (X, AY). Then it is easy to see that A is a normal and skew operator, so its polar decomposition A = 5 J has 5 J = J S with 5 positive definite symmetric and J an isometry. Furthermore, these properties and the skewness of A give

Jt

= At (5- 1)t = -AS- 1 = ­5 J 5­ 1 = -J.

64

SYMPLECTIC STRUCTURES ON NILMANIFOLDS

But J is an isometry, so ]I = J- 1 . Hence, -J = J- 1 and, therefore, J2 := J( _J- 1 ) := -1. Thus J is an almost complex structure. Also, the definition of A and the symmetricness of S give w(J X, JY)

= (J X, AJY) :=

(JX, SJ 2y )

= (lX, -SY) = (SJX,-Y) = (AX, -Y) = -(Y, AX) = -w(Y, X)

=w(X, Y). o

Thus, J is an 'isometry' with respect to w as well.

Remark 3.5. We can define a new scalar product by (X, Y)

:=

(SX, Y)

where S is the symmetric positive definite matrix of the polar decomposition = SJ. The reader may easily show that, in fact, (X, Y) = w(J X, Y) so that the complex structure J is calibrated by w (see [AuI, AuL] for example). The space of calibrated almost complex structures is important because it is known to be connected and contractible. In order to understand the restrictiveness of the requirement of an almost complex structure on a manifold, we now give a beautiful result of Michele Audin. A

Theorem 3.6. [Au2] If M 1 and M 2 are 4-manifolds with almost complex structures, then W := M 1 #M 2 has no almost complex structure. To prove this result, we will modify Audin's original proof by using a lemma brought to our attention by Bob Gompf. Lemma 3.7. If M is a 4-manifold with almost complex structure, then

is even, where b1 is the first Betti number, b+ is the number of positive diagonal entries for the signature form, X is the Euler characteristic and (1 is the signature. Proof. The Euler characteristic is defined to be X := I - b1 + h - b3 + I while the signature is (1 = b; - L with b2 = b+ + L. Hence we have X + (1 = 2 - 2b1 + 2&+. Borel and Hirzebruch showed that, if x is any cohomology class with P2(x) = W2 (P2 is reduction mod 2 and W2 is the Stiefel-Whitney class), then x 2 = (1 mod 8. := An almost complex structure provides Chern classes and P2 (ci) = W2, so (1 mod 8. The Pontryagin class PI (T M) is given by

ci

NILMANIFOLDS

Pl(TM)

65

= (-I)Cl(TM Q\I q = cl(TM EB TM) =

ci -

2C2

since we have total Chern class

c(T M EB T M)

= c(TM)c(T M) = (1 + Cl + c2)(1 = 1 + 2C2 - ci.

The Hirzebruch signature formula gives (ci ci = 2X + 30" = 2(X + 0") + 0" and we have 2(X

Thus , 1 - b1

+ b+-

2(x+o) -4

+ 0")

ci-a -4

=

-

Cl

+ C2)

2C2, [M))

ci - 0" =

= 30"

with

C2

x, so

8£.

- 2£ even .

4 -

0

Proof of Theorem 3.6. Except in degrees 0 and 4, the Betti numbers of the connected sum are the sums of the Betti numbers of the components. Hence,

1- bl(W)

+ b+(W) = 1 = 1-

+ b+(Md + b+(M 2 ) bl(M 1 ) + b+(M l) - bl(M2 ) + b+(M 2 ) .

b1(Md - bl(M2 )

But 1- bl(Md + b+(Md is even by Lemma 3.5 and so is 1 - bdM2 ) + b+(M 2 ) which means -b 1(M2 ) + b+(M 2 ) is odd. Therefore, 1 - bl(W) + b+(W) is odd, so W does not have an almost complex structure. 0 Example 3.8. One of our basic examples of symplectic (in fact, Kahler) manifolds is CP2. By Theorem 3.6, Cp 2#Cp 2 is not symplectic however, because it does not have an almost complex structure.

Now let's consider the case of an almost complex structure J on a nilmanifold. We may define J on a nilmanifold by first defining it on the Lie algebra of the associated nilpotent Lie group. Example 3.9: Generalized Hopf Manifolds [CFL). Take the Lie algebra of Example 1.6 (1) n of dimension 2m + 2 having basis {Xl, ... ,Xm , Y1 , ·

· ·

,Ym , Z, W}

with bracket structure given by

[Xi, Yi) = -Z and all other brackets zero. Let

for all i = 1, .. , ,m

SYMPLECTIC STRUCTURES ON NILMANIFOLDS

66

= Y;,

leX;)

ley;)

= -Xi,

l(Z)

= W,

leW)

= -Z.

This defines a left invariant almost complex structure on the nilpotent Lie group associated to n which then descends to the nilmanifold level. The Nijenhuis tensor of the almost complex structure is then

and this is easily computed to be zero for X, Y, Z, W. Hence, by the NewlanderNirenberg Theorem [KN], the almost complex structure J is integrable and the nilmanifold has the structure of a complex manifold. Of course, as we have seen, by virtue of its being a nilmanifold, this manifold can never be Kahler. Note that in case m = 1 we get the J( odaira- Thurston manifold which is symplectic. In higher dimensions this construction never yields a symplectic manifold. Indeed, taking the minimal model of it in the form given by Example 1.6: (M, d) with dx;

= (A(Xl, '"

Xm, Yl, ... , Ym, ZJ w), d)

= dYi = dw = 0 and m

dz

= 2.:= Xi 1\ Yi, i=1

we can calculate H 2(M). We get 2

H (M , d)

= ([w 1\ Xi], [w 1\ Yi], [Xi 1\ Y]], [Xi 1\ Xj], [Yi 1\ Yj] Ii = 1,

"'J

m, j = 1,

"'J

m)

which implies w

m

+ 1 = 0,

2 for any w E H 2(M).

if m

Since the dimension of the given manifold (i.e. = dim n) is 2m + 2 the latter equality shows that M is not cohomologically symplectic and, hence, the corresponding manifold cannot carry symplectic structures either. This example was discussed in [CFL] in the following context. Let (M, J) be an almost complex manifold with a compatible Riemannian metric (, ) and a fundamental (1, 1)­form given by w = (1 X Y) for X, Y E X(M). Recall that the Lee form () is defined by J

J

()(X)=-(

1

n­ 1

)t5w(lX)

where 6 here denotes the coderivative. A manifold (M, (, ),1) is said to be a locally conformal J( dhler manifold if 1 is integrable and

dw

= w 1\ 8.

If, in addition, the Lee form () is parallel with respect to the Levi­Civitta connection \7, the manifold is called a generalized Hopf manifold. It is shown in [CFL]

NILMANIFOLDS

67

that the nilmanifold Njf corresponding to n is a generalized Hopf manifold. Indeed, the corresponding Lie group can be realized as a direct product N(m, 1) x 5'1 where N(m, 1) consists of matrices A

E JRm and c E JR. The Lie algebra of Xm

Xl

o

F)

0

o o

0 0

Ym

0

and the exponential mapping gives a global coordinate system on N(m, 1),

xi(a) = ai,

Yi(a) = bi , The corresponding global vector fields are [)

[)

z(a) = c. [)

[)

Z="'z' i=l, ... ,nl. UXi UYi oz U Define a left-invariant metric on N(m, 1) by the requirement that all Xi, Y; and Z are orthonormal. The corresponding Levi-Civitta connection then can be calculated on these vector fields: 1 V'XiYi = -V'YiXi = 2Z , V'Xi Z = V'ZXi = V'Yi Z

1

-2 Yi ,

= V'ZYi =

with the other covariant derivatives being zero. The corresponding Riemannian metric on N(m, 1) x 5'1 is defined as a product metric (it is obviously compatible with J). Take a canonical 'length element' on 8 1 , say, 7) and its pull-back ij on N( m, 1) x 8 1 . Everything is left invariant and calculations can be done on the level of the Chevalley-Eilenberg complex (An*, d). We get the following correspondence between the canonical (1, 1)-form w, the Lee form () and elements ofAn*: m

W f--+

W

= I) X i 1\ Yi) + z 1\ w,

() = -7)

f--+

-w .

i=l

This implies

dw=wl\w (local Kahler conformality). The identity V'ij N (m, 1) is a generalized Hopf manifold.

o IS

verified directly. Thus,

68

SYMPLECTIC STRUCTURES ON NILMANIFOLDS

Example 3.10. The reader may carry out the details to show that the CFGManifolds M(p, q) of Example 2.6 have almost complex structures (see [CFG)). The left invariant forms on G(p, q) map to forms as follows:

d.X;

f­­+

ei ,

dYi

f­­+

d.Z, ­ XidYi

7)i ,

f­­+

Wi

and there is then a symplectic form on M(p, q) given by

F'

= tel /\ WI + te2 /\ W2 + 7)1/\ 7)2.

Let Ui , 'T; and Vi be matrices of vector fields dual to to the l­forrn matrices ei, 71i and Wi respectively. Then an almost complex structure may be defined by

To end this section (as well as the chapter), we must note one geometric property of nilmanifolds which will prove essential to our discussion of McDuff's blow­up examples later on. Theorem 3.11. Let G be a Lie group and suppose tbet. I' is a lattice contained in G. Then G If is parallelizable. Recall that a lattice I' is a discrete subgroup of G such that the space of cosets G If is compact. Also recall that a manifold M k is parallelizable if its tangent bundle is trivial TM M x ]Rk. Proof. It is easy to see that a Lie group is parallelizable. Take a basis for the Lie algebra 9 = TeG, {Xl, .. . , Xd, and use left translation in G to extend the Xi to left invariant vector fields on G. Because the translated Xi form a basis for each of the tangent spaces TgG, the tangent bundle is a product G x ]Rk. Now note that the lattice I' acts on G by left multiplication so that the Xi are f­invariant as well. Therefore the Xi descend to vector fields Y; on the quotient manifold G If. Furthermore, because I' is discrete, the projection 7r: G --+ G If is a smooth covering map ­ hence a local diffeomorphism. In particular, we have for every 9 E G, 7r*:

TgG

"" --=-. T

1r(g)G/f.

Now, if at some point 7r(g) E G/f, there exist ai, i = 1, ... ,k not all zero with

then, because

7r*

is an isomorphism, alX l (g)

+ ... + akXdg)

= O.

This contradicts the fact that the Xi are linearly independent. Hence, no such a, exist and the Y; are linearly independent at every point as well. Therefore, G If is parallelizable. 0

NILMANIFOLDS Corollary 3.12. Nilmanifolds are paralJelizable.

69

CHAPTER 3

SOLVMANIFOLDS In this chapter we analyze the existence problem for Kahler and symplectic structures in the class of solvmanlfolds. The first results in this direction were obtained in [BG2 FG2, FLS, Tr2, Tr3]. More generally, in [BG2] the authors raised the problem of describing asphericol K dhler and symplectlc manifolds. The problem described in this section [S much more subtle than the problem descr ibed in Chapter 2 for several reasons. The main one is determined by the fact that there are symplectic solvnuuiijolds which have the rational homotopy type of K iihler manifolds but which are certainly non-Kdhler. These manifolds are distinguished from Kahler manifolds by more refined techniques such as, for example, Kodaira's classification of complex surfaces [BPV J FM]. However, the main stimulating conjecture in this regard extends the Benson-Gordon-Hasegawa theorem (Theorem 2.2.2) to solvmanifolds (see [BG2]). J

Conjecture (Benson-Gordon). The only compact solvmanifolds which carry Kahler structures are tori. One of the main goals of this chapter is to present, in a unified manner, various approaches to this conjecture. In particular, we systematically use the M ostow bundle

Nf/f

--4

G/f

--4

GINf

associated with a solvmanifold G/f. However, if one wants to use this bundle in the rational homotopy context, one meets the difficulty that, in general, the action of the fundamental group of the base 11"1 (G If) on the cohomology of the fiber H*(G/f) is not nilpotent and, therefore, Theorem 1.4.4 is not available. Therefore, we must use the more complicated Theorem 1.4.6 combined with a deeper analysis of the 11"1 (G If)-action. This analysis (Theorems 3.7 and 3.8 below) should be considered as potentially one of the most important results of this chapter. Theorem 3.8 allows us to prove the Benson-Gordon conjecture in dimension four. This result explains all of the known four-dimensional examples of Fernandez-Gray and others. Next we consider examples of symplectic non-Kahler solvmanifolds in higher dimensions. These examples are based on the Hattori theorem. Since this theorem (and its proof) is very important and potentially may be extended to more general situations, we give a complete proof (more in the spirit ofrational homotopy than the original) and construct examples. Finally, we propose a general approach to the problem along the lines of Sullivan's original twisted models, models which are not K.S. extensions, but rather Stasheff's aloebraic fibrations. We make a complete algebraic analysis of these models in the context of the Mostow bundle. Unfortunately, we don't know whether non-nilpotent Mostow bundles admit twisted tensor products as models. However J using algebraic results of this chapter we are J

SOLVMANIFOLDS AND THEIR MOSTOW BUNDLES

71

able to derive the Benson-Gordon theorem which describes Chevalley-Eilenberg complexes of completely solvable solvmanifolds.

1. Solvmanifolds and their Mostow Bundles To begin, let us recall some notions and results in the theory of solvmanifolds. The basic references for this information are [Aus, AGH, VGS]. We restrict ourselves to quotients of simply connected solvable Lie groups over lattices. Definition 1.1. A quotient G If of a solvable simply connected Lie group G and a discrete co-compact subgroup I' (i.e. a lattice) is called solvmanifold. It is important to note that there is a more general notion of solvmanifold (see [Aus, AGH, VGS]) , but we are only considering solvmanifolds as in Definition 1.1. In particular, note that the Klein bottle is not a solvmanifold covered by

this definition. Of course, every nilmanifold is also a solvmanifold. However, there are many solvmanifolds which are not diffeomorphic to any nilmanifold since 7rl (G If) is, in general, not nilpotent. Moreover, it is known that if the fundamental group of a solvmanifold is nilpotent, then this manifold is diffeomorphic to a nilmanifold [VGS]. All solvmanifolds are aspherical. While examples of solvmanifolds are discussed in this section, there is, however, no simple criterion for the existence of a lattice I' in a solvable Lie group G. Therefore, it is not as easy to produce solvmanifolds as it is to produce nilmanifolds. Nevertheless, solvmanifolds have been studied for many years and it is known that they possess a rich structure. The most important points of that structure come from the M ostow structure theorem. Theorem 1.2 (Mostow Structure Theorem). Let G If be a compact solvmanifold. Consider the maximal connected nilpotent subgroup N of G. Then

(i) Nf is a closed subgroup in G; (ii) N n I' is a lattice in N; (iii) G I vr is a torus.

Consequently, each solvmanifold G If can be naturaJIy fibered over a torus with a nilmanifold as a fiber,

NIN

n I'

= Nf/f --> G/f --> GINf

.

(1)

Definition 1.3. The bundle (1) will be called the Mostow bundle. Remark 1.4. In the context of this work, the method of proof of this theorem is not related to the techniques which we want to develop. However, we sketch the proof very briefly now for completeness. Note, first, that the only non- trivial part of the proof is (i). Indeed, having the closedness of Nf we get the compactness of GINf and Nf/f == NIN n f. Hence, NIN n r is a compact nilmanifold as required. Moreover, since GIN == lR. k , the obvious covering GIN --> GINnf has fiber r If n N and the only possibility for the base is GIN n r == T k

SOLVMANIFOLDS

72

Thus, it remains to prove that Nf is closed in G. Take the closure Nf and denote by H an identity component of Nf. Thus, H is a connected Lie subgroup in G. In fact, we shall prove below that H =N.

This equality shows that Nf as a Lie group has the topology induced from G, and, hence, is a Lie subgroup in G. From the well-known Cartan theorem, Nf is closed. So, it remains to prove that H = N. We refer to Mostow's work [Mos], for details. First, observe that for each Lie algebra there exists a minimal ideal gOO C g such that g/gOO is nilpotent. Secondly, for each Lie group and Lie algebra the notion of a regular element is defined. Definition 1.5. Let 'P be any endomorphism of a vector space V. Denote by a-( 'P) the number of eigenvalues of'P different from 0 and by 11"( 'P) the number of its eigenvalues different from 1. Recall that, by definition, rankg = SUPXE9 a-(adX) and rank G = sUPxEG 1I"(Ad x). We say that X (respectively, x) is regular if

at ad X) = rank g,

1I"(Ad x) = rank G.

The required equality is proved by combining the following observations. On one hand, it is proved in [Mos] that for each regular element exp X E G, the endomorphism Ad( exp X) I(g"" /[g"" ,g""]) has no eigenvalues equal to 1. On the other hand, if we assume that H # N, an easy exercise in Lie algebra theory yields two chains of ideals in g with proper inclusions: gOO C n C

C g,

C

C g.

Since the inclusions above are proper, H OO # {l} and, hence I' n H'" is a discrete co-compact subgroup in H OO • Mostow shows (in the most difficult and technical part of the proof) that there exists a regular element s E f n H such that the induced linear mapping Ad has all eigenvalues equal to 1. This is then a contradiction. In what follows we will also need a classification of 3-dimensional Lie algebras. This can be found in practically any book on Lie algebras. Theorem 1.6. All 3-dimensional Lie algebras over If g is nilpotent, then it is either

are listed below.

N 1 3-dimensional abelian or

N2 g

= (X, Y, z, [X, Y] = Z, [X, Z] = [Y, Z] = 0).

If g is solvable non-nilpotent (SN), then g

= (X, Y, Z; [X, Y] = 0, [X, Z] = aX + j3Y, [Y, Z] = "IX + bY)

where the matrix

SOLVMANIFOLDS AND THEIR MOSTOW BUNDLES

is non-degenerate (a,

73

/3, I, 8 E

If g is simple, then it is either

= -2Z, [X, Z] = 2Z, [Y, Z] = X) (X, Y, Z; [X, Y] = Z, [X, Z] = -Y, [Y, Z] = X).

SI (X, Y, Z; [X, Y]

S2

or

Let us mention one important property of lattices in nilpotent Lie groups. Because we will need it only once in the sequel, we will not prove it. Also, note that this property is not valid for arbitrary solvable Lie groups. Theorem 1. 7 (Rigidity Theorem) [VGS]. Let f

and f

be lattices in uilpo"" respectively. Any isomorphism I' 1 --=-. f 2 extends

tent Lie groups N 1 and N 2

1

2

"" N . uniquely to an isomorphism of Lie groups N 1 --=-. 2

Example 1.8: 3-dimensional solvmanifolds. Now we want to present a description of all 3-dimensional solvmanifolds. We discuss it here with the aim of giving the reader a feeling for the structure of solvmanifolds. In what follows we will use the standard notation of a semi direct product of groups. That is, A >41> lP!.2. Now, we must find ¢. Since ¢ is a one-parameter subgroup in GL(lP!.2) , we must look for a one-parameter subgroup in GL(lP!.2) preserving a lattice in lP!.2. For each solvmanifold G/f = (lP!. >41> lP!.2)/f, the natural map 2 'f): G 1lP!.2 - lP!. possesses the property that 'f)(lP!. n f) is a non-trivial closed subgroup of lP!. generated by a single element, say, g. Thus, ¢ : lP!. - GL(lP!.2) is a one-parameter subgroup going through g. This subgroup can be identified with either (i) or (ii) and the proof follows. 0 Detailed Proof. 1) We start with the following observation. If G is a connected, simply connected solvable, non-nilpotent 3-dimensional Lie group with maximal nilpotent subgroup N and a discrete co-compact subgroup I', then dim N = 2. To prove this assertion, note that since G is non-nilpotent, dim N # 3. Since it is solvable, dim N # O. Thus, we must eliminate the case dim N = 1. To do this, assume that dim N = 1. This implies that dim [G, G] = I or dim [G, G] = 0 (since [G, G] eN). Since G is non-abelian, the only possibility is dim [G, G] = 1. Since [G, G] is a normal subgroup in G, we can define a G-action on [G, G] by conjugation

1(g)(a) = gag-I,

9 E G,

a E [G,G].

From the Mostow Structure Theorem 1.2, [G, G] n f = N n f = f is infinite cyclic. Obviously, f is invariant with respect to the f-action on [G, G] induced by the G-action. Moreover, 1(-y) acts on f by automorphisms. But there are only two automorphisms of an infinite cyclic group, so there is a subgroup f' C I' of index 2 such that I" acts trivially on I' n [G, G] = f. (This follows from the Rigidity Theorem 1.7.) Now, if H denotes the maximal subgroup in G which acts trivially on [G, G], then H ::J I" and H is normal from the definition of the G-action and the normality of [G,G]. Since G/f is compact and I" has finite index 2 in r, G If' and, consequently, G I H is compact. Define an action of G I H on [G, G] by gH(a) = gag- 1 for a E [G,G]. From the compactness of GI H it follows that this action is neccessarily trivial and this implies that a E Z(G), the center of G. This means that [G, G] C Z(G) and this implies that G is nilpotent. This is then the required contradiction. 2) Now, since dim N = 2, it follows that N = lP!.2. Therefore, the group G can be included into an exact sequence 2

1-lP!. - G

11"

----->

lP!. -1,

where lP!. = GIN. Since N is abelian, the factor-group GIN acts on N by = gng- 1 and thus the homomorphism

gN(n)

¢: lP!. -

Aut(N),

determines an isomorphism

¢(gN)

= 1(g):

N _ N

(2)

SOLVMANIFOLDS AND THEIR MOSTOW BUNDLES

75

By Theorem 1.2 again, we see that N n I' is a lattice generated by two linearly independent vectors in JR 2. Choosing the appropriate coordinates in JR 2 we can assume that these vectors constitute a standard basis in JR2, ej = (1,0), e2 = (0, I). Since etc is compact, N n r "tJ r and 7I"(f) C JR becomes a non-trivial closed subgroup generated by a single element B. Note that cP(B) preserves a lattice generated by eI and e2. We have either cP(B) E Aut(JR2) and cP(B)(Nnf) c N n r, or

At this point we observe that the subgroup f

r

is completely determined. That is,

= 7lB ) (7leI EB 7le2),

with cP(B)(ed and cP(B)(e2) given by (2). Now, we must determine the JR-action on JR2 by finding a one-parameter subgroup ¢: JR ---+ GL(JR2) for which cP(B) has values on eI and e2 the same as those given by (2). Since ¢( B) preserves an integral lattice in JR2 its matrix with respect to the basis eI and e2 has integral entries and has determinant either 1, or -1. Since cP(t) is a one-parameter subgroup, det cP(B) = 1. Then the possibilities for cP(B) fall into three distinct cases: (i) the eigenvalues of cP(B) are real and positive, (ii) the eigenvalues of ¢(B) are real and negative, (iii) the eigenvalues are complex with non-zero imaginary parts. In case (i), since cP(B) lies in a one-parameter subgroup, there exists a square root = = cP(B). Notice that ¢(B) may be written in a diagonal form ¢( B) = diag( AI, A2), for AI> 0, A2 > 0 and it is easy to check that the square root must be in the diagonal form = diag(v1"I, v1"2). This may be seen by direct calculation. Applying this argument successively and taking the appropriate square roots, we get cP(tB) = diag(e k t , e- k t ) , where ek t + e- k t i- 2 is an integer, as required. Case (ii) can be eliminated because of the impossibility of taking successive square roots of cP( B). In case (iii), ¢(B) is conjugate to diag(A, ),), where A), = 1. Since cP(B) E G L(JR2) has integer entries, the coefficients of its characteristic polynomial are also integers and this implies that A + >. E 7l. The only matrix which satisfies all conditions above is given by condition (ii) of the theorem. The proof is then complete. 0 Now, we can describe a1l4-dimensional solvable Lie groups admitting lattices. This will be used in the proof of the Benson-Gordon conjecture in dimension 4. Theorem 1.10. Any simply connected solvable 4-dimensional non-nilpotent Lie group wbict: admits a lattice has one of the folIowing two forms:

= JR ) N3 , where cP: JR ---+ Aut(N3 ) is a one-parameter subgroup in Aut(N3 ) such that cP(1)(N3 (Q)) c N 3 (Q) or

(i) G

(ii) G = JR )1m. ---->G--.Im.---->1 with 1m. 2 = GIN. Since N is abelian, the factor group GIN acts on N by gN(n) = gng- 1 and, thus, the homomorphism

¢: 1m. 2 ----> Aut(N),

¢(gN)

= I(g):

N ----> N

determines an isomorphism G == 1m. 2 1m. 2 . Again, since N is normal in G, there is an action of r on the subgroup N n r by automorphisms 1(,) for, E r. Again, since G = 1m. 2 1m. 2 , there are one-parameter subgroups ¢(t) going through elements 1(,) which, therefore, preserve a lattice c 1m. 2 . Repeating word for word the argument in the previous theorem, we see that ¢(t) has either form (i) or form (ii) of Theorem 1.9. The first case, however, must be excluded since ¢(t), although it preserves is not an isomorphism of In fact, it is an injective but not surjective homomorphism which, in the appropriate basis, is given by (Zl,Z2) ----> (mzl,mz2), for (Zl,Z2) E IZEBIZ, m E IZ, m # 1. In the second case, we can easily find a subgroup I" of finite index acting trivially on Again, as in the proof of Theorem 1.9, we take the maximal subgroup H in G acting trivially on [G, G] and carry out the proof repeating word for word the argument of Theorem 1. 9. Finally, if N is a 3-dimensional nilpotent Lie group, Theorem 1.6 shows that either (i) or (ii) holds (the conditions regarding ¢(1) are necessary for the existence of a lattice). The proof is then complete. 0

r

r,

r.

r.

We complete this section with the following important property of solvmanifolds either of low dimension (by [AuSz]) or of the form G Ir (by Theorem 2.3.11). Theorelll 1.11 [AuSz]. Any compact orientable solvmanifold which either has dimension less than or equal to 4 or is of the form G Ir is parallelizable. Of course, there are many non-parallelizable solvmanifolds such as the Klein bottle (i.e. it is non-orientable). Auslander and Szczarba also constructed examples of non-parallelizable solvmanifolds in higher dimensions.

COHOMOLOGY OF SOLVMANIFOLDS: HATTORI'S THEOREM

77

2. Cohomology of Solvrnanifolds: Hattori's Theorem First recall that Nomizu's theorem implies that the Chevalley-Eilenberg complex (An*, 6) turns out to be the minimal model of a nilmanifold N If (see Theorem 2.1.3). In this respect, arbitrary solvmanifolds differ essentially from nilmanifolds. In general, the Chevalley-Eilenberg complex (Ag*,6) is not even a model for G If since there are examples of solvmanifolds G If such that H*(Ag*, 6) 'F H*(G/r)·

However, in the special case of completely solvable Lie groups G, the problem of calculating the cohomology of G/f can be settled by Hattori's theorem. The technique which is necessary for proving this result is of great importance. While the original proof is given in [Hat], we present here a more'rational-homotopic' and simplified version. Note that results of Benson-Gordon, Fernandez-Saralegide Leon and others cover the completely solvable case and are heavily based on Hattori's theorem. Definition 2.1. A solvmanifold G If is called completely solvable if the Lie algebra 9 satisfies the condition that all linear operators ad V : 9 --+ 9 have only real eigenvalues for all V E g. In general this condition does not hold (see Example 2.4 below). Example 2.2. In dimension 3, the solvmanifold Gdf l of Theorem 1.9 is completely solvable since the Lie algebra gl is of the form gl

= (X, Y, Z;

[X, Y]

= kY,

[X, Z]

= -kZ,

[Y, Z]

= 0,

k E lR).

Example 2.3. Every nilmanifold is completely solvable. Example 2.4. In dimension 3, the solvmanifold G 2 / f 2 of Theorem 1.9 is not completely solvable since 9

= (X, Y, Z;

[X, Y]

= Z,

[X, Z]

= - Y)

and ad(X) has imaginary eigenvalues. The theorem below is the main result of this section. Hattori's Theorem [Hat]. For any compact solvmanifold with completely solvable simply connected Lie group G tbere exists a quasi­isomorphism

78

SOLVMANIFOLDS

and, therefore, the Chevalley-Eilenberg complex (Ag*, 8) is a model (not minimal in general) ofG/f.

In particular, note that, if G is nilpotent, then Nomizu's theorem is recovered. The proof of this theorem requires the development of the machinery of the Leray-Serre type spectral sequence associated with a locally trivial smooth fiber bundle (cf. [GrM, McC]). To make the proof more comprehensible, we first describe the main idea and then complete the details. Sketch of Proof. Take the Mostow bundle (I) associated with a solvmanifold G /f. Let E = G /f and denote by F and B respectively, the fiber and the base of the Mostow bundle. Consider the Leray-Serre spectral sequence (E;'*, dr ) associated with this bundle. As usual,

Consider now the Chevalley-Eilenberg complex (Ag*, 8) and, using the embedding (Ag*, 8) C r2 D R(E), define a filtration on (Ag*, 8) inherited from r2 D R(E). The corresponding spectral sequence (E;'* , dr ) converges to H* (Ag* ,8). To prove the required isomorphism H*(Ag*, 8) == H*(G/f) we want to use the spectral sequences (E;'*,d r ) and (E;'*,d r ) . It turns out that there is a commutative diagram

where n denotes (as before) the maximal nilpotent subalgebra in g. Now the idea is quite clear. Try to use in.duciion with respect to the dimension of g. Assuming that the required isomorphism has been already proved for all Lie groups of dimensions < n and taking dim G = n, apply the above diagram. The bottom row of the diagram will yield an isomorphism by the induction hypothesis. Thus EP,q E· 2p,q "" 2

for all p, q, and the standard technique of spectral sequences [McC] implies the required isomorphism. However, here we meet a difficulty which requires a rather subtle analysis. Indeed, this is the reason why the theorem is not true for arbitrary solvmanifolds. To apply the induction hypothesis, we must assume that a stronger condition holds. Namely, we must assume H*(Ag*,F) == H*(G/f,F)

for the cohomology with coefficients in a g-module (this is precisely the situation we meet considering the previous diagram where F = H*(An*)). Then we must prove the latter isomorphism for dim g = 1. It zs here that the condition of complete solvability ss used and the proof can be completed. 0

COHOMOLOGY OF SOLYMANIFOLDS: HATTORI'S THEOREM

79

Details of Proof. Let us start with the following version of the Leray-Serre spectral sequence. Consider the locally trivial fiber bundle F

i ------->

E

7f

------->

(3)

B

and let F be a COO (E)-module. Assume that the Lie algebra X(E) of all smooth vector fields on E operates on F. By definition, this means that there is a map X(E) x F ----+ F, (X, u) f--+ Xu satisfying the conditions (1) X(fu) (2) (fX)u

= (Xf)u + f(Xu), for f E cOO(E), = f(Xu), for f E cOO(E), U E F,

U E F,

(3) [X, Y]u = X(Yu) - Y(Xu), for X, Y E X(E). Given a pair (E, F) such that X(E) operates on F, we can define a twisted de Rham complex (O(E,F),d F

)

of differential forms with values in a COO (E)-module F. Namely, OP(E, F) consists of all alternating COO (E)-linear maps w: X(E) x ... x X(E) ----+:F. The differential dF is defined formally by the same rule as the usual de Rham differential in ODR(E),

The difference between d and dF is hidden in the notation XiW since now the value of W is an element of F on which X operates. Given a bundle (3), we define a filtration on the twisted de Rham complex (O(E, F), dF ) as follows:

FP(Op+q (E, F)

= {w E Op+q (E, F) I We (VI, ... whenever

Vi

1 , ..• ,

Vi

p+1

, V p+q)

=a

E Ker( dt:e)}

for all e E E. Here of course, VI, ... ,V p+q E Te(E). Since E is a manifold, OS (E, F) = a for s > dim E and, therefore, the filtration ... =::J •.. =::J

FPOp+q(E,F)

=::J .•. =::J ...

is a bounded decreasing filtration. The standard construction of the spectral sequence associated with a filtration [McC) shows that there exists a first quadrant spectral sequence (E;'*, dr ) converging to the cohomology of the twisted de Rham complex H*(O, F). Thus,

80

SOLVMANIFOLDS

is a spectral sequence which will be called in the sequel the Lerau-Serre spectral sequence associated with a locally trivial smooth fiber bundle (3). This will be the first of two spectral sequences we need to prove the generalized isomorphism (*). The second spectral sequence is associated with the Chevalley-Eilenberg complex with ualues in a g-module U. Here we assume that a g-module U is a finite dimensional vector space on which g operates via a representation A: g ----> g[(U). Define the complex

(A(g*,U),8u ) in the same manner as rl(E,F): AP(g*,U) consists of all alternating IP?,-linear maps (): g x ... x g ----> U. The differential 8u is defined formally by the same expression as d F . Interpreting g as the Lie algebra of invariant vector fields on G, one can define a filtration on A(g*, U) in precisely the same manner as on rl(E, F). Namely,

FPAp+q(g*,U) = {w E Ap+q(g*,U)1 W(Vl, whenever

Vi"

...

P+q) = 0 , Vi p + 1 E Ker(d7l'e)}. ... ,V

Here, of course, E is not arbitrary, but denotes G If and Vi is interpreted as a tangent vector obtained from an invariant vector field on G projected on G If. So, in what follows, (3) denotes the Mostow bundle. Again, we get a first quadrant spectral sequence (E;,*,d r ) converging to H*(A(g*,U),8u ). Finally, we have obtained two spectral sequences which we may compare to yield the proof. We write

(E;'*,d r ) ===> H*(rl,:F),

(E;",d r ) ===> H*(g,U).

Here and in the sequel we write H*(g,U) instead of H*(A(g*,U),8u ). The following lemma is the main tool enabling us to compare these spectral sequences. In order to formulate it, we introduce, first, the following definition.

Definition 2.5. A vector field X on E is called vertical if at each point x in E its value at x is a tangent vector to the fiber going through this point: Xx E Tx(F). Denote by Xv (E) the set of all vertical vector fields in E. Restricting differential forms with values in the COO (E)-module F to vertical vector fields, we get a complex

(rl v ( E, F), dF) . Now W E rlv(E,F) denotes an alternating map w: Xv(E) x ... x Xv(E) ---->:F. Since Xv(E) is an IP?,-Lie subalgebra in X(E), d F is well-defined by the same rule as before. From now until the end of this section, we consider only the particular x(E)-module F, where

F

= COO(E) c>9 U,

and U is a finite dimensional vector space over IP?,.

COHOMOLOGY OF SOLVMANIFOLDS: HATTORI'S THEOREM

81

Lemrna 2.6. The following isomorphisms are valid:

where H" (n, U) is celculeied with respect to the g-module U considered as an nmodule, Hq(flv(E, F)) denotes the cohomology of the twisted de Rham complex of vertical forms on F with values in a Coo (E)-module :F and Hq(flv(E,F)) is endowed with an appropriate structure of a Coo (B)-module on which x( B) operates. The latter structure is determined by a natural embedding p* : Coo (B) --> cOO(E), p*(J) = fop for the bundle map p: E --> B. Proof of Lemma 2.6. The proof, in fact, is based on a direct (although somewhat technical) calculation of E 2 -t erm s in both spectral sequences. We calculate sucdo), dd and E;'* and then repeat the same procedure for cessively (E;'*,d r ) , r = 0,1,2. Introduce any connection in the bundle F --> E --> B (in our approach we assume the existence of a horizontal distribution Q C T(E) satisfying the lifting property for curves in B [Hat]). As usual, this connection yields a COO (B)­linear lifting map

I: X(B) Now, we want to calculate

-->

X(E),

X

-->

X = I(X).

Define, first, a map

where Xi E X(E) for all i. One can verify by direct calculation that the kernel of this map is FP+lfl p+q and that 'Pp,q commutes with the differentials. Denote by !j5P,q the composition of 'PM with the restriction of differential forms in fl(E, F) to vertical vector fields. Then the following map

yields an isomorphism 'Pb,q : Eg,q

-->

Note that this calculation is straightforward and is still valid for any F. Now we begin the calculation of Ef,q. Consider ,* = EB p ,qflP(B, (E, F)). From the definition of the differential do in the spectral sequence (E;'*, dr ) we can deduce

SOLVMANIFOLDS

82

that do is identified with the differential d' on ffip,qflP(B, flZ(E, F)) given by the formula

d'w(X j

, ••.

,Xp )

= (-I)Pd F(w(X

j , •••

,Xp ) ) ,

Xi E X(B), i

= 1, ...

,p.

On the other hand, EBp,qflP(B, flZ(E, F)) has already been endowed with the natural differential. Let (Zp,q)' and (BP,q)' denote the subspaces of cocycles and coboundaries with respect to d'. Obviously, Ef,q = (Zp,q)' j(BP,q),. Introduce also the notation zq for the subspace of cocycles in flv (E, F) and B" for the corresponding subspace of coboundaries with respect to d F . Obviously, the natural projection zq ­ Hq(flv(E,F)) induces a natural map

flP(B, zq) ­

sr;», Hq(flv(E, F))

with kernel flP(B, Bq). Obviously,

Therefore, we get a natural map

x. (HP,q)' = Ef,q Now,

­ flP (B, Hq(fl v (E, F))).

>. is injective if and only if (BP,q)'

and

= flP(B, Bq)

>. is surjective if and only if the map

sri», zq) ­

flP(B, Hq(fl v (E, F)))

is surjective. The proof of the latter two facts is straightforward. For instance, let e: X(B) x ... x X(B) ­ Hq(flv(E,F)) be a differential form with values in Hq(flv(E,F)). Then e(X j , • • . ,Xp) = [w]x" .x.: The problem is, in fact, to find a canonical choice of representatives in all cohomology classes. In general it is not possible, but since F = COO(E) Q?I U, where U is finite dimensional and E is compact, the corresponding cohomology is finite dimensional and we can choose representatives for cohomology classes in an arbitrary way (note that if w E flv(E, COO(E) Q?I U), then w = Li (XiWi, (Xi E JPI., where Wi: Xv(E) x ... x Xv(E) ­ COO(E) is a usual de Rham form for any i). By the same method one can prove the injectivity of >.. Thus, we have calculated More or less the same techniques yield the calculation of E 2 - t er m . We leave the remaining calculations of and to the reader. These calculations are not trivial, but no new methods are used in the proof. 0

E;"

E; "

Remark 2.7. In Hattori's paper [Hat], the proof is more general since he considers cohomology with coefficients in arbitrary sheaves. However, to avoid overly technical considerations, we have dealt directly with the particular case of F = Coo (E) Q?I U.

Now, complete solvability comes into play.

COHOMOLOGY OF SOLVMANIFOLDS: HATTORI'S THEOREM

83

Definition 2.8. Let 9 be a finite dimensional Lie algebra and let U be a finite dimensional vector space over K A representation>. : 9 - l - g£(U) is called iruingular if there exists a sequence U = U o :::l U l :::l ... :::l Un = {O} of subspaces of U such that dim Ui = dim Ui + l + I, 0 i n and each Ui is invariant under >.(g). In this case, the g-module U is called triangular. Let G be a simply connected Lie group and I' be a lattice. Notice that for each finite dimensional g-module U, the x( G)-module II = U 1,

= {u E [11 u = Xu'}.

One can see that

We claim that these maps are bijective for any triangular module U. To prove this, take a basis el, ... , em of U such that

x-, = Laijej. j

(6)

?i

Then elements of U are written uniquely in the form Li j;ei with fi E COO(Sl) and Li fiei E U if and only if fi E Wi.. We prove, first, the surjectivity of U X ---> Assume that X(Li fiei) = O. Since X(Li fiei) = Li(Xf;)ei + j;(X e;), it follows from (4) and (5) that

ax.

X i« + annfn

=0

==> fn E Wi..

By induction (whose possibility is guaranteed by (6)) we easily get j; E Wi. for all i, or Li j;ei E U as required. The injeetivity is obvious. To show the bijectivity we reformulate this condition as follows: of the second map U I XU --+ I X

a a,

(i) for any f E COO(SI) and k, 1 :S k :S p there exist hi E coo(Sl) and Ci E Wi. such that

fek (ii) If X(Li j;ei) Now, since we have

X

(t,

hie i)

=X

= Li c.e,

+

t,

c.e,

+

with fi E COO(Sl) and

Ci

E Wi., then j; E Wi..

= (Xh k + auhk + ck)ek + (Xh k- 1 + ak-l,k-1hk- 1 + ak.k-lhk + ck-I)ek-l + ...

we can determine (by (5)) h k, Ck, h k _ I, Ck -I,

...

inductively to satisfy

X hk + au hk + Ck

Xhk_1

+ ak-I,khk-l + ak,k-1h k + Ck-l

=f = 0,

COHOMOLOGY OF SOLVMANlFOLDS: HATTORI'S THEOREM

85

which implies (i) and (ii) and, hence, the proof of Lemma 2.9 in the case dim g = 1. Assume that Lemma 2.9 holds for all g such that dim g < n. Consider a solvable Lie algebra g of dimension n. Then either g is non­nilpotent or it is nilpotent. In the first case, take a maximal nilpotent sub algebra n, which obviously satisfies the inequality dim n < dim g, and construct the pair of spectral sequences (E;'*, dr ) , (E;'*, dr ) with respect to the modules U and U. The following diagram is commutative (Lemma 2.6).

Here U = Coo(G If) (2)U. Now, we can check by direct calculation using only the definition that H" (n, U) is a triangular gin­module (we leave the details for the reader who may also consult [Hat, p.328]). Since dim(g/n) < n and dim n < n, we can use the induction hypothesis as follows. First,

HP(g/n, Hq(n,U)) == HP(B, Hq(n,U)) where Hq(n,U)

= COO(GINf) (2) Hq(n,U).

We claim that

and note that, if we have proved this, then the whole proof would be complete. Thus, our aim now is the proof of this isomorphism. Note, first, that

for each point b E B and the corresponding fiber Fi over b. Notice that each alternating map w: nx ... x n -> U can be naturally extended to the corresponding alternating map w: X(Fb) x ... x X(Fb) -> Coo(Fb)(2)U (since each vector field X on Fb can be represented as a COO (Fb)­linear combination Li !iXi where Xi are invariant vector fields on Fi = N INn I' determined by Xi E n. Define a map

where

(f

(2) w)e(X 1,,,,

,Xp) = j(b)We(X1IFb'"" ,XpIFb )

and b = 1l"( e). It is easy to verify that C is well defined on the cohomology level. The proof is straightforward and uses only the previous isomorphism between Hs (n, U) and the corresponding cohomology of the fiber (which, in turn, follows from the induction hypothesis). Moreover, the same isomorphism shows that the corresponding map is injective. The surjectivity of is proved as follows.

SOLVMANIFOLDS

86

Let [0] be an element in Hq(rlv(E, Coo(E) &JU)). Then, for each b E B, [O!Fb] can be identified with [w](b) E Hq(n,U). Since Hq(n,U) is a finite dimensional vector space, [w](b) = Li j;(b)[W]i, where the [W]i constitute a basis of Hq(n,U). Since 0 is smooth, the functions fi are in Coo (B) and we get a well-defined map (: Hq(rlv(E, COO (E) &J U)) ---> Hq(n,U), [0] 1--+ Li f;(W]i' Obviously, 0 ( and ( 0 are identity maps and the required proof follows. Finally,

EP,q 2

-

EP,q -'2

l

for all

p, q.

Standard comparison theorems [McC] show that the limits of the given spectral sequences are also isomorphic. H*(Ag*,U)

== H*(Gjf,U)

Note that, if g is nilpotent, we can lower the dimension by taking the center n = 3(g) (which is always non-trivial for nilpotent algebras) and repeat the argument above for gjn. The proof of Lemma 2.9 is now complete. 0 Conclusion to the Proof of Hattori '05 Th.eorem.. To complete the whole proof, it

only remains to notice that H*(Gjf, Coo(Gjf))

== H*(Gjf, IFI,).

o In the general case, instead of Hattori's isomorphism we have a weaker statement which is also important and which will be used in the next section.

Theorem 2.10. For any solvmanifold G jf there is a natural injection H*(Ag*, 8)

--->

H*(Gjf).

The proof of this theorem can be found in [Rag]. We omit it here because the methods of proof are beyond the scope of this book.

Example 2.11. Let G2/f 2 be the solvmanifold (ii) given in Example 1.8 with = n = 2. A straightforward calculation (using the isomorphism 71"1 (G2/f 2) == ,22; 'P ,22;3) shows that

p

which means that Hattori's theorem does not hold for this solvmanifold.

RATIONAL MODELS OF SOLVMANIFOLDS

87

3. Rational Models of Solvmanifolds with Kahler Structures In Chapter 2 we have shown that rational homotopy models allowed us to effecti vely detect symplectic and (non)­ Kahler nilmanifolds because the ChevalleyEilenberg complex (An*, 8) (which turned out to be the minimal model of N If) contains all necessary information. In the solvmanifold case, however, the problem appears to be much more difficult. In fact, examples show that one must look for more refined invariants. To reveal the difficulties, we begin with an example constructed by Fernandez and Gray in [FG2]. Example 3.1. The following theorem shows that there are symplectic solvmanifolds which carry no Kahler structures but which non-detectable from Kahler manifolds by cohomological invariants or by the property offormality. Let Gdr 1 be a compact solvmanifold given by Example 1.8. Recall that G l = and kt 4;(t) is a one­parameter subgroup defined by the formula 4;(t) = diag( e , e- k t ) where e k + e- k # 2 is an integer. Theorem 3.2. Let M 4 denotes the solvmanifold (Gdf d x 51. Then (i) M 4 is a compact symplectic solvmanifold; (ii) M 4 has no complex structures (and, therefore, has no Kahler structures either); (iii) M 4 is formal; (iv) is a Lefschetz algebra.

Proof. We begin with the proof of part (i). Calculating the Chevalley­Eilenberg complex for the Lie algebra 9 = gl EEl we get (Ag*,8)

8z

= bh. = 0,

8x

= -hx,

8y

= hy,

= A(z,h,x,y) deg(z)

= deg(h) = deg(x) = deg(y) = 1

(7), since, without loss of generality, 9 = (Z, H, X, YI [H, X] = X, [H, Y] = -Y) (with all other brackets zero as in Theorem 1.10). Now we see that the element w=zl\h+xl\y represents a cohomologically symplectic element. Recall that (Ag*, 8) represents invariant differential forms on G which implies that w 2 is nowhere vanishing invariant form and, hence, w is non­degenerate. Finally, w descends to a symplectic form on G/f. Proof of (ii). We want to show the non­existence of complex structures on M 4 . This is done by appealing to classification theorems of Kodaira and Yau that are specific to complex dimension 2. Suppose the contrary. Since M 4 is parallelizable (Theorem 1.11), its Euler characteristic and the first Pontryagin

SOLVMANIFOLDS

88

number vanish and this implies the vanishing of the Chern numbers and the geometric genus Pg of M 4 : (8)

Then, from the Kodaira classification of complex surfaces [BPV], it follows that M 4 is an algebraic variety. Indeed, from the classification table in [BPV, table 10, p. 188], we notice that this is the only possibilty compatible with (8). However, the same table yields b}(M 4 ) = 2, but the latter condition together with (8) is not compatible with the Yau classification of compact complex surfaces in [Y]. Proof of (iii). Note that M 4 is completely solvable and hence the Hattori theorem is applicable. This means that the Chevalley-Eilenberg complex (Ag*, b) is a model (not minimal) for GIl'. However, the minimal model of (Ag*, b) can be calculated explicitly by taking (M, d) to be defined by the formulas

(M, d) = (A(x, h) C9lR!.[X] C9 A(Y), d) dz = dh = 0, dX

= 0,

dy = X 2 ,

We also have a morphism 7jJ: (M, d)

1/J(z)

= z,

'Ij!(h)

= h,

deg(z)

= deg(h) = 1,

deg(X) ---->

= 2,

deg(y)

= 3.

(Ag*, b) as follows:

1/!(X) = x

1\

y, 1/J(fj)

= o.

A straightforward calculation shows that 1/J is a quasi-isomorphism. Thus, we have found the minimal model of GIl'. This graded differential algebra is obviously formal, since the DGA-map p: (M,d) ----> H*(M,d),p(z) = [z],p(h) = [h], p(X) = [X]' p(fj) = 0 is a quasi-isomorphism. (MS2XT2, d) (M 4 has rational homotopy To prove (iv), note that (M, d) type of a K iihler manifold T 2 x 8 2 ) . (We note that the terminology 'rational homotopy type' in the non-simply connected situation essentially means 'has the same minimal model'.) This means that H*(M, d) is a Lefschetz algebra. The proof of (iv) and the whole theorem is now complete. 0 This example reveals all the essential difficulties which can be met when considering the solvmanifold case. Notice that minimal models themselves, as well as the Hard Lefschetz Theorem, are not sufficient to settle the problem. More refined invariants are necessary. In dimension 4, these invariants may come from the Kodaira classification of compact complex surfaces. Also, the ChevalleyEilenberg complex in general contains less rational homotopic information than a 'free model' of GIl'. Moreover, because our solvmanifolds are, in general, not completely solvable, the calculation of their cohomologies instead must be made using a theorem of Y. Felix and J .C. Thomas from rational homotopy theory. This type of 'model' will allow us to continue our attack on the problem. The model is provided by Theorem 1.4.6 as applied to the Mostow bundle. We shall

RATIONAL MODELS OF SOLVMANIFOLDS

89

use our calculation of solvmanifold cohomology via the Felix-Thomas theorem to verify the Benson-Gordon conjecture for solvmanifolds of dimension four. Consider the Mostow bundle associated with a solvmanifold G If. For convenience, we will use the simpler notation

where N denotes thc fiber (which is a nilmanifold) and T denotes the base which IS a torus. The action of 71"1 (T) on H* (N) is non-nilpotent, so the Serre spectral sequence is of little use in calculating H* (5). Also, as we mentioned above, Hattori's result does not apply to solvmanifolds in general, so our approach to applying the Felix-Thomas theorem should find wide use. The main difficulty in trying to apply the result is the possible lack of information about the action of the fundamental group of the base on the cohomology of the fibre. We show that this problem can be overcome in the case of the Mostow fibrations associated to our constructions by understanding actions on the Lie algebra n (and its dual n*) of the nilpotent Lie group N. Exarnp le 3.3: The Klein Bottle. One of the simplest examples of a solvmanifold is the Klein bottle K, The Mostow fibration associated to the Klein bottle IS

with 71"1 (51) = z:: acting on HI (51) = z:: as the negative of the identity. This action is clearly non-nilpotent so that U = {OJ. Theorem 1.4.6 then gives the rational (or real) cohomology of the Klein bottle as that of a circle. Of course this agrees with standard calculations. As we mentioned before, however, it should be noted that the Klein bottle is not a solvmanifold of the form GIf. Now, we will present a general construction of certain types of solvmanifolds and analyze the action of the fundamental group of the base on the fibre in the Mostow fibration. We start with the simplest case. Recall that by Theorem 2.12 in [Rag, Chapter II] a simply connected nilpotent Lie group N admits a lattice I' if and only if the Lie algebra n has a basis in which all structural constants are rational. In the latter case, denote by no the vector space generated over Q by this basis and take a lattice L of maximal rank in n lying in no. Then, exp L is a lattice in N. Conversely, if I' is a lattice in N, a Z::-submodule in n generated by exp -1 (r) is a lattice of maximal rank in n. Example 3.4: The Construction of Certain Solvrnarrifolds. Let ¢: -+ Aut( n) denote a one-parameter subgroup of the automorphism group of the nilpotent Lie algebra n such that each ¢(t) preserves the lattice L. Because the corresponding nilpotent Lie group N is simply connected, ¢ may be 'lifted' uniquely to a one-parameter subgroup -+ Aut(N) with = ¢(t) for each t ([War Theorem 3.27, 3.57]). Furthermore, each J(t) will preserve the lattice f. Also, note that automorphisms of n correspond to derivations of n under the exponential map, exp: Der(n) -+ Aut(n), so we can take 1/)(t) E Der(n) with

SOLVMANIFOLDS

90

exp( 1/)( t)) product

= cf;(t),

a fact we shall make use of shortly. Now form the semi-direct

G=ill!.)

so that the holonomy action is naturally related to the structure group action on the fibre F. By considering the adjoint formulation fJB

FF

->

II

1 H

FF

taking components and noting that Jro (fJB) Jr](B)

= Jr] (B),

we obtain

[F, F]

->

II

1 Jro( H)

[F, F]

->

where [F, F] denotes the set of free homotopy classes of self maps of F. Thus, Proposition 3.6. If F

-> E -> B is a fibre bundle with structure group H, then the action of the fundamental group of the base on the homology of the fibre factors through the component group Jro( H) of the structural group of the bundle.

There is yet another more algebraic way to understand this action. Consider an embedded circle 5] -> T which gives a generator for Jr] (T) and pull back the Mostow fibration to a fibration over 5] N

5

=1

1

N

->

->

T

1

5'

where it is easy to see that both S' -> Sand Jr](5') pullback fibration has an exact homotopy sequence

->

Jr](5) are injective. The

SOLVMANIFOLDS

92

with a splitting s: IZ ---+ 71"1(5') since IZ is free. This gives 71"1(5') the structure of a semi-direct product and the splitting s provides an outer action of IZ on 71"1 (N). Specifically, if 0: E IZ, then for any x E 71"1 (N), we may take 0: . X = s( 0: )xs( 0: )-1, conjugation of x in the group 71"1(5') 71"1(5). This 'action' is not well defined however because it depends on the particular splitting s chosen. The action is defined, however, up to conjugation in 7I"1(N). To see this, let r : IZ ---+ 71"1(5') be another splitting and let s(o:) = (), r(o:) = T where 0: generates IZ. Then there is some g E 71"1(5') with T = g () and, because () and T both project to 0: in LZ, by exactness it must be the case that g = n E 7I"1(N). Hence, for any x E 7I"1(N),

TXT- 1 = (n (})x(n (})-1) n((}x(}-I)n- 1 and the conjugations are the same up to a conjugation inside 71"1 (N) itself. Now, N = K (71"1 (N), 1) is an Eilenberg- Mac Lane space and it is well known that the free homotopy classes of self maps of such spaces correspond to conjugacy classes of self homomorphisms of the fundamental groups. Thus, again the action of the fundamental group of the base is reflected in a free homotopy action on the fibre. Further, every element in a free homotopy class has the same effect on (co)homology, so the action of 7I"1(T) on H*(N) is well defined. Now, because 7I"1(T) is the fundamental group of a product of circles and each pullback has 71"1(5') ---+ 71"1(5) injective, it is clear that we can understand the action in the entire Mostow fibration as conjugation in 71"1 (5). We state this as

Theorem 3.7. For the Mostow fibration N ---+ 5 ---+ T associated to a solvuuuiifold 5, the action Of7l"1(T) on H*(N) is through conjugation in 71"1(5). IZ

Of course, we also point out that if we start with a semi-direct product 71"1 (5) = ¢ I', then conjugation of I' by IZ in 71"1 (5) corresponds to the action of the

)4

automorphism

J;

(0:,1)(0, ,)(0:,1)-1 = (0:, 1)(0,,)(-0:,1)

= (0:, J(o:)(t))( -0:,1) = (0, J(o:)(,)). An important thing to realize in the discussion above is that the structure group action for the Mostow fibration is given by left translation

Nf Ifax .vrIf

---+

.vrIf

and, for the solvmanifolds constructed in 3.4, this may be detected at the Lie algebra level in a simple way. Hattori's theorem gives an isomorphism H*(N) '= H*(An*,8) and the transpose of the ¢ action on n corresponds to the action of 71"1(5 1). For a solvmanifold constructed as in 3.4,

RATIONAL MODELS OF SOLVMANIFOLDS

93

where I' is a lattice in Nand 1rl(S) ==;z, >4J I', this follows because the action J extends uniquely to a lattice preserving action on N and this in turn provides the original chosen action ¢J on the Lie algebra n. In particular, because a homomorphism of Lie groups induces a homomorphism of (left) invariant forms ([GHV vol. II 4.7]), we obtain the following diagram for each t. A¢(t)t

An*

An*

-----+

Qinv (N)

--+

Qinv(N)

---+

1 O(N)

J*

1

O(N)

where Q(N) is the De Rham algebra on the nilmanifold Nand Qinv(N) is the algebra of left invariant forms on the nilpotent Lie group N. Note that this algebra of invariant forms is naturally isomorphic to An*, the differential graded algebra of cochains on the nilpotent Lie algebra n. If we apply cohomology to the diagram above, we obtain H*(An*,6)

(A¢(t)'r

H*(An*,6)

==1

1==

H*(N)

H*(N)

==1

1==

We state this as follows. Theorem 3.8. For the examples S constructed in example 3.4, the action of 1rl(Sl) =;z, on H*(N) = H*(1rl(N)) = H*(f) is given by:

(1) restricting ¢J: lFI.

-+

Aut(n) to 1J:;z,

-+

(2) taking the dual automorphisms ¢Jt: ;z,

Aut(n) -+

Aut(n*)

(3) extending to the exterior algebra A¢t: ;z,

-+

Aut(An*) as differential

graded algebra maps (4) and taking the induced automorphisms on cohomology (A¢t)*: ;z, Aut(H*(An*,6)).

-+

SOLVMANIFOLDS

94

Instead of looking at the action on H*(An*, 8) directly, we observe that the differential graded algebra (An*, 8) is generated as an exterior algebra by n*. Thus, knowing the action ¢ on n" will determine the action completely. Before we consider more general cases, let us look at two particular examples. Example 3.9 [BG2]. Let IR*e(1) has no eigenvalues equal to 1 and the same argument as in subcase 1) yields b1 (G If) = 1 and the proof follows. If (X = 1, we get 11 2 + v 2 = 1. However, in this case n;; = o" EB !R'.e3, where e;; is a vector dual to e3 (which is in the center of n3, and, therefore, does not generate non-zero cohomology classes). Again, H 1 (AY) = H 1(An;;,b) = (vi,v2) and [jl = UnH 1 (AY ) = {O}, which implies b1(G/f) = 1. Consider now the case G =!R'. :xl¢!R'.3. This case is easier, since R" is an abelian Lie algebra and one can use the Jordan matrix directly that is, without loss of generality, the matrix representing A = 1>(1 )*, can be represented as follows: (x l

either

A

=

(

*

or

SOLVMANIFOLDS

100

In each case, H*(AY, dy ) = A(Xl, X2, X3), where all Xi constitute a base in which A has one of the two forms above. In the first case above we get dim( U n H1(AY)) 2, in the second case, respectively, dim(U n H1(AY)) 1. This means that either b1(G/r) 3, or b1(G/f) 2 (from (10)). The cases b1 = 1 and h = 3 are eliminated by the Hard Lefschetz Theorem, the case b1 = 2 by Theorem 3 in [FG2, p.298]. D

:s

:s

:s

:s

Remark 4.2. [FG2, Theorem 3, p. 298] is, in fact, a corollary to the Kodaira and Yau classification of compact complex surfaces [BPVJ, [Y].

5. Higher dimensional examples and twisted tensor products We have already mentioned that, fortunately enough, the 4-dimensional case could be settled by the use of an additional information specific to complex compact surfaces [BPV]. There is no such information in higher dimensions. Therefore, the aim of this section is twofold. First, we try to collect as much information as we can coming from rational homotopy theory. With this purpose, we analyse the algebraic properties of the Chevalley-Eilenberg complexes associated with solvmanifolds (completely solvable or not, Theorem 5.1). These results enable us to construct several higher dimensional examples. Secondly, we try to develop an alternative approach along the lines of Sullivan's original twisted models. Unfortunately, we don't know whether the total spaces of Mostow bundles admit twisted tensor products as rational models.

Theorem 5.1. Let M = G/f be a compact solvmanifold carrying a Kii.hlerian structure. Then the cochain complex (Ag*, 6) of the Lie algebra 9 satisfies the property the: all triple Massey products and higher order Massey products of the pair (Ag* , H* (Ag*) vanish as cohomology classes in H* (Ag*). That is, there is a choice of cocheins in Ag* representing all triple Massey products and higher order Massey products so that these cochains are exact. Remark 5.2. This condition is stronger then saying simply that all Massey products vanish, since for instance, a triple Massey product ([aJ, [bJ, [c]) is defined with an indeterminancy lying in the ideal generated by [a] and [c]. Recall that for introducing Massey products one needs a pair (A, H*(A)) (a differential graded algebra A together with its cohomology algebra). It is important for us to stress this fact, since in the sequel we will change algebras without changzng cohomologies and, therefore, Massey products will also vary for different pairs. For this reason, we will use a slightly different terminology considering Massey products of the pair (A, H*(A)). Proof of Theorem 5.1. Theorem 2.9 implies that the natural map (Ag*, 6) ---+ (ODR(G/f), d) induces a monomorphism on the cohomology level and the proof of the theorem then follows from Lemma 5.3 below. D Lemma 5.3. Let

HIGHER DIMENSIONAL EXAMPLES

101

(6) be a morphism in !Pl.-DCA, inducing monomorphism in cohomology. Then the foJJowing implication holds: if (B, dB) is a formal consequence of its cohomology algebra, then aJJ triple Massey products and higher order Massey products of the pair (A, H*(A)) vanish as cohomology classes in H*(A). That is, there is a choice of cochains in A representing aJJ triple Massey products and higher order Massey products so that these cochains are exact. Proof. Consider, first, triple Massey products. Of course, we could proceed with Massey products of an arbitrary order, but we prefer to accomplish the proof separately for triple and then for quadruple Massey products presenting all essential moments of the proof but avoiding clumsy notation. Note that for each morphism 'P : (A, dA ) ­­­+ (B, de).

'P* ([a), [b), [c])

= ('P*[a), 'P*[b), 'P*[c]),

where the right­hand side of this identity is considered as a cohomology class in H*(B, dB) and the right­hand product (,,) is taken with respect to the pair (B, H*(B)). Indeed, the cohomology class [y] representing the triple Massey product is determined by the following procedure. If

then

Here and in the sequel x denotes (-I)P x for x E AP. We take an arbitrary cocycle y obtained by this procedure and fix it, We don't care about the indeterminancy lying in ([a), [cD. Then

and

'P(d Ax 12 ) = dB'P(x 12 ) = 'P(a)'P(b) 'P(d Ax 23 ) = dB'P(x 23 ) = 'P(b)'P(c), so that 'P(Y) represents, by definition, a triple Massey product

(['P(a)), ['P(b)), ['P(c)]) of the pair (B, H*(B)). Finally,

'P*[Y]

= (['P*[a), 'P*[b), 'P*[c])

SOLVMANIFOLDS

102

as expected. Now, let us prove the lemma for triple Massey products. Let (MB, d) be the minimal model of (B, dB) and p be the corresponding quasi-isomorphism

Assume that there exists a non-zero cohomology class [y] E H*(A, dA ) represented as a triple Massey product [y] = ([a]' [b], [cD. We have already shown that 1/'*[y] = (1/'*[a], 1/'*[b], 1/'* [c]). Since 1/'* is injective, 1/'*[y] represents a nonzero cohomology class in H*(B) which is represented as a triple Massey product of the pair (B, H*(B)). Since p" is an isomorphism, there exist cocycles as«, bi«, CM E MB such that

Recall that

1/;* [a] 1/'* [b]

= 0,

1/'* [b]1/'*[c]

= 0,

Hence,

dZ 12 for cochains

z12, z23

= aMb M,

dz 23

= bMcM

E MB. Take

which is a cocycle representing the triple Massey product ([aM], [b M], [CM]) of the pair (M B, H*(MB) ::= H*(B)). Thus,

P*[YM]

= p*([aM], [b M], [CM]) = (p* [aM], p*[b M], P*[CM]) = (1/'*[a], 1/)*[b], 1/'* [c]) =1/'*[y]fO.

But, since (MB, d) is formal, [YM] must vanish as a cohomology class in H* (MB) because of the following general fact proved in [DGMS, p.262](also see Remark

1.3.2): if (MB, d) is formal, then uniform choices can be made so that the cochains representing all Massey products and higher order Massey products are exact. This contradiction implies ([a]' [b], [c]) = 0 in H*(A). Consider now quadruple Massey products. Recall the procedure of constructing the corresponding cohomology classes. Again, we assume that all cohomology classes representzng triple Massey products of the pair (A, H*(A)) vanish. This

HIGHER DIMENSIONAL EXAMPLES

103

is the condition under which quadruple Massey products are well-defined. Sometimes this condition is called 'simultaneous vanishing'. Assume that [w] E H*(A) is a quadruple Massey product,

[w]

= ([a], [b], [c], [u]).

Again, we begin with the general observation that

M

rank(1t 0 [2Iw) :::; rank(1t 0 [21'rd (J can only decrease the rank of the apropriate 2-form). Therefore, a fat connection can only be obtained in the case when one exists in the appropriate universal bundle. The remaining part of this section is devoted to explicit examples of fat bundles. We construct an example of an O-fat connection in universal SO(n)-bundles and a non-homogeneous It(F)-fat connection. Invariant Fat Connections. Let P(M, G) be a principal bundle and let K be a connected Lie group acting on M transitively so that M = K / H. We always assume that the action of K can be extended to a fibertransitive action on P by automorphisms of P(M, G). For example, this is always true for G-structures, subbundles of the frame bundle L(M, GL(n, Indeed, in this case one can define a K -action on P by the rule Exarnp le 5.6:

TOTAL SPACES OF BUNDLES

166

x E M,

k E K.

Define the isotropy representation ,\: H --+ G as follows: fix Uo E P and take an element '\(h) E G such that h(uo) = R)"(h)UO' An easy calculation shows that ,\ is a well-defined homomorphism. Assume that M = K/ H is a reductive homogeneous space. That is, assume there exists a decomposition

=

t

Ad(H)(m) C m.

EB m,

Since K acts by automorphisms on P(M, G), it makes sense to consider Kinvariant connections in P(M, G). These are described by the Wang theorem. Wang's 'I'Iieorern [KN]. Under tbe above assumptions there is a one-to-one correspondence between K-invariant connections in P(M, G) and linear maps Am: m --+ 9 such that Am(Ad(h)X)

= Ad('\(h)(Am(X)),

for each X E m and h E H. This correspondence is given by the formula

where B is a connection form and X is a fundamental vector field generated by X E t on P. Here A is given by

A: t

--+

g,

A(X) = ,\*(X),

if X

A(X) = Am(X),

if X Em.

The curvature form of B is expressed by fluo(X, Y)

= [Am(X), Am(Y)] -

Am([X, Y]m) - ,\*([X, Yh)

where X, Y E m and [X, Y]m and [X, Yh denote the m- and h-cornponent respectively.

=0

In particular, the map Am theorem and, therefore,

obviously satisfies the conditions of the Wang

Am

=0

defines an invariant connection in P(M, G) which is called the canonical connection. Thus, we get the following result. 'I'heor-em 5.7. Let F be a symplectic manifold wit]: a Hamiltonian G-action. A canonical in varian t connection is J.1( F)-fat if and only if the alternating 2-form (X, Y)

is non-degenerate for each

t->

f

J.1 (f) (,\ * ([X, Yh),

X, Y Em

(7)

E F.

Proof The proof follows from the expression of the curvature form given by the Wang theorem applied to the case Am = O. Indeed, in this case, the condition

SYMPLECTIC FAT BUNDLES

167

of the non-degeneracy of 1-l(J) 0 n on each horizontal space is equivalent to the non-degeneracy of (7) since the horizontal distribution in P is generated by the reductive complement m. 0 Theorem 5.7 yields explicit examples of symplectic fatness. 'I'heor-em 5.8. There exists a co-adjoint orbit 0

C SO(2n)* such that the

canonical invariant connection in the principal bundle SO(m

+ 2n)jSO(m)

SO(2n)

SO(m

+ 2n)(SO(m)

x SO(2n))

is O-fat for all integers m and n. Proof. Under the assumption of the theorem, the moment map is an embedding 0 ---+ g* and, therefore, the fatness condition is expressed as follows: an alternating 2-form

(X, Y)

Y]ry))

f-+

is non-degenerate for an element E 0. Further, the isotropy representation A: SO(m) x SO(2n) ---+ SO(2n) is a natural projection. Take the reductive complement m consisting of matrices

x= where A is an arbitrary (m x 2n)-matrix. Now, taking UQ = SO(m) in the total space SO( m+ 2n)j SO( m) and applying the formula for the curvature form given by the Wang theorem we get

t

nuo(X, Y) = -A*([X, Y]so(m)x5o(2n)) = -[X, Y]so(2n) = AB - BAt

(8)

where X, Y E m and A, B are matrices representing X and Y respectively. Consider the basis {Eij} of m which consists of matrices in so( m + 2n) having all entries zero except entries in the positions (i, j + m) and (j + m, i), for 1 :::; i :::; m, 1:::; j :::; 2n (the latter entries are 1 and -1 respectively). Consider also the basis {%} of the Lie algebra so(2n) consisting of matrices with zero entries everywhere except entries in positions (i + m, j + m) and (j + m, i + m) which are 1 and -1 (here 1 :::; i < j :::; 2n. Obviously, [Eij, Ekdso(2n)

= -qjl,

if i

= k,

and

[Eij, Ekl]

= 0,

if i:f 0

= 1. A straightforward calculation for j < l. Define E so(2n)* by the rule shows that, for the basis {E i j } , the 2-form (7) has a matrix of the form o U

o

0 ...

TOTAL SPACES OF BUNDLES

168

where U is a (2n x 2n)-matrix

0

1

o

-1

(

-1

1 1

-1

o

-1

-1

o

which is non-degenerate.

Example 5.9: A Non-Homogeneous Fat Bundle. Let F be the Hirzebruch surface F

= {([a, bJ, [x, y, z]) E Cpl

X

Cp2

I

ay - ox = O} .

Since F is a non-singular algebraic variety in Cpl x CP2, it is a complex submanifold of a Kahler manifold and, therefore, is itself a Kahler manifold with respect to the induced Kahler structure. Theorem 5.10. Let M = 8 4 be a 4-dimensional sphere represented as quaternion projective space !HIp1

8 4 =:: !HIpl =:: 8p(2)j(Sp(1) x S'p(1)). Consider the principal bundle

(9)

Sp(2)(S4, Sp(1) x Sp(l)).

There exists a symplectic and Hamiltonian Sp( 1) x 8p( 1 )-action on the Hirzebruch surface F endowed with a standard symplectic structure such that the canonical connection in the principal bundle (9) is J-l(F)-fat for the moment map W F ---+ (5.p(1) x 5.p(1))* of this action. Proof. Define the required action of Sp( 1) x Sp( 1) on F via a natural projection onto the first component and the well-known isomorphism S'p(1) =:: S'U(2). Namely, set

g([a, bJ, [x, y, z]) = ([g(a, b)J, [g(x, y), z]),

9 E SU(2).

This action is symplectic, since the action of U(2) x U(3) is symplectic on Cpl x CP2. Since SU(2) is semisimple, this action is Hamiltonian [Ki] and, therefore, admits a moment map. Obviously, for each Lie group homomorphism 2, without loss of generality we can assume that M is a product of two symplectic manifolds of positive dimension. Now, the Leibniz rule for L * completes the proof. 0

sl(2)-Representations. These representations are studied in detail in [GrH]. Note that for evident reasons we need to consider the case of infinite-dimensionol representations. Therefore, we introduce, first, the following definition.

SURVEY

178

Definition 1.5. Let V be the (infinite-dimensional) vector space of a Lie algebra

representation p: s[(2)

--->

g[(V).

We say that V is an s[(2)-module of finite H -specirum if the following two conditions are satisfied: 1) V is a direct sum of eigenspaces of H; 2) H has only finitely many distinct eigenvalues.

Now, the basic algebraic result concerning s[(2)-representations is the following Proposition 1.6. Let V be an s[(2)-module of finite H -spectrum. Let, as be-

fore, H, X, Y denote the canonical generators of s[(2). The representation p has the following properties: 1) All eigenvalues of H are integers; 2) the maps

are isomorphisms for all k :::: 0; 3) the following equality holds: {v E Vk

I p(X)v = O} = {v

E Vk

I p(y)k+ 1 = O}

(6)

This proposition obviously implies the basic fact we need, the duality on forms, since DDR(M) obviously admits the structure of an s[(2)-module of finite Hspectrum (recall (2) and the definition of A = p(H)). Corollary 1. 7. The maps

are isomorphisms for all k :::: O. Identities (1) imply also that the structure of s[(2)-module of finite H -spectrum is inherited by the subspace

1t.,DDR(M)

=

of all symplectzcally harmonic forms. Corollary 1.8. The maps

Lk

:

u;»

--->

k:::: 0

(5)

are isomorphisms for all k :::: O. Thus, to complete this part of the proof, we must show that Proposition 1.6 holds. It follows from the following observation: the formula

BRYLINSKI'S CONJECTURE AND SYMPLECTIC INVARIANTS

p(X)(p(y)k(V))

= p(y)k(p(X)(v)) + k(A -

k

+ 1)p(y)k1(v),

179

VE V

holds for all egenvectors of p(H) corresponding to eigenvalue A. This formula can be verified by a straightforward calculation and we leave it to the reader. Now, as a corollary, we get the fact that, for v E V an eigenvector of p(H) and W the JJ[(2)-submodule generated by v, W is a finite dimensional vector space. Finally, this fact then implies Proposition 1.6 since, for finite dimenszonal JJ[(2)representations this result is well known [GrH]. 0 The Final Step. Assume, first, that the Brylinski condition holds. That is, each de Rham cohomology class contains a symplectically harmonic representative. Then the natural map

must be surjective. Now, the commutative diagram

tc:" s

1 Hn-k(M)

L

k

H ns +k

1

k

L ---+

Hn+k(M)

has two surjective vertical arrows and a bijective upper horizontal arrow (Corollary 1.8) which imply the 'if' part of the proof. Now we assume that L k : H n- k ­­­" Hn+k is surjective for all k :::; n. First we claim that Hn-k can be decomposed into a sum (which is direct in the compact case)

Indeed, by the surjectivity assumption, for any B E Hn-k(M) there exists ( E Hn-k-2(M) with Lk+1(B) = Lk+2((). This is equivalent to

B- (

1\ w E

Pn -

k

or

B = (B ­ (

1\ w)

+ ( 1\ w E 1m L + Pn ­ k

as required. Now we complete the proof by induction with respect to degrees of forms. Obviously, O­cocycles and l­cocycles are symplectically harmonic. Assume that when r < n - k . any class B E H" (M) contains a symplectically harmonic representative. We show that any class in H n - k also contains such a cocycle. By the induction hypothesis, any class in Im L contains a symplectically harmonic

SURVEY

180

cocycle. Therefore, it suffices to show that the same is true for cohomology classes in Pn - k . Let v = [z] E Hn-k(M). Since [z 1\ w] = 0, there exists, E D';)tt+1(M) such that z I\w = d,. Since Lk+ 1: D';)Rk-1(M) ----> D';)tt+1(M) is surjective, we can choose 0' E D';)Rk-1(M) such that, = (¥ 1\ w k + 1. Then (z - dO') 1\ wk+ 1 = O. It means that for w = z - do: we get

[w]

= [z] = v,

an d

L k+ 1 ( w)

= O.

Now, an application of (I) completes the proof. Indeed, Lk+1(w) = 0 implies L*(w) = 0 because of (6) (recall once more that we have an 5[(2)-representation in DDR(M) such that p(X) = L* and p(Y) = L). Now, using the fourth identity in (1) we get 6(w)

= [L*,d](w) = -d(L*(w)) = 0

which means that w is syrnplectically harmonic.

o

Theorem 1.4 shows that symplectic harrnonicity in the sense of Brylinski's conjecture is equivalent to the Hard Lefschetz condition. Therefore, we obtain a tool for distinguishing between Kahler and non-Kahler symplectic manifolds. It seems quite natural that further analysis will produce new homotopic invariants related stnctly to symplectic structures. At present our knowledge is not satisfactory. Of course, Theorem 1.4 yields a new proof of the fact that the only Kahler nilmanifolds are tori since the Hard Lefschetz condition is violated for non-toral nilmanifolds. (This fact was used in [FIL3], where the first explicit example of a symplectic 'non-Brylinski manifold was given. As the reader might expect, the example was the Kodaira-Thurston manifold). We complete this section by mentioning several results 'around' the Brylinski conjecture. It would be useful and interesting to analyze them from the rational homotopy point of view. In [Bo], Bouche introduced a natural differential complex related to symplectic manifolds as follows. Let (M, w) be a symplectic manifold of dimension 2n. Let

be the subspace of coeffective forms on M. Obviously the complex

is a differential sub complex of the de Rham complex. Its cohomology HP (A( M)) is called coeffective cohomology of the symplectic manifold. It was proved by Bouche that the coeffective complex is elliptic for p 2': n+ 1 (that is, the coeffective cohomology groups HP(A(M)) are finite dimensional for p 2': n+l). On the other hand, it is known from [Bo] that the truncated de Rham cohomology groups

APPLICATIONS TO THE ARNOL'D CONJECTURE

181

are isomorphic to the coeffective cohomology groups in case the symplectic manifold is actually Kahler. Therefore, again we get a tool for distinguishing between Kahler and symplectic non­Kahler manifolds. The authors of [FIL2] obtained an analogue of the Nomizu­Hattori theorem (see Chapter 3.2) for the coeffective cohomology of nilmanifolds and completely solvable solvmanifolds (the defini tion of the coeffective cohomology for the Chevalley­Eilenberg complex is left to the reader as an exercise). In [FIL3] the authors considered two natural spectral sequences associated with the J( oszul differential 0 = i( G) 0 d ­ do i(G) introduced in Theorem 1.4 and the de Rham differential d. It was shown in [FIL3] by example that the degeneracy of the first spectral sequence is, in a sense, independent of the Brylinski conjecture. In [11] R. Ibanez considered the Brylinski conjecture for Poisson manifolds. Note that the Koszul differential can be defined for any Poisson manifold and, consequently, harmonicity of f} means that of} = df} = O. It was shown in [11] that the Brylinski conjecture holds for compact cosymplectic manifolds. Recall that a (2n + 1)­dimensional manifold M is called cosymplectic if it is endowed with an almost contact structure (tP, 7/) which is compatible with a Riemannian metric g and satisfies the following conditions: (i) a 2­form defined by the formula (X, Y)

= g(1)(X), Y),

X, Y E X(M)

is closed; (ii) 7J is closed; (iii) \l1> = 0 for the Levi­Civita connection \l. In [12] a version of the Hodge decomposition for the coeffective cohomology was given for Kahler manifolds. Some results were obtained also for indefinite Kahler metrics.

2. Applications to the Original Arnol'd Conjecture Arnol'd's conjecture is a well­known long­standing problem which has stimulated the development of symplectic topology and geometry. Recall that Arnol'd conjectured (see [HZ]) that the number of fixed points of for a Hamiltonian selfdiffeomorphism on a symplectic manifold M is at least as large as the number of critical points for any smooth function on M. Conley and Zehnder proved the conjecture for tori, but the lack of appropriate invariants forced a modification of the conjecture (so that the lower bound is replaced by the cup­length of M) and this modification was solved by Floer [HZ] through the development of his homology theory. Ii is not the aim of this section to present here the difficult analytical and topological aspects of the theory. We refer to [HZ], for instance, for a complete modern account. We present here only the rational homotopy aspect of the problem. Although the original Arnol'd conjecture remains open

182

SURVEY

in general, the description of rational homotopy in terms of the algebraic minimal model allows for the definition of new and powerful algebraic invariants which then must be related to geometry. One of these, Toomer's invariant eo was studied in [MO] and led to the affirmation of the original Arnol'd conjecture for nilmanifolds. Because this result was proved by means of rational homotopy theory, we discuss it briefly here. The following result was announced by Poincare shortly before his death and proved by G. D. Birkhoff in 191:3 [Hi]. The result is sometimes called Poincare's Last Geometric Theorem or Theorem 2.1: Birkhoff's Twist Theorem. Any area-preserving diffeomorphism of the annulus which rotates the inner and outer boundary circles in opposite directions has at least two fixed points. In the 1960's Arnol'd saw how to generalize this theorem [Ar, Appendix 9]. First note that, for dimension 2, area­preserving is the same as symplectic. Also, note that, the Lusternik­Schnirelmann (LS) category obeys cat(Annulus) + 1 = cat(8 1 ) + 1 = 1 + 1 = 2 :S #(critical points of any function on the Annulus). Hence, the number of fixed points of a 'symplectic' map on a manifold should have something to do with the number of critical points of smooth functions on the manifold. Finally, the generalization of the twist condition is to consider exact or Hamiltonian diffeomorphisms. These are the diffeomorphisms of a symplectic manifold which arise from a time­dependent Hamiltonian flow as follows. Suppose (M,w) is a symplectic manifold and H: M x 8 1 ­­­4 !PI. is a time dependent Hamiltonian which is l­periodic in time. Denote the dependence on time t by writing H, for H restricted to time t. Then the symplectic form gives a time­dependent Hamiltonian vector field XH defined by

w(X H

,- )

= dH t ( -

)

and has associated to it a flow ¢t with d dt¢t = XH(¢t)

and

¢o = 1M·

Let ¢ = ¢I be the time 1 map of the flow and say that any such map arising as a time 1 map of a Hamiltonian flow is an exact or Hamiltonian diffeomorphism. Arnol'd Conjecture 2.2. Let MC(M) denote the minimum number of critical points for any function f: M ­­­4 !Pl.. Then the number of fixed points of an exact symplectic diffeomorphism on (M, w) is at least MC(M). This is equivalent to saying that, for a time-dependent Hamiltonian which is I-periodic in time H: M x 8 1 ­­­4 !Pl., the associated Hamiltonian flow has at least MC(M) I-periodic orbits. Remark 2.3. If M is aspherical, then MC(M) = dimM + 1. Takens [Tak] showed that M C( M) ::; dim M + 1 and Eilenberg­Ganea [EG] showed that cat(M) = dim(M) for such a space. Since, by the original theorem of LusternikSchnirelmann, cat(M) + 1 ::; MC(M), the equality follows.

A large step in understanding the Arnol'd conjecture came in 1983 [CZ] due to Conley and Zehnder.

APPLICATIONS TO THE ARNOL'D CONJECTURE

183

Theorem 2.4. The Arnol'd Conjecture is true for T 2 n . That is, there are at least 2n+ 1 fixed points for any Hamiltonian diffeomorphism on the torus. This is equivalent to saying that, for a time-dependent Hamiltonian which is I-periodic in time, the associated Hamiltonian flow llas at least 2n + 1 I-periodic orbits. Conley and Zehnder first reduce the problem to finite dimensions by lifting the symplectic structure and Hamiltonian from M = T 2n to M = !PI. 2n . Periodic orbits on !PI. 2n project to contractible orbits on M and these are what are going to be counted. While the analysis is too complicated to present here, suffice it to say that the (L 2 ) gradient of the action functional (where x(t) is a I-periodic loop)

OH(X) is doH(x)

= -Jx -

=

t' 2

- H(x,t)dt

Jo

\7 H(x, t). Therefore, a critical loop x gives

0=

-]X -

\7 H(x, t)

J» = -\7H(x,t)

x= x

_(1)-1\7 H(x, t)

= J\7 H(x, t)

by Proposition 1.2.1

So, periodic solutions of the Hamiltonian system correspond to critical points of 0H. Conley and Zehnder reduce the problem of finding these critical points to finding the critical points of a special kind of function g: M x !PI. 2m --+!PI.. All critical points of 9 are in a compact invariant set S (contained in an isolating neighborhood) of the gradient flow \7g(x) = x. Let cg denote the number of critical points of g. Conley and Zehnder show that cg 2: cup(T 2 n) + 1 = 2n + 1 and therefore prove the original Arnol'd conjecture for tori. As we mentioned above, subsequent to the Conley-Zehnder proof, the Arnol'd conjecture was modified to take account of the difficulty in estimating the number of critical points on a manifold. The modification consisted of focussing on the more computable cuplength of a manifold and applying the known result that the number of critical points is bounded below by cup length plus 1 (see 2.5 and 2.6 below). In order to understand the rational approach to the problem of estimating critical points, let's recall the notions of LS category and cuplength. Recall that a space X is said to have LS category n if there exist n + 1 open sets U, C X such that n+l

X=

UUi

and each U, is contractible in X

i=1

and n is the least integer for which this is true. The second condition means that there is a homotopy H : U, X I --+ X such that H a is the inclusion and HI is the constant map. One thing that we can say right away is that category is related to cuplength. Recall that the cuplength of X, denoted cup(X), is the largest

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184

integer k so that there exist X; E Hn'(X;R), for i = 1,···,k and a nontrivial cup product 0 i- XIX2'" Xk. Here R is a ring of coefficients. The following result is well-known and is the basis of many calculations of category. Proposition 2.5. cup(X)::; cat(X).

Proof. Suppose cat(X) = k. choose any k + 1 elements X; E Hn,(X; R), i = 1 ... k + 1 with deg(x;) = i. and form the cup product XIX2" ·Xk+l. Now, because the inclusion U; '----> X is nullhomotopic, the exact cohomology sequence of the pair gives

which guarantees that each X; has a preimage X; E Hj,(X, U;) and, therefore, XIX2" 'Xk+1 has a preimage XIX2" ·Xk+l. The properties of relative cup products then give (for N = L ji) XIX2 ... Xk+1

E H

N

(X,

U;)

= H N (X, X) = O.

Since XIX2 ... Xk+1 maps to XIX2'" Xk+l, the latter is zero as well. Hence, any cup product of length greater than k is trivial and cup(X) ::; cat(X). 0 Properties of Category 2.6. Other important properties of category are given by the following: (1) Category is an invariant of homotopy type. (2) If C, = Y u, CX is a mapping cone, then cat(Cf ) ::; cat(Y) + l. (3) If X is a CW-complex, then, by induction on skeleta and property (2), cat (X) ::; dim X . (4) In fact, (3) may be generalized: If X is (r-l)-connected, then cat(X) ::; dimX/r. (5) The category of a map f: X ----4 Y is cat(J) = n if n is the least integer such that there is a covering of X by n + 1 open sets on each of which f is nullhomotopic. If f: X C Y, then cat(J) = catj- X, the relative category of X in Y. (6) The category provides a lower bound for the number of critical points of a function on a compact manifold. That is, if M is compact, then cat(M) + 1 ::; MC(M). The proofs of these properties may be found in [Whi] and the excellent survey [J] for example.

Examples 2.7. (1) (2) (3) (4) (5)

cat(X) = 0 if and only if X is contractible. cat(Sn) = 1. More generally, cat(X) = 1 if and only if X is a nontrivial co-H space. cat(T n ) = n (this follows from Proposition 2..5 and property (3) above). If (M 2 n , w) is a simply connected compact symplectic manifold, then cat(M)

APPLICATIONS TO THE ARNOL'D CONJECTURE

185

:= n := dim M. First, observe that the volume form is not exact since it represents a nontrivial fundamental class of M. Because w n In! := vol, the non degenerate closed 2-form w cannot be exact either. Hence, w n represents a nontrivial cup product of length n in IR-cohomology. By property (4) above, cat(M) :::; (dimM)J2 := n. Hence, n :::; cup(M) :::; cat(M) :::; dimM := n and the result follows.

In order to study category from the point of view of rational homotopy theory, it is convenient to use another, more homotopical, approach. Let Mn+l denote x M (n + 1 times) and let the product M x M x T n+l M:= {(Xl,

X2,

Xn+l)! at least one Xi is a specified basepoint *}.

denote the fat wedge. Include Minto M n + 1 via the diagonal map 6,( x) (x, x, ..., x) and denote the inclusion Tn+l M '-> M n+l by j. Then cat(M) := n if and only if n is the least integer for which 6, may be deformed into T n +l M. That is, there is a map 6,': M -> T n + l M and a homotopy commutative diagram, M 6,' -,

Tj Tn+IM.

This is, of course, the Whitehead approach to category. Similarly, if I: M -> M is a map of spaces, then cat(J) := n if and only if n is the least integer for which there is a map 6,': M -> T n +l M and a homotopy commutative diagram, M

IT M

---+

M n+l

TJ Tn+IM.

Now, in order to translate these diagrams into rational homotopy theory, we need minimal models for M, M, M n+ l and Tn+l M. If we denote the model of M by (AV,d), then the model for M n + 1 is simply AV0 n + 1 with d on each factor and extended by the Leibniz rule. Moreover, the model of the diagonal map is simply multiplication, denoted by u : Ay0 n + 1 -> AY. In [FH] it is shown that a minimal model for the fat wedge T n + l M is given by a minimal model ¢J: AY --> n of the quotient DCA Ay0 n + 1 n -- .A+Y0 ,. . . ,. .,. . . ,.n+ ,.----,-I:- '

where A+ Y consists of all elements of positive degree. If 7T: AV0 n +1 -> n denotes the projection, then, since ¢ is a quasi-isomorphism, there is a lift AV0 n + 1 -> AY with ::::: 7T (see Corollary 1.2.8). The DCA map is, in fact, a minimal model for the inclusion j of the fat wedge into the product. A direct translation from spaces to minimal DGA's provides (with the notation above and where we denote the model of M by A V and the model of I by F: A V --> A V)

SURVEY

186

Definition 2.8 [FH]. The rational category of a space M, denoted cato(M), is the least integer n so that there exists a DG A map p: AY ----+ AV with ::: tJ. The rational category of a map f: M ----+ M, denoted cat.; (J), is the least integer n so that there exists a DGA map p: AY ----+ AV with FtJ· Remark 2.9. Since a spatial diagram induces a diagram of minimal models, clearly cato(J) :; cat(J). Also, it is important to note that Felix and Halperin worked in the general situation of all path connected spaces. Thus, the minimal model approach to category may be done for these spaces, but inferences are generally only good for nilpotent spaces since it is in this situation that minimal models are a true reflection of homotopy theory. It is the presence of algebraic structure which allows a more accessible criterion for the determination of rational category. Let A?:n+l V denote the differential ideal of AV consisting of sums of monomials Xi, ... Xik having k ;::: n + 1 and let p: AV ----+ AV/A?:n+lv be the projection. Take a minimal model (): AZ ----+ AV/ A?:n+lV and note that, since () is a cohomology isomorphism, there is a lift p: AV ----+ AZ with (}p::: p. For details, see [FH, Definition 4.3 and Theorem 4.7]. Multiplication induces an obvious map Ii: n ----+ AV/A?:n+lV which in turn has a lift to minimal models il: AY ----+ AZ with (}il ::: Ii¢;. The following fundamental diagram commutes up to DGA homotopy. AZ

AY

Tp AV

F f--

AV

Now, according to [FH, Theorem 4.7] and [Fe, p.126], we have

Theorem 2.10. cato(M) = n if and only if if n is the least integer such that there exists a DGA map r : AZ ----+ AV with rp::: idAv. cato(J) = n if!nd only ififn is the least integer such that there exists a DGA map r : AZ ----+ AV with rp::: F. The theorem allows us to define invariants which may be used to estimate cato(J). One of these is Toomer's eo invariant. The eo invariant is well known for spaces [To] and minimal models [FH]. It was introduced in [MO] for maps.

Definition 2.11. Define eo( M) to be the largest integer k so that there exists a cocycle 0: E A?:kV with [0:] -# 0 in H*(AV) H*(M;!Q). Define eo(J) (= eo(F)) to be the largest integer k so that there exists a cocycle 0: E A?:kV with F*[o:] -# O.

APPLICATIONS TO THE ARNOL'D CONJECTURE

187

Proposition 2.12. Let f: M --> M be a map of path-connected spaces with minimal model F: AV --> AV. Then 1. If f* = 0, then eo(f) = 0. 2. eo(f) eup(Imagef*). 3. eo(f) :::; eo(M), eo(M). 4. eo(f) :::; cato(f)· 5. If f* is injective, then eo(f)

= eo(M).

Proof. Parts 1, 2 and 3 are obvious, so we prove only 4 and 5 For 4, suppose cato(F) = n and let 0: E A2: n +1 V be a cocycle. We have a diagram, AZ r /

AV Now, p(a) = 0, so (J*p*[a] p*[a] = 0. Finally, F*[a]

F f--

T P "-." (J

AV

p --->

= p*[a] = 0. But (J is a cohomology isomorphism, so = rp*[a] = 0. Therefore, no cocycle in A2: n + 1 V has

nontrivial image in cohomology. Hence, eo(f) :::; n = cato(f). For 5, let 0: E A2: e o ( M ) V be a cohomologically nontrivial cocycle. Because F is a DGA map it cannot decrease length, so F(o:) E A2: e o ( M ) V . Further, since f* = F* is injective, F( 0:) is also cohomologically nontrivial. Hence, eo(f) eo(M). Together with part 3, this implies the result. 0

Now that we understand cuplength and category, we can understand the statement of Floer's theorem. Floer's methods are deep and introduce completely new techniques - indeed a completely new homology theory suited for such problems as the Arnol'd conjecture. Floer took the Conley-Zehnder variational approach with a more general action functional defined on a type of loop space. Indeed, one reason that a hypothesis (i.e. 7r2M = 0) such as the one in the theorem below is necessary is to ensure that his action is well defined. Again critical points of the action correspond to contractible periodic solutions, but now the estimate must be made in an infinite dimensional space - and this requires a method different from that of Conley and Zehnder. Floer invents his homology to handle this estimation. But note that, since Floer works with homology, he is restricted to a cuplength estimate. Good references for all of this - especially the variational analysis - are [ABKLR], [HZ] and [McD3]. Floer proved the Theorem 2.13 [Fl]. For any symplectic manifold (M, w) with 7r2M modified Arnol'd conjecture is true. That is, there are at least CUP2M

= 0,

the

+ 1 fixed

points for any Hamiltonian diffeomorphism, where CUP2M is the !Z/2-cuplength ofM. Remark 2.1.4. Floer actually proved more general results which require weaker hypotheses than the vanishing of 7r2M, but the result above is the easiest to state. In the past several years, even more refined results have appeared in this vein (e.g. [Ono)).

188

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Some time ago, Chris McCord had the idea to stay closer to the ConleyZehnder approach when considering the Arnol'd conjecture on manifolds one step away from tori ­ nilmanifolds. The variational analysis briefly outlined above goes through in virtually the same way, but there is one difference later on. Namely, while Conley and Zehnder could achieve the Arnol'd conjecture using cup length because cup(T 2n ) = 271 = dim T 2n , for nilmanifolds this was no longer the case. The cuplength and dimension of a nilmanifold may be very different. Indeed, while there is a finite algorithm for calculating the cuplength of a nilmanifold, there does not seem to be known a formula which gives the cuplength of a nilmanifold in terms of invariants of the finitely generated torsionfree nilpotent group say. So, what can take the place of the cup length as an estimator of critical points? Certainly the LS category works. In fact, Proposition 2.15 [MO]. Let M be a nilmanifold in the setting of the Arnol'd conjecture. The number of critical points c9 of the special function g: M x m. 2m ---> obeys c9 catxS + I = cat(i) + I

where X = M x m. 2m , S is the compact maximal invariant set of the gradient flow of 9 and i: 5' ---> X is inclusion. Therefore, we are in the position of estimating the category of the inclusion map i: S ---> X. Some technical details arise here which we will ignore, but we can also say that the inclusion i induces an injection on cohomology H* (M; Q) =: H*(X; Q) ---> H*(S; Q). First, let's note

Lemma 2.16 [02]. If M is a nilmanifold, then eo(M) = dim M. Proof. By the definition of eo (Definition 2.11), we are looking for the 'longest' product cocycle which represents a nontrivial cohomology class. But we have seen in our discussion of nilmanifolds that the top class is represented by the longest product in the whole minimal model ­ the product of all generators. Since the number of generators is the dimension of the nilmanifold, we are done. D

Since category is trapped between eo and dimension, we have Corollary 2.17. For a nilmanifold M,

(I) eo(M) = cato(M) = cat(M) = dimM. (2) If f: S ---> M induces an injective homomorphism f* on cohomology, then eo(J) = dim M. Proof. The first part is clear. The second part follows from Proposition 2.12 (5) and the preceding discussion. D

As a result, we can prove the anginal Arnol'd conjecture for nilmanifolds. Corollary 2.18 [MO]. If M is a nilmanifold, then every I-periodic Hamiltonian system has at least dimM + I = MC(M) I-periodic orbits. That is, the Arnol'd conjecture is true for nilmanifolds.

DOLBEAULT HOMOTOPY THEORY Proof. Let i: S M map the invariant set S to M c:: M X jR2m an injection on cohomology. We have the following estimate: MC(M)

= dimM + 1 =co(M) + 1

by Corollary 2.17

=eo(i)+1

by Corollary 2.17

::; cat( i)

+1

189

=X

inducing

by [Tak] [EG]

by Proposition 2.12 (4)

=cat xS+ 1

by definition of relative category

::; cg .

by Proposition 2.15

o In fact, we can prove more. The important points about the proof above are the facts that we can do the Conley-Zehnder analysis on a universal covering space which is jRN for some N and that we have an eo-invariant. Therefore we have 'I'hcorern 2.19 [MO]. If (M, w) has universal cover a Euclidean space, then every I-periodic Hamiltonian system has at least eo(M) + 1 I-periodic orbits.

Unfortunately, for non-nilpotent spaces, just as for any minimal model invariant, the eo-invariant is not a very good estimator of category. There may be hope for this approach however, for twisted minimal models are now making their appearance with the goal of algebraicizing non-nilpotent homotopy theory. If all of the rational category theory can be done for these twisted models, then the twisted eo-invariant should play the same role as the untwisted one. Namely, it should happen that, for aspherical manifolds covered by Euclidean space, the twisted Co should be the dimension of the manifold and the Arnol'd conjecture would hold exactly as before. Beyond the question of the Arnol'd conjecture, there seems to be much to be discovered about the role rational homotopy can play in dynamical systems. For some new exciting work relating rational homotopy to Conley's index, see [Co].

3. Dolbeault Homotopy Theory Dolbeault homotopy theory was developed in [NT] and can be characterized as a version of Sullivan's theory for complex manifolds and their Dolbeault cohomology. We outline briefly the contents of the Neisendorfer-Taylor paper since results there have much potential for application, but as yet have not been utilized to their utmost. We hope our exposition leads to a re-examination of this subject. Throughout, M is a complex manifold of complex dimension n, 0/:':/ is the space of complex valued differential forms of type (p, q) on M. As in Chapter 1,

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190

Definition 3.1. We define a differentzal bigraded algebra (DBA) to be a bigraded commutative algebra over (( with a differential a of type (0,1) which is a derivation with respect to the total degree. In this way we obtain the category CC-DBA of complex DBA's by requiring the maps to be bidegree preserving and commuting with the derivation a. Now, n:;.r* is equipped with the differential in a standard way and we see that (n:;.r*, a) E ((-DBA as well. By definition,

a

H*'*(n:;.r*, a)

= H*'*(M)

is the usual Dolbeault cohomology. The usual de Rham cohomology is related to H*'*(M) by the Frolicher spectral sequence.

Definition 3.2. (i) A bigraded algebra is called BBA-algebra (bigraded, bidifferential) if it is equipped with two differentials and of types (1,0) and (0,1) respectively, such that

a

a

aa = -aa; (ii) The cohomology of BBA is defined a/ways with respect to a­differential. Observe that (n:;.r*, a, a) E ((-BBA. Also, the reader should not confuse this notion of 'bigraded' with the one involving a lower grading. We have chosen to use the original terminology because, at this point, no better terminology presents itself to us. The authors of [NT] prefer to use the Bousfield­Guggenheim version of rational homotopy theory [BGu], using cofibrant models instead of Sullivan's minimal models. They define a cofibrant DBA as follows.

Definition 3.3. A DBA­algebra (A, aA) is cofibrant if, given a map of DBA's g: (B,

a

B ) ­­>

(C, Be)

which is both a surjection and a quasi­isomorphism and a map of DBA's

I . (A, DB) there exists a DBA-map h: (A, aA)

­­>

­­>

(C, ae),

(B, BB) such that go h

= f.

By definition, a cofibrant model for a DBA (D, a'O) is a cofibrant DBA-algebra (M, aM) which is quasi­isomorphic to (D, 0'0). Using these notions, we can also define cohomotopy of DBA's and BBA's [NT].

DOLBEAULT HOMOTOPY THEORY

191

Example 3.4. We want to find a cofibrant DBA-model for Cl?"; that is, for the Dolbeault complex 8). Now, the minimal model of Cl?" is given by

(Mrcpn, d) d(X)

= 0,

d(x)

=X

n

= (C[X] @ A(x), d)

+1 ,

deg(X)

= 2,

deg(x)

= 2n + 1 .

Since cpn is Kahler,

Thus,

H 2(cpn)

= H1,1(cpn)

and the only possibility for assigning bidegrees to Mcpn is bideg(X) which implies 8(x) DBA

(C[X]

= (1,1)

X n + 1 . Since minimal models are cofibrant [BGu], the

A(x), 1J

= d),

bideg(X)

= (1,1),

bideg(x)

= (2n, 1)

is the cofibrant and minimal DBA-model for the Dolbeault complex

8).

Definition 3.5. A DBA or BBA-algebra (A,8A) is formal if there exists a cofibrant algebra (B, 813) with maps (in the appropriate category)

(A, 8A ), (B, 813) ----> H*'*(A, 8A) inducing isomorphisms on the cohomology level. (B, 813)

---->

Now, we shall introduce the notions of Dolbeault homotopy theory which are most relevant for our work. Definition 3.6. A bigraded bidifferential graded algebra (A, aA, 8A) is said to be a-degenerate ifthere exists aBBA (M, a, 8) such that

(i) a = 0; (ii) there is a map in the category C-BBA

(M, 8, 0= 0)

---->

(A, 8A, aA);

(iii) (M,8) is a cofibrant model for (A, 8A) in the category C-DBA. Definition 3.7. (i) A bigraded bidifferential algebra (A, 8A, aA) is strictly formal if it is a-degenerate and formal as an object in C-DBA; (ii) A complex manifold M is Dolbeault formal if a, 8) is formal as an object in C-DBA. M is called strictly formal if 0, 8) is strictly formal as an object in C-DBA.

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192

Theorem 3.8 [NT]. Any strictly formal complex manifold is also Dolbeault formal and de Rham formal.

The relations between Dolbeault formality and de Rham formality are rather subtle as examples in [NT] show. For example, Stein manifolds appear to be Dolbeault formal, but not de Rham formal in general. Also, Calabi-Eckmann manifolds are all Dolbeault formal, but not all of them are strictly formal (see [NT] for details). The most important results of [NT] can be summarized as follows. Theorem 3.9.

(1) Any compact Kahler manifold is strictly formal; (2) A a-degenerate manifold has a natural Hodge decomposition. This means that, in the category of a-degenerate manifolds, there exists a natural isomorphism of cohotaotopy theories and a natural isomorphism of cohomology theories. Outline of Proof. (i) Recall from Chapter 2 that any compact Kahler manifold a, 8)). Now, given any M satisfying the a8satisfies the a8-lemma for lemma, define K*M* to be the subalgebra (in the category C­DBA)

(K*M*,8) c (r2*M* , 8, 8), K*M*

= Ker

0.

The above inclusion is a quasi-isomorphism (by the same approach as in Chapter 1). Choose a cofibrant model (M, 8M ) for (K*M*, 8) and notice that the sequence of maps

(M*'*,

8M , a = 0)

--+

(K*M*, 8, a

= 0)

--+

8, a)

is a sequence of quasi-isomorphisms in C­BBA. Now, exactly as in Chapter 1, we obtain strict formality from the sequence of quasi-isomorphisms

8, 8) is a-degenerate and thus there exists a To prove (ii), notice that cofibrant model (M, 8M , 0) and a quasi-isomorphism in C­BBA

Since this is a map in C­BBA, we get a map

(M, d = 8)

--+

d)

which the Frolicher spectral sequence shows to be a quasi-isomorphism.

0

Remark 3.10. It is possible to prove that isomorphisms between cohomology theories are independent of the choice of a cofibrant model [NT].

MISCELLANEOUS EXAMPLES

193

Corollary 3.11. Let M be any a-degenerate complex manifold. Then the Sullivan minimal model of M, (M, d), is simultaneously a cofibrant (minimal) model for 0, 8) in the category C.-DBA. Remark 3.12. This corollary leads to the following natural question: Are there compact symplectic manifolds which are not strictly formal? This question is quite natural since the known examples are based on less refined invariants. It would be quite interesting to find a complex compact symplectic manifold for which non-Kahlerness follows from Dolbeault homotopy theory.

4. Miscellaneous Examples In this section we will present several important examples which have some specific features and are not covered completely by techniques developed in earlier chapters.

Example 4.1 (Yamato [Yam]). This example exploits Thurston's idea of constructing symplectic non-Kahler manifolds as bundles over symplectic manifolds with 2-dimensional fibers (e.g. the Kodaira-Thurston manifold) together with the calculation of triple Massey products (see Chapter 1). To formulate the main result, we need several notions related to the Dehn twist dijJeomorphisms of surfaces. Thus, let 2: g be an oriented closed surface of genus 1. Let {aI, ... ,a g ; h, ... ,b g } be closed simple oriented curves on 2:g satisfying the following conditions: ai

n aj = b, n bj = 0 for i i- j

and a; intersects bi at one point with intersection number (+ 1) for i = 1, ... ,g.

Definition 4.2. In the sequel we will call the system of curves defined above a symplectic system.

Let T Z be a 2-dimensional torus with a coordinate system (exp(Ol yCT), exp(OzR)) and let a and b denote the closed curves such that a( 0) exp( oR. Define a neighborhood of a U b by

U

= {( exp 01 H, exp oR) E T Z ,

-3f

< 01 < 3f,

where e > 0 is a small fixed number such that 3f < hoods U, for all pairs ai Ubi and diffeomorphisms

i«: U; for all i.

U,

f(ad

= a,

h(b i )

exp oR, b(0)

or - 3f

< Oz < 3f}

Thus, we have neighbour-

=b

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194

Definition 4.3. A map T(a;) (resp. T(b i)) is called the Dehn twist diffeomorphism along ai (resp. bi) if it is a diffeomorphism T(ai): 2:g

->

2:g

T(b i): 2:g

resp.

->

2:g

such that supp (T(ai)) C U, (resp. supp (T(b i)) C U;) and on Ui,

or, respectively,

where ,(B) is a smooth function on IP? satisfying the conditions

(1) l(B + 211') = ,(B) + 211'; (2) ,(B) = 0 for e - 211'::; (}::; -e and ,(8) (3) ,is strictly increasing on [-e,eJ.

= 211' for

e::; ()::; 211' - e,

Note that 2:g is an oriented surface and therefore is endowed with a volume form v. One can show that this form can be considered as invariant with respect to all Dehn twist diffeomorphisms. In what follows, we fix the volume form v.

Theorem 4.4 (Yamato, [Yam]). Let (N, fl) be a closed symplectic manifold admitting a homomorphism

such that

p( lTl(N))

= (T(cd, .. · ,T(cn)1 supp(T(c;)nT(cj) =fl

Define a 11'1 (N)­action on a(x, z)

fori

=p 1, c, E {al, . . ,ag,h, ... ,b g}}.

Nx

2:g by

= (O'(a)(x), p(a)(z)),

for

a E lTl(N)

where 11': N N is a universal covering of Nand 0'( a) is a covering transformation corresponding to a E lTl(N). Let

be the quotient space with respect to the projections

11'1 (N)­action.

Consider the natural

MISCELLANEOUS EXAMPLES

195

Then we may conclude (1) the 2-form

w = pi(7T*n) + p;v is closed, non-degenerate and 7T1 (N)-invariant; (2) the 2-form w thus projects to a natural symplectic form w on M; (3) (M,w) is a compact symplectic manifold which is non-Kiibler since the ptut

(nDR(M), H*(M)) has non-vanishing triple Massey products. Example 4.5 (Ferllandez-Gray-Morgan [MG]). This method also provides examples of symplectic manifolds with non-vanishing triple Massey products. These manifolds are constructed from lower dimensional symplectic manifolds by taking circle bundles over mapping tori built over a compact symplectic manifold (and constructed with respect to a given symplectomorphism). Let (M, w, 'P) be a triple consisting of a compact manifold, a symplectic structure wand a symplectomorphism 'P. Let N