ñòð. 1 |

103 Surveys in combinatorics 1985, I. ANDERSON (ed)

104 Elliptic structures on 3-manifolds, C.B. THOMAS

105 A local spectral theory for closed operators, I. ERDELYI & WANG SHENGWANG

106 Syzygies, E.G. EVANS & P. GRIFFITH

107 Compactification of Siegel moduli schemes, C-L. CHAI

108 Some topics in graph theory, H.P. YAP

109 Diophantine Analysis, J. LOXTON & A. VAN DER POORTEN (eds)

110 An introduction to surreal numbers, H. GONSHOR

III Analytical and geometric aspects of hYPerbolic space, D.B.A.EPSTEIN (ed)

112 Low-dimensional topology and Kleinian groups, D.B.A. EPSTEIN (ed)

113 Lectures on the asymptotic theory of ideals, D. REES

114 Lectures on Bochner-Riesz means, K.M. DAVIS & Y-C. CHANG

115 An introduction to independence for analysts, H.G. DALES & W.H. WOODIN

116 Representations of algebras, P.I. WEBB (ed)

117 Homotopy theory, E. REES & J.D.S. JONES (eds)

118 Skew linear groups, M. SHIRVANI & B. WEHRFRITZ

119 Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL

121 Proceedings of Groups - St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds)

122 Non-classical continuum mechanics, R.I. KNOPS & A.A. LACEY (eds)

124 Lie groupoids and Lie algebroids in differential geometry, K. MACKENZIE

125 Commutator theory for congruence modular varieties, R. FREESE & R. MCKENZIE

126 Van der Corput's method for exponential sums, S.W. GRAHAM & G. KOLESNIK

127 New directions in dynamical systems, T.I. BEDFORD & J.W. SWIFf (eds)

128 Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU

129 The subgroup structure of the finite classical groups, P.B. KLEIDMAN & M.W.LIEBECK

130 Model theory and modules, M. PREST

131 Algebraic, extremal & metric combinatorics, M-M. DEZA, P. FRANKL & LG. ROSENBERG (eds

132 Whitehead groups of finite groups, ROBERT OLIVER

133 Linear algebraic monoids, MOHAN S. PUTCHA

134 Number theory and dynamical systems, M. DODSON & J. VICKERS (eds)

135 Operator algebras and applications, 1, D. EVANS & M. TAKESAKI (eds)

136 Operator algebras and applications, 2, D. EVANS & M. TAKESAKI (eds)

Analysis at Urbana, I, E. BERKSON, T. PECK, & J. UHL (eds)

137

138 Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UHL (eds)

139 Advances in homotopy theory, S. SALAMON, B. STEER & W. SUTHERLAND (eds)

140 Geometric aspects of Banach spaces, E.M. PEINADOR and A. RODES (eds)

141 Surveys in combinatorics 1989, J. SIEMONS (ed)

142 The geometry of jet bundles, D.J. SAUNDERS

143 The ergodic theory of discrete groups, PETER J. NICHOLLS

144 Introduction to uniform spaces, I.M. JAMES

145 Homological questions in local algebra, JAN R. STROOKER

146 Cohen-Macaulay modules over Cohen-Macaulay rings, Y. YOSHINO

147 Continuous and discrete modules, S.H. MOHAMED & B.I. MULLER

148 Helices and vector bundles, A.N. RUDAKOV et al

149 Solitons, nonlinear evolution equations and inverse scattering, M.A. ABLOWITZ & P.A.

CLARKSON

150 Geometry of low-dimensional manifolds 1, S. DONALDSON & C.B. THOMAS (eds)

151 Geometry of low-dimensional manifolds 2, S. DONALDSON & C.B. THOMAS (eds)

152 Oligomorphic permutation groups, P. CAMERON

153 L-functions in Arithmetic, J. COATES & MJ. TAYLOR

154 Number theory and cryptography, J. LOXTON (ed)

155 Classification theories of polarized varieties, TAKAO FUJITA

Twistors in mathematics and physics, T.N. BAILEY & R.l. BASTON (eds)

156

London Mathematical Society Lecture Note Series. 151

Geometry of Low-dimensional

Manifolds

2: Symplectic Manifolds and Jones-Witten Theory

Proceedings of the Durham Symposium, July 1989

Edited by

S. K. Donaldson

Mathematical Institute, University ofOxford

C.B. Thomas

Department ofPure Mathetmatics and mathematical Statistics,

University ofCambridge

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CAMBRIDGE UNIVERSITY PRESS

Cambridge

New York Port Chester Melbourne Sydney

Published by the Press Syndicate of the University of Cambridge

The Pitt Building, Trumpington Street, Cambridge CB2 lRP

40 West 20th Street, New York, NY 10011, USA

10, Stamford Road, Oakleigh, Melbourne 3166, Australia

Â© Cambridge University Press 1990

First published 1990

Printed in Great Britain at the University Press, Cambridge

Library ofCongress cataloguing in publication data.available

British Library cataloguing in publication data av.ailable

ISBN 0 521 40001 5

COl'

vi

Contents of Volume 1

vii

Contributors

viii

Names of Participants

xi

Introduction

r

˜ xiv

Acknowledgements

!

1

PART 1: SYMPLECTIC GEOMETRY \

"

,˜

3

Introduction

Rational and ruled symplectic 4-manifolds 7

()usa McDuff

Syrnplectic capacities 15

II. Hofer

'rhe nonlinear Maslov index 35

J\. B. Givental

45

I˜'i II ing by holomorphic discs and its applications

Yakov Eliashberg

69

PART 2: JONES/WITTEN THEORY

71

Inlroduction

73

New results in Chern-Simons theory

Edward Witten, notes by Lisa Jeffrey

97

( icometric quantization of spaces of connections

N.J. Hitchin '

(˜valuations of the 3-manifold invariants of Witten and

Reshetikhin-Turaev for sl(2, C) 101

Robion Kirby and Paul Melvin

Representations of braid groups 115

M.F. Atiyah, notes by S.K. Donaldson

123

PART 3: THREE-DIMENSIONAL MANIFOLDS

Int roduction 125

1\11 introduction to polyhedral metrics of non-positive curvatur˜. on 3-manifolds 127

I.R. Aitchison and I.H. Rubinstein .

163

I,'illite groups of hyperbolic isometries

('.H. Thomas

!'i,/ structures on low-dimensional manifolds 177

I{.('. Kirby and L.R. Taylor

CONTENTS OF VOLUME 1

Contents of Volume 2 vi

vii

Contributors

viii

Names of Participants

ix

Introduction

Acknowledgments xiv

PART 1: FOUR-MANIFOLDS AND ALGEBRAIC SURFACES 1

Yang-Mills invariants of four-manifolds 5

S.K. Donaldson

On the topology of algebraic surfaces 41

Robert E. Gompf

=Pg = 0

The topology of algebraic surfaces with q 55

Dieter Kotsehick

63

On the homeomorphism classification of smooth knotted surfaces in the 4-sphere

Matthias Kreck

Flat algebraic manifolds 73

F.A.E. Johnson

PART 2: FLOER'S INSTANTON HOMOLOGY GROUPS 93

Instanton homology, surgery and knots 97

Andreas Floer

Instanton homology 115

Andreas Floer, notes by Dieter Kotschick

Invariants for homology 3-spheres 125

Ronald Fintushel and Ronald J. Stern

On the FIoer homology of Seifert fibered homology 3-spheres 149

Christian Okonek

Za-invariant SU(2) instantons over the four-sphere 161

Mikio Furuta

PART 3: DIFFERENTIAL GEOMETRY AND MATHEMATICAL PHYSICS 175

Skynne fields and instantons 179

N.S. Manton

Representations of braid groups and operators coupled to monopoles 191

Ralph E. Cohen and John D.S. Jones

Extremal immersions and the extended frame bundle 207

D.H. Hartley and R.W. Tucker

Minimal surfaces in quatemionic symmetric spaces 231

F.E. Burstall

Three-dimensional Einstein-Weyl geometry 237

K.P. Tod

Harmonic Morphisms, confonnal foliations and Seifert fibre spaces 247

John C. Wood

CONTRIBUTORS

I. R. Aitchison, Department of Mathematics, University of Melbourne, Melbourne, Australia

M. F. Atiyah, Mathematical Institute, 24-29 St. Giles, Oxford OXl 3LB, UK

F. E. Burstall, School of Mathematical Sciences, University of Bath, Claverton Down, Bath, UK

Ralph E. Cohen, Department of Mathematics, Stanford University, Stanford CA 94305, USA

S. K. Donaldson, Mathematical Institute, 24-29 S1. Giles, Oxford OXl 3LB, UK

Yakov Eliashberg, Department of Mathematics, Stanford University, Stanford CA 94305, USA

Ronald Fintushel, Department of Mathematics, Michigan State University, East Lansing,

MI 48824, USA

A. Floer, Department of Mathematics, University of California, Berkeley CA 94720, USA

Mikio Furuta, Department of Mathematics, University of Tokyo, Hongo, Tokyo 113, Japan, and,

Mathematical Institute, 24-29 St. Giles, Oxford OXl 3LB, UK

A. B. Givental, Lenin Institute for Physics and Chemistry, Moscow, USSR

Robert E. Gompf, Department of Mathematics, University of Texas, Austin TX, USA

(). H. Hartley, Department of Physics, University of Lancaster, Lancaster, UK

N. J. Hitchin, Mathematical Institute, 24-29 St. Giles, Oxford OXl 3LB, UK

II. Hofer, FB Mathematik, Ruhr Universitat Bochum, Universitatstr. 150, D-463 Bochurn, FRG

(jsa Jeffrey, Mathematical Institute, 24-29 St. Giles, Oxford OX} 3LB, UK

I:. A. E. Johnson, Department of Mathematics, University College, London WCIE 6BT, UK

J. D. S. Jones, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Robion Kirby, Department of Mathematics, University of California, Berkeley CA 94720, USA

I )icter Kotschick, Queen's College, Cambridge CB3 9ET, UK, and, The Institute for Advanced

Study, Princeton NJ 08540, USA

Matthias Kreck, Max-Planck-Institut rUr Mathematik, 23 Gottfried Claren Str., Bonn, Gennany

N. S. Manton, Department of Applied Mathematics and Mathematical Physics, University of

( 'ambridge, Silver St, Cambridge CB3 9EW, UK

I )usa McDuff, Department of Mathematics, SUNY, Stony Brook NY, USA

I)aul Melvin, Department of Mathematics, Bryn Mawr College, Bryn Mawr PA 19010, USA

( 1hristian Okonek, Math Institut der Universitat Bonn, Wegelerstr. 10, 0-5300 Bonn I, FRG

.J. II. Rubinstein, Department of Mathematics, University of Melbourne, Melbourne, Australia,

(/11Â£1, The Institute for Advanced Study, Princeton NJ 08540, USA

R()nald J. Stem, Department of Mathematics, University of California, Irvine CA 92717, USA

I R. Taylor, Department of Mathematics, Notre Dame University, Notre Dame IN 46556, USA

I.

( c. B. Thomas, Department of Pure Mathematics and Mathematical Statistics, University of

(1˜Hnbridge, 16, Mill Lane, Cambridge CB3 9EW, UK

K. P. Tad, Mathematical Institute, 24-29 St. Giles, Oxford OXI 3LB, UK

I˜. W. Tucker, Department of Physics, University of Lancaster, Lancaster, UK

I':dward Witten, Institute for Advanced Study, Princeton NJ 08540, USA

'()hn (˜. Wood, Department of Pure Mathematics, University of Leeds, Leeds, UK

Names of Participants

L. Jeffreys (Oxford)

N. A'Campo (Basel)

M. Atiyah (Oxford) F. Johnson (London)

J. Jones (Warwick)

H. Azcan (Sussex)

M. Batchelor (Cambridge) R. Kirby (Berkeley)

D. Kotschick (Oxford)

S. Bauer (Bonn)

I.M. Benn (Newcastle, NSW) M. Kreck (Mainz)

R. Lickorish (Cambridge)

D. Bennequin (Strasbourg)

J. Mackenzie (Melbourne)

W. Browder (Princeton/Bonn)

N. Manton (Cambridge)

R. Brussee (Leiden)

P. Bryant (Cambridge) G. Massbaum (Nantes)

F. Burstall (Bath) G. Matic (MIT)

D. McDuff (SUNY, Stony Brook)

E. Corrigan (Durham)

M. Micallef (Warwick)

S. de Michelis (San Diego)

S. Donaldson (Oxford) C. Okonek (Bonn)

P. Pansu (Paris)

S. Dostoglu (Warwick)

J. Eells (Warwick/Trieste) H. Rubinstein (Melbourne)

Y. Eliashberg (Stanford) D. Salamon (Warwick)

G. Segal (Oxford)

D. Fairlie (Durham)

R. Fintushel (MSU, East Lansing) R. Stern (Irvine, CA)

A. Floer (Berkeley) C. Thomas (Cambridge)

K. Tod (Oxford)

M. Furuta (Tokyo/Oxford)

G. Gibbons (Cambridge) K. Tsuboi (Tokyo)

R. Tucker (Lancaster)

A. Givental (Moscow)

R. Gompf (Austin, TX) . C.T.C. Wall (Liverpool)

C. Gordon (Austin, TX) S. Wang (Oxford)

4J_C. Hausmann (Geneva) R. Ward (Durham)

N. Hitchin (Warwick) P.M.H. Wilson (Cambridge)

H. Hofer (Bochum) E. Witten (lAS, Princeton)

J. Hurtebise (Montreal) J. Wood (Leeds)

D. Husemoller (Haverford/Bonn)

P. Iglesias (Marseille)

INTRODUCTION

In the past decade there have been a number of exciting new developments in an

area lying roughly between manifold theory and geometry. More specifically, the

l)rincipal developments concern:

(1) geometric structures on manifolds,

(2) symplectic topology and geometry,

(3) applications of Yang-Mills theory to three- and four-dimensional manifolds,

(4) new invariants of 3-manifolds and knots.

Although they have diverse origins and roots spreading out across a wide range

()f mathematics and physics, these different developments display many common

f(˜atures-some detailed and precise and some more general. Taken together, these

developments have brought about a shift in the emphasis of current research on

luanifolds, bringing the subject much closer to geometry, in its various guises, and

)hysics.

()ne unifying feature of these geometrical developments, which contrasts with some

˜(\ometrical trends in earlier decades, is that in large part they treat phenomena in

specific, low, dimensions. This mirrors the distinction, long recognised in topology,

I)ptween the flavours of "low-dimensional" and "high-dimensional" manifold theory

(n.lthough a detailed understanding of the connection between the special roles of

t1)(˜ dimension in different contexts seems to lie some way off). This feature explains

t.he title of the meeting held in Durham in 1989 anq in turn of these volumes of

Pl'oeeedings, and we hope that it captures some of the spirit of these different

cI(-velopments.

It, tnay be interesting in a general introduction to recall the the emergence of some

of t.hese ideas, and some of the papers which seem to us to have been landmarks.

(We postpone mathematical technicalities to the specialised introductions to the

Hix separate sections of these volumes.) The developments can be said to have

1.(˜˜t1n with the lectures [T] given in Princeton in 1978-79 by W.Thurston, in which

1)(' developed his "geometrisation" programme for 3-manifolds. Apart from the

illll)(˜tus given to old classification problems, Thurston's work was important for

foil<' way in which it encouraged mathematicians to look at a manifold in terms of

various concomitant geometrical structures. For example, among the ideas exploited

ill I'f] the following were to have perhaps half-suspected fall-out: representations of

liuk groups as discrete subgroups of PSL 2 (C), surgery compatible with geometric

tdol'tlet.ure, rigidity, Gromov's norm with values in the real singular homology, and

1l1oHt important of all, use of the theory of Riemann surfaces and Fuchsian groups

to develop a feel for what might be true for special classes of manifolds in higher

Ii lilt -llsions.

f

f\1(Â·H.llwhile, another important signpost for future developments was Y. Eliashberg's

proof in 1981 of "symplectic rigidity"- the fact that the group of symplectic diffeo-

IllOl'phisrns of a symplecti(˜ l'unJ1ifo]<! is Cfo-elosed in the full diffeomorphism group.

Introduction

x

This is perhaps a rather technical result, but it had been isolated by Gromov in

1970 as the crux of a comprehensive "hard versus soft" alternative in "symplec-

tic topology": Gromov showed that if this rigidity result was not true then any

problem in symplectic topology (for example the classification of symplectic struc-

tures) would admit a purely algebro-topological solution (in terms of cohomology,

characteristic classes, bundle theory etc.) Conversely, the rigidity result shows the

need to study deeper and more specifically geometrical phenomena, beyond those

of algebraic topology.

Eliashberg's original proof of symplectic rigidity was never fully published but there

are now a number of proofs available, each using new phenomena in symplectic

geometry as these have been uncovered. The best known of these is the "Arnol'd

Conjecture" [A] on fixed points of symplectic diffeomeorphisms. The original form

of the conjecture, for a torus, was proved by Conley and Zehnder in 1982 [CZ]

and this established rigidity, since it showed that the symplectic hypothesis forced

more fixed points than required by ordinary topological considerations. Another

demonstration of this rigidity, this time for contact manifolds, was provided in 1982

by Bennequin with his construction [B] of "exotic" contact structures on R 3 .

Staying with symplectic geometry, but moving on to 1984, Gromov [G) introduced

"pseudo-holomorphic curves" as a new tool, thus bringing into play techniques

from algebraic and differential geometry and analysis. He used these techniques

to prove many rigidity results, including some extensions of the Arnol'd conjecture

and the existence of exotic symplectic structures on Euclidean space. ( Our "low-

dimensional" theme may appear not to cover these developments in symplectic

geometry, which in large part apply to symplectic manifolds of all dimensions: what

one should have in mind are the crucial properties of the two-dimensional sunaces,

or pseudo-holomorphic curves, used in Gromov's theory. Moreover his results seem

to be particularly sharp in low dimensions.)

We turn now to 4-manifolds and step back two years. At the Bonner Arbeitstagung

in June 1982 Michael Atiyah lectured on Donaldson's work on smooth 4-manifolds

with definite intersection,form , proving that the intersection form of such a manifold

must be "standard". This was the first application of the "instanton" solutions of

the Yang-Mills equations as a tool in 4-manifold theory, using the moduli space of

solutions to provide a cobordism between such a 4-manifold and a specific union

of Cp2,s [D]. This approach again brought a substantial amount of analysis and ,

differential geometry to bear in a new way, using analytical techniques which were '

developed shortly before. Seminal ideas go back to the 1980 paper [SU] of Sacks and

Uhlenbeck. They showed what could be done with non-linear elliptic problems for

which, because of conformal invariance, the relevant estimates lie on the borderline

of the Sobolev inequalities. These analytical techniques are relevant both in: the

Yang-Mills theory and also to pseudo-holomorphic curves. Other important and

influential analytical techniques, motivated in part by Physics, were developed by

C.Taubes [Tal.

Introduction xi

Combined with the topological h-cobordism theorem of M. Freedman, proved shortly

before, the result on smooth 4-manifolds with definite forms was quickly used to

deduce, among other things, that R4 admits exotic smooth structures. Many differ-

ent applications of these instantons, leading to strong differential-topological con-

clusions, were made in the following years by a number of mathematicians; the

other main strand in the work being the definition of new invariants for smooth

4-manifolds, and their use to detect distinct differentiable structures on complex

algebraic surfaces (thus refuting the smooth h-cobordism theorem in four dimen-

sions).

From an apparently totally different direction the Jones polynomial emerged in a

series of seminars held at the University of Geneva in the summer of 1984. This was

a. new invariant of knots and links which, in its original form [J], is defined by the

traces of a series of representations of the Braid Groups which had been encountered

in the theory of von Neumann algebras, and were previously known in statistical

luechanics. For some time, in spite of its obvious power as an invariant of knots

and links in ordinary space, the geometric meaning of the Jones invariant remained

rather mysterious, although a multitude of connections were discovered with (among

other things) combinatorics, exactly soluble models in statistical physics and con-

rormal field theories.

[n the spring of the next year, 1985, A. Casson gave a series of lectures in Berkeley

()1) a new integer invariant for homology 3-spheres which he had discovered. This

Casson invariant "counts" the number of representations of the fundamental group

ill SU(2) and has a number of very interesting properties. On the one hand it gives

an integer lifting of the well-established Rohlin Z/2 J-t-invariant. On the other hand

Casson's definition was very geometric, employing the moduli spaces of unitary

r<˜presentations of the fundamental groups of surfaces in an essential way. (These

luoduli spaces had been extensively studied by algebraic geometers, and from the

point of view of Yang-Mills theory in the influential 1982 paper of Atiyah and Bott

lAB].) Since such representations correspond to flat connections it was clear that

(˜asson's theory would very likely make contact with the more analytical work on

Yang-Mills fields. On the other hand Casson showed, in his study 9f the behaviour

the invariant under surgery, that there was a rich connection with knot theory

()f

and more familiar techniques in geometric topology. For a very readable account of

(˜assons work see the survey by A. Marin [M].

Around 1986 A. Floer introduced important new ideas which applied both to sym-

pl(˜ctic geometry and to Yang-Mills theory, providing a prime example of the in-

t.˜˜raction between these two fields. Floer's theory brought together a number of

I)()we:rÂ£ul ingredients; one of the most distinctive was his novel use of ideas from

Morse theory. An important motivation for Floer's approach was the 1982 pa-

I)('r by E. Witten [WI] which, among other things, gave a new analytical proof of

I.hc' Morse inequalities and explained their connection with instantons, as used in

<.Jl1a.ntum Theory.

xii Introduction

In symplectic geometry one of Floer's main acheivements was the proof of a

generalised form of the Arnol'd conjecture [FI]. On the Yang-Mills side, Floer

defined new invariants of homology 3-spheres, the instanton homology groups [F2].

By work of Taubes the Casson invariant equals one half of the Euler characteristic

of these homology groups. Their definition uses moduli spaces of instantons over

a 4-dimensional tube, asymptotic to flat connections at the ends, and these are

interpreted in the Morse theory picture as the gradient flow lines connecting critical

points of the Chern-Simons functional.

Even more recently (1988), Witten has provided a quantum field theoretic interpre-

tation of the various Yang-Mills invariants of 4-manifolds and, in the other direction,

has used ideas from quantum field theory to give a purely 3-dimensional definition

of the Jones link invariants (W2]. Witten's idea is to use a functional integral in-

volving the Chern-Simons invariant and holonomy around loops, over the space of

all connections over a 3-manifold. The beauty of this approach is illustrated by the

fact that the choices (quantisations) involved in the construction of the represen-

tations used by Jones reflect the need to make this integral actually defined. In

addition Witten was able to find new invariants for 3-manifolds.

It should be clear, even (roln this bald historical summary, how fruitful the cros-

fertilisation between the various theories has been. When the idea of a Durham

conference on this area was first mooted, in the summer of 1984, the organisers

certainly intended that it should cover Yang-Mills theory, symplectic geometry and

related developments in theoretical physics. However the proposal was left va.gue

enough to allow for unpredictable progress, sudden shifts of interest, new insights,

and the travel plans of those invited. We believe that the richness of the contribu-

tions in both volumes has justified our approach, but as always the final judgement

rests with the reader.

References

[A] Arnold, V.I. Mathematical Methods of Classical Mechanics Springer, Grad-

uate Texts in Mathematics, New York (1978)

[AB] Atiyah, M.F. and Batt, R. The Yang-Mills equations over Riemann surfaces

Phil. Trans. Roy. Soc. London, Sere A 308 (1982) 523-615

[B] Bennequin, D. Entrelacements et equations de Pfaff Asterisque 107-108

1983) 87-91

[CZ] Conley, C. and Zehnder, E. The Birkhoff-Lewis fixed-point theorem and a

conjecture of V.I. Arnold Inventiones Math.73 (1983) 33-49

[D) Donaldson, S.K. An application ofgauge theozy to four dimensional topology

Jour. Differential Geometry 18 (1983) 269-316

[Fl] Floer, A. Morse Theozy for Lagrangian intersections Jour. Differential

Geometry 28 (1988) 513-547

Introduction xiii

[F2] Floer, A. An instanton invariant for 3-manifolds Commun. Math. Phys.

118 (1988) 215-240

[G] Gromov, M. Pseudo-holomolphic curves in symplectic manifolds Inventiones

Math. 82 (1985) 307-347

(J] Jones, V.R.F. A polynomial invariant for links via Von Neumann algebras

BulL AMS 12 (1985) 103-111

[M] Marin, A. (after A. Casson) Un nouvel invariant pour les spheres d'homologie

de dimension trois Sem. Bourbaki, no. 693, fevrier 1988 (Asterisque 161-162 (1988)

151-164 )

[SU] Sacks, J. and Uhlenbeck, I{.K. The existence of minimal immersions of 2-

spheres Annals of Math. 113 (1981) 1-24

[T] Thurston, W.P. The Topology and Geometry of 3-manifolds Princeton Uni-

versity Lecture Notes, 1978

[Ta] Taubes, C.H. Self-dual connections on non-sell-dual lour manifolds Jour.

Differential Geometry 17 (1982) 139-170

[WI] Witten, E. Supersymmetry and Morse Theory Jour. Differential Geometry

17 (1982) 661-692

[W2] Witten, E. Some geometrical applications of Quantum Field Theo:cy Proc.

IXth. International Congress on Mathematical Physics, Adam Hilger (Bristol) 1989,

pp. 77-110.

Acknowledgements

We should like to take this opportunity to thank the London Mathematical So-

ciety and the Science and Engineering Research Coun,cil for their generous support

of the Symposium in Durham. We thank the members of the Durham Mathemat-

ics Department, particularly Professor Philip Higgins, Dr. John Bolton and Dr.

Richard Ward, for their work and hospitality in putting on the meeting, and Mrs.

S. Nesbitt and Mrs. J. Gibson who provided most efficient organisation. We also

thank all those at Grey College who arranged the accommodation for the partici-

pants. Finally we should like to thank Dieter Kotschick and Lisa Jeffrey for writing

up notes on some of the lectures, which have made an important addition to these

volumes.

PART 1

SYMPLECTIC GEOMETRY

Introduction 3

In this section we gather together papers on symplectic and contact geometry.

Ilecall that a symplectic manifold (M, w) is a smooth manifold M of even dimension

2n with a closed, nondegenerate, 2-form w i.e dw = 0 and w n is nowhere zero. A

contact structure is an odd-dimensional analogue; a contact manifold (V, H) is a pair

(O()llsisting of a manifold V of odd dimension 2n+ 1 with a field H of 2n-dimensional

Huhspaces of the tangent bundle TV which is maximally non-integrable, in the sense

that if a is a I-form defining H, then dan 1\ a is non-zero (i.e. da is non-degenerate

H).

(Hl

III t.heir different ways, all the articles in this section are motivated by the work

of M. Gromov, and in particular by his paper [G2] on pseudo-holomorphic curves.

11(˜rc the idea is to replace a complex manifold by an almost-complex manifold with

I\. ("ompatible symplectic structure, and to study the generalisations of the complex

(,Ilrvcs-defined by this almost-complex structure. The paper of McDuff below

v;i V(˜H a direct application of this method by showing that a minimal 4-dimensional

Hylllplectic manifold containing an embedded, symplectic, copy of 8 2 = cpt is

C'ither Cp2 or an S2 bundle over a Riemann surface, with the symplectic form

l-e'inp; non-degenerate on fibres. The uniqueness of the structure in the minimal

4'U.:'H\ can be thought of as an example of rigidity.

AIlother important symplectic notion investigated by Gromov is that of "squeezing" .

11(0 proves for example that the polycylinder D (1) xÂ·Â·Â· x D (1) (n factors) cannot

2 2

1"0 ˜ymplectically embedded in D 2 (R) X R 2 n-2 if the radius R of the disc is less

t.han 1. H. Hofer (see below) approaches this and other questions from the point of

vic'w of a symplectic capacity: we can summarise his definition as follows.

A capacity c is a function defined on all sub8ets of symplectic manifold8 of a given

tli17u˜nsion 2n taking values in the positive real numbers augmented by 00 and 8at-

' .â€¢Iying the axioms;

( ˜()) If f is a symplectic DEM defined in a neighbourhood of a subset S C (M, w)

,It"fl. c(S,w) = c(f(5), f*(w)).

( ˜ I )(Conformality) If A > 0 then c(S, AW) = AC( S,w).

(˜2)(Monotonicity) If (S,w) C (T,w) then c(S,w):5 c(T,w).

(˜:J)(Normalisation)c(D2n(1) = c(D 2 (1)xD 2n-2(r)) = 7r for allr ˜ 1, with respect

en = C X Cn- 1.

/" lh(˜ standard symplectic form on

All ('xample of a capacity is provided by the "displacement energy"-heuristically,

#J.iv(IU two disjoint bounded subsets S, S' of R 2 n how much energy does one need

dpform 8 into 5'? As Hofer shows, this capacity can be used to prove the

tu

'Ic(lH"('hing theorem above, and furthermore Axioms CO-C2 suffice to recover the

t 11,.idit,y theorem (mentioned in the general introduction) namely that the symplectic

J )11:M p;roup is Co closed in 4Diff(M).

'1'11(' ('xi˜tence of periodic solutions for Hamiltonian systems leads to a whole family

â€¢If independent capacities, which provide a framework in which to discuss the ex-

Introduction

4

istence of a closed integral curves for the vector field associated with an arbitrary

contact structure on S2n-l. More generally still there is the possibility of using

Floer's symplectic instanton homology groups to define sequence-valued capacities.

These homology groups were the tool used by Floer to prove Arnol'd's conjecture

on the fixed points of symplectic diffeomorphisms for a wide class of manifolds. We

refer to the notes by Kotschick in the first volume of these proceedings for the def-

inition of the symplectic instanton homology groups, and their relation to Floer's

homology groups for 3-manifolds. Gromov's paper provides many parallels between

the theory of Yang-Mills instantons and pseudo-holomorphic curves, and between

the results derived from the two. For example, the existence of a symplectic form

on R2n which is not the restriction of the standard form under some embedding of

R2n in itself is reminiscent of the exotic smooth structures on R 4 â€¢

The Arnol'd conjecture can also be discussed in the framework of the "Nonlinear

Maslov Index" developed by A. Givental and described in his article below. The

definition can be summarised as follows: the linear Maslov index of a loop 'Y in the

=z.

manifold An of linear Legendrian subspaces of Rp2n-l is its class in the 7rl(An)

The nonlinear invariant is obtained by replacing An by the infinite-dimensional

homogeneous space of all Legendrian embeddings of Rpn-l in Rp2n-l.

The various fixed-point theorems now in the literature illustrate the "hard" aspect

of symplectic geometry (for this terminology see [G3]). This rests on the Eliashberg-

Gromov rigidity theorem to which we have referred above, and in the general in-

troduction. In another striking parallel with Coo theory Gromov has shown that

4-dimensional symplectic theory has some of the flavour of smooth surfaces. For

example, if M = 8 2 X 8 2 has the standard symplectic structure w EB w coming from

the Kahler forms on the factors, then DiffW(jjw(M) contracts onto the isometry

group of M, which is a Z/2Z extension of SO(3) X SO(3). Question: does rigidity

give rise to similar results for contact manifolds in dimension 5, for example 8 5 or

S2 X S3 ?

Perhaps the most important test between the hard and soft approaches to symplectic

geometry lies in the problem of the existence and classification of symplectic forms

on a closed manifold M 2 n. At present, no counterexample is known to the obvious

soft conjecture that a global symplectic form exists whenever T M 2 n. has structural

group reducible to U(n) and one prescribes a class x E H 2 (M;R) such that x n is

compatible with the complex orientation. Given the early work of Gromov [GI] on

geometric structures on open manifolds, the problem is to find obstructions to ex- '

tending a symplectic structure defined near the boundary 8D 2 n over the whole ball.

This shows the importance of what Eliashberg has called ''fillable'' structures; some

of the difficulties which arise are illustrated as follows. Let E be a 2-dimensional sur-

face embedded in the three-dimensional boundary of an almost-complex manifold

M 4 , and consider the natural foliations of the 3-dimensional cylinder Z = D 2 X [0, 1]

(or 8 3 minus two poles) by holomorphic discs. If E is diffeomorphic to 8Z or 8D 3 ,

and if the embedding of E in 8M can be extended to an embedding of Z or D 3 in

5

Introduction

A.J 4 which is holomorphic on the leaves of the foliation, then we say that E is fillable

I)y holomorphic discs. Under certain conditions, explained below in the article of

11˜liashberg, these extensions exist and this can be used, for example, to provide a

lu'cessary condition (not "overtwisted") for a contact structure on 8M to bound a

!Iylllplectic structure on M.

W(, conclude this introduction with a few remarks about contact manifolds. Here

lH one reason for believing that contact manifolds may be "softer" than symplec-

tiC": if Vi and V2 are contact, then their connected sum admits a contact structure

"˜J'(˜cing with the original forms on VI, V2 outside small discs. This cannot oc-

('III" for symplectic manifolds-the basic difference being that the odd-dimensional

,'pltcre S2n+l admits a contact form a such that da is the pull-back of the standard

nylllplectic form on cpn. It follows that O-surgeries can be performed on contact

IlIH.uifolds, and 2n surgeries seem to fit into the same framework. The situation for

:˜u + 1 surgeries is more delicate--at least under certain conditions a I-surgery can

I.., (Â°a.rried out, as Thurston and Winkelnkemper sho\ved in dimension 3. And by

tHiiIJp; a rag-bag of special tricks it is possible to prove the soft realisation conjecture,

ill the contact case, for a large class of n - I-connected 2n + I-manifolds. Thus at

I,ll(' time of writing the situation is tantalisingly similar to that for codimension-I

I'c)lint.ions before the major contribution of Thurston; see [T] for a summary of what

WItS known a few years back.

AN li˜liashberg's paper shows, much work has been done in dimension 3, stimulated

111)t ()nly by Gromov but also by the work of Bennequin. It would be very interesting

t ˜ S('C if contact geometry can be applied to classification problems in 3-dimensional

I

topology. For example, A. Weinstein conjectures that if HI (V 3 ; Z) is finite (and in

I'll rt.icular if V 3 has universal cover diffeomorphic to S3), then the characteristic

foliat.ion of an s-fillable contact form contains at least one closed orbit. Under what

,Â·,,"dit.ions is it possible to use the existence of such orbits to construct a Seifert

Ii I.. "ring of V? Such a manifold would then be elliptic, completing part of the

˜J.' '. )1) letrisation programme.

1(: 1] Gromov, M. Partial differential relations Springer Berlin-Heidelberg (1986)

1< ;2] Gromov, M. Pseudoholomorphic curves in symplectic manifolds Inventiones

˜1"tl1. 82 (1985) 307-347

1< ;:˜] Gromov, M. Soft and hard Symplectic Geometry Proc. Int. Congress Math-

.<tllilticians, Berkeley 1986, Vol. I, 81-98

1'1'1 Thomas, C.B. Contact structures on (n - I)-connected (2n + I)-manifolds

IIIIIIHell Centre Publications 18 (1986) 254-270

7

I{ational and Ruled Symplectic 4-Manifolds

I.>USA McDUFF<*)

State University of New York at Stony Brook

I. INTRODUCTION

I'his note describes the structure of compact symplectic 4-manifolds (V, w) which contain

sYlnplectically embedded copy C of S2 with non-negative self-intersection number.

n

(Such curves C are called "rational curves" by Gromov: see [G].) It turns out that there is

II concept of minimality for symplectic 4-manifolds which mimics that for complex

"lIrl˜lces. Further, a minimal manifold (V, (0) which contains a rational curve C is either

NYlnplectomorphic to CP2 with its usual Kahler structure T, or is the total space of a

ttPliynlplectic ruled surface" i.e. an S2-bundle over a Riemann surface M, with a symplectic

f. Â»nn which is non-degenerate on the fibers. It follows that if a (possibly non-minimal)

(V,w) containsamtionalculVe C with CÂ·C> 0, then (V,w) maybe blown down either

tu S2 x S2 with a product form or to (CP2, T), and hence is birationally equivalent to

G' P 2 in Guillemin and Sternberg's sense: see [GS]. (In analogy with the complex case,

we will call such manifolds rationaL) Moreover, if V contains a rational curve C with

( '. (' = 0, then V may be blown down to a symplectic ruled surface. Thus, symplectic 4-

lIutnifolds which contain rational curves of non-negative self-intersection behave very much

It kl- rational or ruled complex surfaces.

It Is natural to ask about the uniqueness of the symplectic structure on the manifolds under

â€¢ nnsideration: more precisely, if 000 and WI are cohomologous symplectic forms on V

which both admit rational curves of non-negative self-intersection, are they

IIYlnplectomorphic? We will see below that the answer is "yes" if the manifolds in

.,u("stion are minimal. In the general case, the most that is known at present is that any two

-.. rh forms may be joined by a familyÂ· Wt, 0 s t s 1, of (possibly non-cohomologous)

"vlllpicctic forms on V. Since the cohomology class varies here, this does not imply that

Ihc' fonns WQ and WI are symplectomorphic: cf[McD 1]. Similarly, all the symplectic

partially supported by NSF grant no: DMS 8803056

( +)

I tlKO Mathematics Subject Classification (revised 1985): 53 C 15, 57 R 99

˜fV words: symplectic manifold, 4-manifolds, pseudo-holomorphic curves, almost

...... plcx manifold, blowing up.

8 McDuff: Rational and ruled symplectic 4..manifolds

fonns under consideration are Kihler for some integrable complex structure J on V,

provided that V is minimal. In the general case, we know only that w may be joined to a

KIDder fonn by a family as above. Note also that there might be some completely different

symplectic forms on these manifolds which do not admit rational curves.

The present work was inspired by Gromov's result in [G] that if (V,w) is a compact

symplectic 4-manifold whose second homology group is generated by a symplectically

embedded 2-sphere of self-intersection +1, then V is Cp2 with its usual Kahler

stmcture. Our proofs rely heavily on his theory ofpseudo-holomorphic curves. The main

innovation is a homological version of the adjunction fonnula which is valid for almost

complex 4-manifolds. (See Proposition 2.9 below.) This gives a homological criterion

for a pseudo-holomorphic curve in an almost-complex 4-manifold to be embedded, and is a

powerful mechanism for relating the homological properties of a symplectic manifold V to

the geometry of its pseudo-holomorphic curves. We also use some new cutting and

pasting techniques to reduce the ruled case to the rational case.

Proofs of the results stated here appear in [McD 3,4]. I wish to thank Ya. Eliashberg for

many stimulating discussions about the questions studied here. I am also grateful to MRSI

for its hospitality and support during the initial stages of this work.

2. STATEMENT OF RESULTS

We will begin by discussing blowing up and blowing down. All manifolds considered will

be smooth, compact and, unless specific mention is made to the contrary, without

boundary.

By analogy with the theory of complex surfaces, we will say that (V,w) is minimal if it

contains no exceptional CUNes, that is, symplectically embedded 2-spheres I: with self-

r.. r. = -1.

intersection number We showed in [McD 2] Lemma 2.1 that every exceptional

curve E has a neighbourhood N E whose boundary (ONE' (0) may be identified with the

boundary (OB4(A + e), (00) of the ball of radius A + e in CP2, where 11"A2 = w(E) and

e > 0 is sufficiently small. Hence E can be blown down by cutting out NE and gluing in

the ball B4(A + e), with its standard form 000. It is easy to check that the resulting

manifold is independent of the choice of f" so that there is a well-defined blowing down

operation, which is inverse to symplectic blowing up.

The following result is not hard to prove: its main point is that one blowing down operation

suffices.

9

McDuff: Rational and ruled symplectic 4-manifolds

2.1 Theorem

I!'vcry symplectic 4-manifold(V,00) covets a minimal symplectic manifold(V', (a)') which

'lilly be obtained from V by blowing down a finite collection of disjoint exceptional

('lIlves.. Moreover, the induced symplectic form 00' on V' is unique up to isotopy..

There is also a version of Theorem 1 for manifold pairs (V, C) where C is a

Nynlplectically embedded compact 2-manifold in V. We will call such a pair minimal if

V . C contains no exceptional curves.

2.2 'rheorem

I,'very symplectic pair (V, C, (0) covers a minimal symplectic pair (V', C, 00') which

"'/I.Y be obtained by blowing down a finite collection ofdisjoint exceptional CULVes in V-C.

Alol"cover, the induced symplectic form 00' on V' is unique up to isotopy (reI C).

J..J Note

If (., is a closed subset of V, two symplectic fonns 000 and 001 are said to be isotopic

(1"("1 <:) if they can be joined by a family of cohomologous symplectic forms whose

I rst rictions to C are all equal. If C is symplectic, Moser's theorem then implies that there

IN un isotopy g t of V which is the identity on C and is such that gl *((01) = 000.

It Is well-known that the diffeomorphism type of V' is not uniquely detennined by that of

V. For example, because (S2 x 8 2) #Cp2 is diffeomorphic to CP 2 # CP2 # CP2, the

.nunifold V = (S2 x 8 2 ) # CP2 may be reduced to Cp2 as well as to S2 x S2.

Ilowcver,this is essentially the only ambiguity, and V'is detennined up to diffeomorphism

If we fix the homology classes of the curves which are blown down.

( 'onvcrsely, one can ask to what extent the minimal manifold (V',w') determines its

hlowing up (V, (0). Since each exceptional curve L in (V, (0) corresponds to an

r.uhcdded ball in V' of radius A, where w(L) = ll'A2, this question is related to properties

li B(Ai) is the

tile space of symplectic embeddings of llB(Ai) into (V', 00'), where

tt'-

.U!ijoint union of the symplectic 4-balls B()q) of radius Ai. We discussed the

â€¢ Â«.r responding question for manifold pairs in [McD 2]. We showed there that, if CÂ·C = 1,

fUld if V is diffeomorphic to CP2 with k points blown up, there is a unique symplectic

IlI.lIcture on (V, C) in the cohomology class a if and only if the space of symplectic

'-UlhtÂ·ddings of li B(Ai) into Cp2 - CP 1 is connected, where 1l'A1 2, .â€¢ ˜ ,ll'Ak2 are the

vuhJt˜s of a on the exceptional curves in V. Unfortunately nothing is known about this

'1pUC:tÂ· of embeddings per see In fact, the information we have goes the other way: we

proved in [McD 2] that the structure on (V, C) is unique when k = 1, which implies that

10 McDuff: Rational and ruled symplectic 4-manifolds

the corresponding space of embeddings is connected. (Because CP 2 # CP2 is ruled, this

uniqueness statement is closely related to the results in Theorem 2.4 below.) It is not clear

what happens when k ˜ 2. However, because any two embeddings of li B()\j) into

(V',oo') are isotopic when restricted to the union llB(Ei) of suitably small subballs, any

V which blow-down to

two forms on may be joined by a family of non-

(a)'

cohomologous forms.

Thus, the problem is essentially reduced to understanding the minimal case. The next

ingredient is a result on the structure of symplectic 5 2-bundles (symplectic ruled surfaces).

2.4 Theorem Let V be an oriented5 2-bundle V -+ M over a COllJpact oriented

1Â£:

surface M with fiber F.

(i) The cohomology class a ofany symplectic [onn on V which is non-degenerate on

each fiber of 1t' satisfies the conditions:

(a) a(F) and a2(V) are positive, and

(b) a 2( V) > (a(F))2 ifthe bundle is non-trivial.

(ii) Any cohomology class a E H 2(V; Z) which satisfies the above conditions may be

represented by a symplectic (onn tL) which is non-degenerate on each fiber of 1T.

Moreover, this {ann is unique up to isotopy.

The existence statement in (ii) above is well-known. It is obvious if the bundle is trivial. If

it is non-trivial, one can think of V as the suspension of a circle bundle of Euler class I

with the corresponding Sl-action, and can then provide V with an invariant symplectic

form in any class a which satisfies (i) (a,b) since these conditions correspond to requiring

that a be positive on each of the two fixed point sets of the 5 I-action. (See [Au].) The

other statements are more delicate. Consider first the case when the base manifold M is

52. Gr˜mov showed in [G] that any symplectic fonn on 52 x 52 which admits

symplecticallyembedded spheres in the classes [52 x pt] and [pt x 52] is isotopic to a

product (or split) form. (In fact, Gromov assumed that the form has equal integrals over the

two spheres, but it is not hard to remove this condition.) A corresponding uniqueness

result when V = CP2 # CP2 (which is the non-trivial SLbundle over 52) was proved in

[McD 2J. Here one requires the existence ofjust one symplectically embedded sphere, but

it must be in the class of a section of self-intersection 1, not of the fiber. Gromov showed

that this hypothesis also implies that there must be a symplectically embedded sphere in the

class of the blown-up point, which is equivalent to condition (i)(b). In the present

situation, we have less infonnation since we start with only one symplectically embedded

sphere, the fiber. Following an idea of ˜liashberg's,we can construct a symplectic section

of 1T (i.e. a section on which the symplectic form 00 does not vanish) and so reduce to the

previously considered case. In the process, we have t˜ change the form w by adding

11'*(0) where 0 is a 2-fooo on M such that o(M) > o. The argument is then completed

by the following lemma, the proof of which uses the theory of hoiomorphic curves.

11

McDuff: Rational and ruled symplectic 4-manifolds

1.S Lemma

I,et V be an S2-bundJe over a Riemann surface M and suppose that oot, 0 s t s 1, is a

lill"ily of(non-cohomologous) symplectic forms on V which are non-degenerate on one

I1blÂ·( of V. Then, if 001 admits a symplectic section, so does 000.

II' (he general case, one cuts the fibration open over the I-skeleton of M in order to reduce

10 (he case M = S2.

We can now state the classification theorem for minimal symplectic pairs.

Theorem

J..()

I,et (V, C, (0) be a minimal symplectic4-dimensional pair where C is a 2-sphere with

,.rU:inlersection C.C=p˜Â·O. Then (V, (0) issymplectomorphiceitherto CP2 ortoa

lfr,"plectic S2-bundle over a compact surface M. Further, this symplectomolphism may

,.t' chosen so that it takes C either to a complex line or quadric in CP2, or to a fiber ofthe

S.' 1>ulldJe, or (if M is S2) to a section ofthis bundle.

I hus, jf p is odd and ˜ 3, (V, (0) is symplectomorphic to the Kahler manifold Cp2 # CP2;

II P == 1, (V, (0) is Cp2 with its standard Kahler form; if p is even and ˜ 2, (V, (0)

I,. the product S2 x S2 with a product symplectic form (or, if p = 4, it could be CP 2);

Iliul. if p = 0, (V, (0) is a symplectic S2-bundle. From this, it is easy to prove:

(˜orollary

J ../

II' (V, C, (0) is as above, the diffeomorphism type ofthe pair (V, C) is detennined by

( I)

II ,.rovided that p :1=. 0, 4.

When p 4, there are two possibilities for (V, C): it can be either (CP 2, Q) or

(II) =

(SJ x S2, [2), where Q is a quadric and where [2 is the graph ofa holomorphic self-

""'1' of S2 ofdegree 2. When p = 0, C is a fiber ofa symplectic S2-bundle.

by the cohomology class of

(Ill) (V) C, (0) is determined up to symplectomolphism 00.

J. H <:orollary

, '" illilllaJ symplectic 4-manifold (V,w) which contains a rational curve C with CÂ· C > 0

i˜ .. "flip/ectomorphic either to CP 2 or to S2 xS 2 with the standard form.

Ill(' Illain tool in the proof of Theorem 2.6 is the following version of the adjunction

'ullllula. We will suppose that J is an almost complex structure on V with first Chern

.I,I.\S cl,andthat f:S2˜ V isaJ-holomorphicmap (ie dfoJo=Jodf,where Jo is

tluÂ· usual almost complex structure on S2) which represents the homology class A E

12 McDuff: Rational and ruled symplectic 4-manifolds

H2(V; I). The assumption that f is somewhere injective lUles out the multiply-covered

case, and implies that f is an embedding except for a fmite number ofmultiple points and a

finite number of "critical points" , i.e. points where dfz vanishes.

2.9 Proposition

If f is somewhere injective, then

AÂ·A˜Cl(A)-2

with equality ifand only if f is an embedding.

This is well-known if J is integrable: the quantity 1/2 (A. A - cl(A) + 2) is known as

the "virtual genus" of the curve C = 1m f. It is also easy to prove if f is an immersion.

For in this case cl(A) = 2 + Cl(VC) where vc is the nonnal bundle to C, and Cl(VC) S

AÂ·A, with equality ifand only if f is an embedding. In the general case, one has to show

that each singularity of C contributes positively to AÂ·A. This is not hard to show for the

simplest kind of singularities, and, using the techniques of [NW], one can reduce to these

by a rather delicate perturbation argument.

With this in hand, we prove Theorem 2.6 by showing that V must contain an embedded

I-simple curve of self-intersection + 1 or o. (J-simple curves do not decompose, so that

their moduli space is compact.) It then follows by arguments of Gromov that V is CP 2

in the former case and a symplectic S2-bundle in the latter.

2.10 Note

Given an arbitrary symplectic 4-manifold one can always blow up some points to create a

manifold (W, 00) which contains a symplectically embedded 2-sphere with an arbitrary

negative self-intersection number. Hence, the existence of such a 2-sphere gives no

infonnation on the structure of (W, w).

Corollary 2.7 may be understood as a statement about the uniqueness of symplectic fillings

of certain contact manifolds. Indeed, consider an oriented (2n-l)-dimensional manifold Ii

with closed 2-fonn o. We will say that (A,o) has contact ˜ if there is a positively

oriented contact form a on !J. such that da = o. It is easy to check that the contact

structure thus defined is independent of the choice of a. Following Eliashberg [E], we

say that the symplectic manifold (Z,oo) fills (Ii,o) if there is a diffeomorphism f: oZ -+

A such that f*(o) = wloZ. Further the filling (Z, (0) is said to be minimal if Z contains

no exceptional curves in its interior.

As Eliashberg points out, information on symplectic fillings provides a way to distinguish

between contact structures: if one constructs a filling of (A, 02) which does not have a

McDuff: Rational and ruled symplectic 4-manifolds 13

certain property which one knows must be possessed by all fillings of (a,ol), then the

contact structures on a defined by 01 and 02 must be different. In particular, it is

interesting to look for manifolds of contact type which have unique minimal fillings.

()bvious candidates are the lens spaces Lp , p > 1, which are obtained as the quotients of

rp

S3 c C 2 by the standard diagonal action of the cyclic subgroup c SI of order p on

4˜2, and whose 2-form 0 is induced by 000.

It is not hard to see that if (Z,oo) fills (Lp, a) we may quotient out az = Lp by the Hopf

Inap to obtain a rational curve Cp with self-intersection p in a symplectic manifold (V,0)

without boundary. Hence Corollary 2.7 implies:

2.11 Theorem

The lens spaces Lp, p ˜ 1, all have minimal symplectic fillings. If p :/=. 4, minimal fillings

(Z,oo) of (Lp,a) are unique up to diffeomorphism, and up to symplectomorphism ifone

lixes the cohomology class [00]. However, (L4, 0) has exactly two non-diffeomorphic

11linimaJ fillings.

In higher dimensions, one cannot hope for such precise results. However, in dimension 6

there are certain contact-type manifolds (such as the standard contact sphere S5) which

impose conditions on any filling (Z,oo), even though they may not dictate the

diffeomorphism type of minimal fillings. In dimensions> 6, one must restrict to "semi-

positive" fillings to get analogous results. See [McD 5]. .

References

[Au] Audin, M. : Hamiltoniens periodiques sur les varietes symplectiques compactes de

dimension 4, Preprint IRMA Strasbourg, 1988.

[E] Eliashberg, Ya.: On symplectic manifolds which are bounded by standard contact

spheres, and exotic contact structures of dimension> 3, preprint, MSRI, Oct. 1988.

[G) Gromov, M.: Pseudo-holomorphic curves in symplectic manifolds, Invent. Math.

82 , 307-347 (1985)

[GS] Guillemin, V. and Sternberg, S.: Birational Equivalence in the symplectic category,

preprint 1988

McDuff, D.: Examples of symplectic structures, Invent. Math 89, 13-36

[McD 1]

( 1987).

McDuff: Rational and ruled symplectic 4-manifolds

14

IMcD 2] McDuff, D.: Blowing up and symplectic embeddings in dimension 4, to appear

in To.pology (1989190).

[McD 3] McDuff, D.: The structure of rational and ruled symplectic 4-manifolds,

preprint, Stony Brook, 1989.

[McD 4] McDuff, D. : The local behaviour of holom0rphic CUlVes in almost complex 4-

manifolds, preprint, Stony Brook, 1989.

[McD 5] McDuff, D.: Symplectic manifolds with contact-type boundaries, in prepamtion,

1989.

[NW] Nijenhuis, A. and Woolf, W.: Some integration problems in almost-complex and

complex manifolds, Ann. ofMath. 77 (1963), 424 - 489.

Symplectic Capacities

H.HOFER

FB Mathematik

Ruhr Universitat Bochum

Federal Republic of Germany

August 3, 1990

Introduction

1

In 1985 M. Gromov proved in his seminal paper Pseudoholomorphic Curves in

Symplectic Geometry, [1], a striking rigidity result, the so-called squeezing theorem.

=

Consider the symplectic vectorspace R 2 n := (R2 )n with the coordinates z

(Zl' â€¢.. ,Zn),Zi = (Xi,Yi), and the symplectic form (j, defined by

n

= L:XiY˜ - X˜Yi.

u(z,z')

'. i=l

Denote by B2n(r) the Euclidean r-ball in R2n and by z2n(r) the symplectic cylinder

defined by

= {z E R 2n II Zl 1< r}.

z2n(r)

Gromov proved, that given a symplectic embedding

we necessarily have the inequality r :5 r'. His proof of this fact relied on his existence

theory for pseudoholomorphic curves, [1].

Coming from the variational theory of Hamiltonian Dynamics, I. Ekeland and

the author observed in [2,3], that using Hamiltonian dynamics the squeezing theo-

rem not only can be proved as well, but in fact many interesting symplectic invari-

ants, so-called Symplectic Capacities can be constructed.

Hofer: Symplectic capacities

16

In this way a fruitful hook up between Symplectic Geometry and Hamiltonian

Dynamics - with their different methods - has been achieved. This merger for

example allows to see the nonobvious but very deep relationship between certain

aspects of Hamiltonian Dynamics and Symplectic rigidity. The aim of this paper is

to survey this relationship. We start with an axiomatic approach, then we survey

several constructions of a symplectic capacity.

Acknowledgement

I would like to thank 1. Ekeland for many stimulating discussions.

AxioIns and Consequences

2

Consider the category S ymp2n., consisting of all (2n )-dimensional symplectic man-

ifolds, together with the morphisms being the symplectic embeddings. We denote

by S a subcategory of S ymp2n. We do not (!) assume that S is a full subcategory.

The following scaling axiom is imposed:

If (M,w) E 8 then also (M,OlW) E S for a E R \ {O}

(8)

Denote by r = (0, +00) U {+oo} the extended positive half line. The obvious

ordering on r is denoted by ˜. In the usual way we consider (r, S) as a category.

r

˜

Definition 2.1 A symplectic capacity for 8 is a covariant functor c : S satis..

fying the axioms

C(Z2n(1),u) < +00

(N)

c(M,aw) =1 a Ie (M,w).

(C)

Here (N) stands for nontriviality and (C) for conformality.

Several remarks are in order.

Remark 2.2 1. The map (M,w) --. (fwn)'k sati8fie8 (C) but not (N) ifn 2:: 2.

M .

So if n ˜ 2 the nontriviality axiom excludes volume-related invari˜nts. However, if

n = 1, then (M,w) --. vol(M,w) is a symplectic capacity.

2. Besides S = S ymp2n there are several other interesting subcategories. FOT

example let (V, w) be a symplectic manifold. Denote by Op (V, w) the collection of

all symplectic manifolds of the form (U, OlW I U) where U is a nonempty open subset

of V and a E R \ {OJ. As morphisms we take the symplectic embeddings. Clearly

S = Op (V,w) is an admissible category. Next assume G is a compact Lie group.

17

Hofer: Symplectic capacities

l)(˜noteby Sa the collection of all smooth (2n )-dimensional symplectic G-spaces,

'where G acts Jymplectically. As morphisms we take the symplectic G-embeddings.

3. One might look for functors c : S ˜ r which transform differently. For

t˜xample c (M,aw) =1 a Ik c (M,w) for some k f:.1. To exclude certain pathologies

Let k E {l, ... ,n}. A symplectic

tine could impose the following axiom system.

r

˜˜.capacity is a covariant functor c : S -+ satisfying

(C)k c(M,aw)=lalkc(M,w), aiO

c(B2(k-l)(1) X R 2(n-k+l),u) = +00

(N)k { C (B2k(1) X R 2(n-k),u) < +00

We note that for k = 1 we precisely recover our definition of a symplectic capacity.

[n fact the first part of (N)! follows from (C)l and the fact that c (B2n(1), 0') > o.

l˜or k = n the volume map is an example of a n-capacity. So far no examples

n.re known for intermediate capacities (1 < k < n). It is quite possible that they

<10 not exist. Some evidence for this possibility are given by the fact that there

iH an enormous amount of flexibility for symplectic embeddings M C-...+ N with

climM ˜ dimN + 2, see Gromov's marvellous book [4]. For example, the existence

()f a symplectic embedding

B 2 (e) x R 2n - 2 ˜ B 2n - 2 (1) R2

X

for some e > 0, would immediately contradict the axioms for a k-capacity 1 <

k < n. As a consequence of results in [3J, concerning symplectic capacities, the

inequality Â£-2 ˜ n -1 is a necessary condition for the existence of such a symplectic

(\lnbedding.

= S ymp2n

Next we derive some easy consequences of the axioms. Set S and

assume c : S ˜ r is a symplectic capacity.

Lemma 2.3 Assume there ezists a symplectic embedding 'I1 : Z2n(rl) ˜ Z2n(r2).

Then rl ˜ r2.

First we note that the map

PROOF:

is a symplectic diffeomorphism. Hence

18 Hofer: Symplectic capacities

Here T E (0,+00) as a consequence of the axioms. If now W : z2n(rl) ˜ Z2n(r2)

we obtain, using the axioms, Tr˜ $ rri.

An even stronger result is

Lemma 2.4 There exists a constant R o E (0, +00) such that there is no symplectic '

embedding

If B 2n(R) embeds symplectica11y into z2n(1) we obtain

PROOF:

Hence .,˜

˜1

˜

.˜

â€¢ .˜˜

i˜˜

Remark 2.5 Gromov's Squeezing Theorem implies in fact, that there is a capacity

.˜

cw˜ 1

˜

2n

(B (1)).

(Z2n(1) =

C C

1

Hence R o = 1 is the best possible choice. .˜

Â·i

.˜

So far we have only studied trivial consequences. The following corollaries are

l

substantially deeper. We call a subset E C R 2n an ellipsoid if there exists a positive

definite quadratic form q such that E = {z E R 2 n I q(z) < I}. Assume c is a

I

R2n

capacity defined on S = Op(R2n, u). We extend c to a map 2 X (R \ {O}) -+

[0, +00] by defining

I V-:;U,

(1) c(U,a) = inf{c(V,au) (V,au)ES} .,˜

::˜

2

Let us also write c (U) := c (U, 1) for U C R It has been proved in [2,3]: ..˜

n.

':!J'

..˜

2n 2n

Proposition 2.6 Let T : R R be a linear map_ Assume

-+

c (E) = c (T(E)

for every ellipsoid. Then T is symplectic or antisymplectic, i. e.

or

T*wo = Wo T*wo = -woo

19

Hofer: Symplectic capacities

(B271(1),0') = (Z2n(1),0') =

III [2,3] the normalisation had been assumed

c 1r

C

throughout the paper. However, ii had not been used for the above proposition,

whieh only depends on (N) and (C). By the definition (1) we have meanwhile

obt,ained a map still denoted by c:

that for every q, E Diff (R2 n, 0'), t E R:

Nuch

c (q,(S) = c (S)

c (tS) = t 2 c (8)

(2)

{ c (S) :5 c (T) if SeT

(Z2n(1)) <

D< C (B2n(1Â») :5 C +00

w(˜ call this map again a symplectic capacity. Next we obtain the following local

O

rigidity result, see [3] or [5].

(,f

'I'heorem 2.7 (Local Rigidity) Assume epk : B 2 n(1) is a sequence of

-+ R2n

,'tyrnplectic embeddings, converging uniformly to a continuous map q> : B 2n (1) -+

It 2n , which is differentiable at 0 E B 2 n(1) with derivative tP'(O). Then Â«p/(D) is

Â·Â·1I1n plectic.

Using this theorem we recover the famous Eliashberg - Gromov rigidity result,

r˜,G].

(˜orollary 2.8 Let (M,w) be a symplectic manifold. Then Diff w(M) is CO-closed

in Diff(M).

The local rigidity result can be proved only using (2). One shows that <p'(O) :

Il.˜n -+ R 2 n and ep'(O) X Id R 2 : R 2 n+2 -+ R2n+2 preserve capacities CR2n and CR2n+2

f(.1' ellipsoids. The linear algebra proposition then implies the desired result.

So far we surveyed results which can be obtained from the existence of a ca-

I)lu˜ity. The following results are different. Their proofs depend on capacities with

,ulclitional features.

:1 Some Special SYIIlplectic Capacities

I )isplacenlent Energy

The first capacity we consider here is very closely related to Hamiltonian Dy-

IIlUllics. Let us introduce some notation. We denote by C the vector space of all

tHJ)()oth Hamiltonians H : [0,1] X R 2n -+ R having compact support. Denote by

Hofer: Symplectic capacities

20

X Ht the time-dependent Hamiltonian vector field associated to H by the formula

dHt(x)(*) = u(XHt(x), *). Denote by WH the time-I-map associated to the Hamil-

tonian System

II * II on C by

We define a norm

IIHII = Co'1˜':t2" H) - CO.l˜t1R2n H) Â·

c.

By V we denote the collection of all 'ItH, H E One easily verifies that V is a

group.

Let S be a bounded subset of R 2n. We study the importance of energy by

analyzing the following problem: How much energy do I need to deform S into a

set S' which is disjoint from 8? To make this precise we define the displacement

energy d(S) by

deS) = inf {IIHIII WH(S) n S = <p}.

For an unbounded set T c R 2n let

= sup {deS) I SeT, S bounded}.

d(T)

It is clear that d(˜(T)) = d(T) for all ˜ E V and T C R2n. What is not clear is

that d(T) :I O. In order to stress this point consider S = B2(1) X B2n-2(R), R 2: 1.

To disjoin S from itself it is enough to disjoin B2(1) C R 2 from itself. Now B2(1) is

via an element in V symplectomorphic to an almost square with round corners and

sidelength approximately Vi. By taking a translation parallel to one side, which

Vi and

disjoints the square from itself we need a Hamiltonian which has slope f'V

is increasing along lines orthogonal to the above side. By cutting this Hamiltonian .˜

off smoothly we see that we can separate the square from itself by a Hamiltonian ,I

I

H with II HII'" Jr. Hence, a simple corollary:

1

B 2n - 2 (R)) ˜

d(B 2 (1) VR > 0

X 7r

1

R 2n - 2 )

D(B 2 (1) X ::; Jr.

˜

Implementing the above construction for a disjoint union of small cubes via a suit-

able highly oscillatory Hamiltonian we can prove the following.

Proposition 3.1 Assume U and V are open bounded sets in R 2n with smooth

boundary and equal volume. Given e > 0 there exists He E C (With IIHÂ£II < e and a

subset MÂ£ C U with meas (Me) ˜ e such that '11 := '11 HÂ£ satisfies

So with other words we can displace a set S from itself modulo sets of small measure

with arbitrary small energy. However, the surprising fact is

21

Hofer: Symplectic capacities

'I'heorem 3.2

In particular the above construction is the best one can do! Clearly, the map d

("Ul be used to define a capacity c : Op(R2n, u) --+ r such that in addition to the

n.xioms (N) and (C) we have

c(B 2n (1),u) = c(Z2n(1),u) = 'Fr.

'I'his capacity for example implies Gromov's Squeezing Theorem. It also has some

vt˜ry interesting consequences for the topological group V. Define the energy of a

IUn.p W E V by

= in! {I(HIII WH = 'II}

E :V R: E(w)

--+

easily verifies, [7], that

()uc

E(w) + E(˜).

E(w˜) ˜

i= ide

AHHume now \11 Then

> O. Hence

E R 2n and e

for some point Xo

ñòð. 1 |