ñòð. 1 |

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

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

A local spectral theory for closed operators, L ERDELYI & WANG SHENGWANG

105

Syzygies, E.G. EVANS & P. GRIFFITH

106

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

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

Diophantine Analysis, 1. LOXTON & A. VAN DER POORTEN (eds)

109

An introduction to surreal numbers, H. GONSHOR

110

111 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

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

114

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

115

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

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

117

Skew linear groups, M. SHIRVANI & B. WEHRFRITZ

118

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

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

121

Non-classical continuum mechanics, R.J. KNOPS & A.A. LACEY (oos)

122

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

124

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

125

Van der Corpufs method for exponential sums, S.W. GRAHAM & O. KOLESNIK

126

New directions in dynamical systems, T.1. BEDFORD & J.W. SWIFf (eds)

127

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

128

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

129

130 Model theory and modules, M. PREST

Algebraic, e˜tremal & metric combinatorics, M-M. DEZA, P. FRANKL & LO. ROSENBERG (eds)

131

132 Whitehead groups of fmite groups, ROBERT OLIVER

133 Linear algebraic monoids, MOHAN S. PUTCHA

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

134

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

135

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

136

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

137

Analysis at Urbana, II, E. BERKSON, T. PECK, & 1. UHL (eds)

138

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

139

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

Surveys in combinatorics 1989, J. SIEMONS (ed)

141

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

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

144 Introduction to unifonn spaces, I.M. JAMES

145 Homological questions in local algebra, JAN R. STROOKER

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

Continuous and discrete modules, S.H. MOHAMED & BJ. MOLLER

147

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

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

149

CLARKSON

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

150

Geometry of lOW-dimensional manifolds 2, S.K.:DONALDSON & C.B. THOMAS (eds)

151

152 Oligomorphic permutation groups, P.J. CAMERON

L-functions in Arithmetic, J. COATES & M.J. TAYLOR

153

Number theory and cryptography, J. LOXTON (ed)

154

155 Classification theories of polarized varieties, TAKAO FUJITA

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

156

I Atndnn

Geometry of Low-dimensional

Manifolds

1: Gauge Theory and Algebraic Surfaces

I)roceedings of the Durham Symposium, July 1989

I!dited by

S. K. Donaldson

Mtlthematicallnstitute, University o/Oxford

(˜.Il. Thomas

IJepartment o/Pure Mathematics and Mathematical Statistics,

llnlversity o/Cambridge

The righl 0/ ,he

Uni.,ersily of Combrit!ge

10 print and sell

(1/1 nl4InRtr oj books

WQS g'CUlIed by

Henry V/If in UJ4.

The Un/llers;,)' has prinled

llnd published ronlimlOusly

since /j84.

CAMBRIIX3E UNIVERSITY PRESS

Cambridge

New Yark Port Chester Melbourne Sydney

Published by the Press Syndicate of the University of Cambridge

The Pitt Building, Trompington 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 p,ublication data available

ISBN 0 521 39978 5

CONTENTS

vi

( Â·c'Illents of Volume 2

( 'c.ntributors vii

viii

Nnlnes of Participants

IUlroduction ix

xiv

Acknowledgments

1

JlAKrr 1: FOUR-MANIFOlDS AND ALGEBRAIC SURFACES

I

5

! Ynng-Mills invariants of four-manifolds

SI K. Donaldson

( )n the topology of algebraic surfaces 41

l˜c)hCI1 E. Gompf

'Ithe topology of algebraic surfaces with q = Pg = 0 55

I)Icler Kotsehick

( )n the homeomorphism classification of smooth knotted surfaces in the 4-sphere 63

Mnllhias Kreck

It'lnt algebraic manifolds 73

II,A.E. Johnson

ItAI˜T 2: FLOER'S INSTANTON HOMOLOGY GROUPS 93

97

I..stunton homology, surgery and knots

Andreas Floer

hununton hontology 115

Andreas Floer, notes by Dieter Kotschick

125

Invariants for homology 3-spheres

I˜()nuld Fintushel and Ronald J. Stem

()n the Floer homology of Seifert fibered homology 3-spheres 149

( ˜hristian Okonek

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

Mikio Furuta

175

PARrr 3: DIFFERENTIAL GEO:METRY AND MATHEMATICAL PHYSICS

Skynne fields and instantons 179

N.S. Manton

I˜cpresentations of braid groups and operators coupled to monopoles :' '˜ 191.

.˜

I(nlph E. Cohen and John D.S. Jones . 0 0

207

I ˜xlremal immersions and the extended frame bundle

1>.11. Hartley and R.W. Tucker

Minimal surfaces in quatemionic symmetric spaces 231

I", E. Burstall

237

"three-dimensional Einstein-Weyl geometry

K.P. Too

247

Ilannonic Morphisms, confonnal foliations and Seifert fibre spaces

John C. Wood

CONTENTS OF VOLUME 2

Contents of Volume 1

Contributors

vi˜

Names of Participants

I

Introduction XJ

xi˜

Acknowledgements

1

PART 1: SYMPLECIlCGEOMElRY

Introduction

1

Rational and ruled symplectic 4-manifolds

DusaMcDuff

1

Symplectic capacities

H. Hofer

The nonlinear Maslov index 3

A.B. Givental

4

Filling by holomorphic discs and its applications

Yakov Eliashberg

PART 2: JONES/WITfEN THEORY

Introduction

New results in Chern-Simons theory

Edward Witten, notes by Lisa Jeffrey

Geometric quantization of spaces of connections

N.J. Hitehin

Evaluations of the 3-manifold invariants of Witten and

Reshetikhin-Turaev for sl(2, C) 10

Robion Kirby and Paul Melvin

11

Representations of braid groups

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

PART 3: THREE-DIMENSIONAL MANIFOlDS 12

Introduction 1

An introduction to polyhedral metrics of non-positive curvature on 3-manifolds 1

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

Finite groups of hyperbolic isometries 1

C.B. Thomas

1:

Pin structures on low-dimensional manifolds

R.C. Kirby and L.R. Taylor

CONTRIBUTORS

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

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

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

I<alph E. Cohen, Department of Mathematics, Stanford University, Stanford CA 94305, USA

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

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

1< t ulald Fintushel, Department of Mathematics, Michigan State University, East Lansing,

MI 48824, USA

1\. 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 S1. Giles, Oxford OXI 3LB, UK

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

I˜. )hert E. Gompf, Department of Mathematics, University of Texas, Austin TX, USA

I), II. Hartley, Department of Physics, University of Lanc'aster, Lancaster, UK

N. .I. Hitchin, Mathematical Institute, 24-29 S1. Giles, Oxford OXI 3LB, UK

II. Ilofer, PB Mathematik, Ruhr Universitat Bochum, Universitatstr. 150, D-463 Bochum, FRG

I I"a Jeffrey, Mathematical Institute, 24-29 S1. Giles, Oxford OXI 3LB, UK

I". 1\. E. Johnson, Department of Mathematics, University College, London WCIE 6BT, UK

I I), S. Jones, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

1˜()I)i()n Kirby, Department of Mathematics, University of California, Berkeley CA 94720, USA

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

"llIdy, Princeton NJ 08540, USA

1\t1nllhias Kreck, Max-Planck-Institut fUr Mathematik, 23 Gottfried Claren Str., Bonn, Germany

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

C 'nluhridge, Silver St, Cambridge CB3 9EW, UK

I )11 ...61 McDuff, Department of Mathematics, SUNY, Stony Brook NY, USA

Itnlll Melvin, Department of Mathematics, Bryn Mawr College, Bryn Mawr PA 19010, USA

( 'III i˜lian Okonek, Math Institut der Universitat Bonn, Wegelerstr. 10, D-53oo Bonn 1, FRG

I II. Ruhinstein, Department of Mathematics, University of Melbourne, Melbourne, Australia,

"".1, 'rhe Institute for Advanced Study, Princeton NJ 08540, USA

I˜. uI˜II( I .J. Stem, Department of Mathematics, University of California, Irvine CA 92717, USA

J I˜. 'I'aylor, Department of Mathematics, Notre Dame University, Notre Dame IN 46556, USA

c U. 'I'homas, Department of Pure Mathematics and Mathematical Statistics, University of

c 'ullllu-idgc, 16, Mill Lane, Cambridge CB3 9EW, UK

1\ I', 'I'()d, Mathematical Institute, 24-29 S1. Giles, Oxford OXI 3LB, UK

H W. 'I'ucker, Department of Physics, University of Lancaster, Lancaster, UK

t 11\\'.1nl Witten, Institute for Advanced Study, Princeton NJ 08540, USA

II th .. ( '. Wood, Department of Pure Mathematics, University of Leeds, Leeds, UK

Names of Participants

L. Jeffreys (Oxford)

N. A'Campo (Basel)

F. Johnson (London)

M. Atiyah (Oxford)

H. Azcan (Sussex) J. Jones (Warwick)

R. Kirby (Berkeley)

M. Batchelor (Cambridge)

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)

G. Massbaum (Nantes)

P. Bryant (Cambridge)

G. Matic (MIT)

F. Burstall (Bath)

D. McDuff (SUNY, Stony Brook

E. Corrigan (Durham)

M. Micallef (Warwick)

S. de Michelis (San Diego)

C. Okonek (Bonn)

S. Donal9son (Oxford)

P. Pansu (Pa.ris)

S. Dostoglu (Warwick)

H. Rubinstein (Melbourne)

J. Eells (Warwick/Trieste)

D. Salamon (Warwick)

Y. Eliashberg (Stanford)

G. Segal (Oxford)

D. Fairlie (Durham)

R. Fintushel (MSU, East Lansing) R. Stem (Irvine, CAl

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

M. Furuta (Tokyo/Oxford) K. Tod (Oxford)

K. Tsuboi (Tokyo)

G. Gibbons (Cambridge)

A. Givental (Moscow) R. Tucker (Lancaster)

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

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

J-C. Hausmann (Geneva) R. Ward (Durham)

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

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

J. Wood (Leeds) .

J. Hurtebise (Montreal)

D. Husemoller (Haverford/Bonn)

P. Iglesias (Marseille) I

INTRODUCTION

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

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

principal developments concern:

(1) geometric structures on manifolds,

(2) symplectic topology and geometry,

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

new invariants of 3-manifolds and knots.

(4)

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

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

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

(l(˜velopments have brought about a shift in the emphasis of current research on

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

I˜hysics.

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

˜(˜()metrical trends in earlier decades, is that in large part they treat phenomena in

HI)(˜(˜ific, low, dimensions. This mirrors the distinction, long recognised in topology,

IH,t,ween the flavours of "low-dimensional" and "high-dimensional" manifold theory

(ltlt,hough a detailed understanding of the connection between the special roles of

the dimension in different contexts seems to lie some way off). This feature explains

t.ll(˜ title of the meeting held in Durham in 1989 and in turn of these volumes of

Proceedings, and we hope that it captures some of the spirit of these different

l˜v(˜lopments.

t

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

lURy

c.1" these ideas, and some of the papers which seem to us to have been landmarks.

(W.' postpone mathematical technicalities to the specialised introductions to the

Nix Heparate sections of these volumes.) The developments can be said to have

ht'Ktlll with the lectures [T] given in Princeton in 1978-79 by W.Thurston, in which

II... (l(˜veloped his "geometrisation" programme for 3-manifolds. Apart from the

illll)(˜t,US given to old classification problems, Thurston's work was important for

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

vn...ic)us concomitant geometrical structures. For example, among the ideas exploited

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

liul< I(roups 88 discrete subgroups of PSL 2 (C), surgery compatible with geometric

r41.1'1icture, rigidity, Gromov's norm with values in the real singular homology, and

InUMt. irnportant of all, use of the theory of Riemann surfaces and Fuchsian groups

t.c˜ ((('vclop a feel for what might be true for special classes of manifolds in higher

.1 ill)( 'Hsions.

M('u.llwhile, another important signpost for future developments was Y. Eliashberg's

I l 1'C)c.f in 1981 of "symplectic rigidity"- the fact that the group of symplectic diffeo-

IIlol'phisms of a symplectic manifold is CO-closed 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 "Amol'd

Conjecture" [AJ 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 â€¢

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 Amol'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 ..dimemional surfaces,

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 o˜

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

solutions to provide a cobordism between such a 4-manifold and a specific unio˜Â

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

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

developed shortly before. SeminalÂ·ideas go back to the 1980 paper [SU] of Sacks and1

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

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

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

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

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

C.Taubes [Ta].

xi

Introduction

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

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

(leduce, among other things, that R 4 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

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

Hions).

(Â·\Â·om an apparently totally different direction the Jones polynomial emerged in a

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

It 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

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

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

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

eÂ»t.her things) combinatorics, exactly soluble models in statistical physics and con-

re )rmal field theories.

Iâ€¢â€¢ the spring of the next year, 1985, A. Casson gave a series of lectures in Berkeley

011 a new integer invariant for homology 3-spheres which he had discovered. This

( ˜n.sson invariant "counts" the number of representations of the fundamental group

i.1 SU(2) and has a number of very interesting properties. On the one hand it gives

h,ll integer lifting of the well-established Rohlin Z/2 It-invariant. On the other hand

(˜lLsson's definition was very geometric, employing the moduli spaces of unitary

n'))fcsentations of the fundamental groups of surfaces in an essential way. (These

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

point of view of Yang-Mills theory in the infiuential1982 paper of Atiyah and Bott

'A Ill.) Since such representations correspond to flat connections it was clear that

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

Y,ulg-Mills fields. On the other hand Casson showed, in his study of the behaviour

of t. he invariant under surgery, that there was a rich connection with knot theory

"'lid Inore familiar techniques in geometric topology. For a very readable account of

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

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

pl.Â·ctic geometry and to Yang-Mills theory, providing a prime example of the in-

t."rnct,ion between these two fields. Floer's theory brought together a number of

pow('lful ingredients; one of the most distinctive was his novel use of ideas from

rvJ()rs˜ theory. An important motivation for Floer's approach was the 1982 pa-

pf'" hy E. Witten [WI] which, among other things, gave a new analytical proof of

1.1 ... Morse inequalities and explained their connection with instantons, as used in

(˜ .. u.ntum Theory. 0

xii Introduction

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

generalised form of the Arnol'd conjecture [Fl]. 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 oVe!;

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 from 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 vague

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 Bott, 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 Birkboff-Lewis fixed-point theorem and a

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

[D] Donaldson, S.K. An application ofgauge theory to four dimensional topology :

Jour. Differential Geometry 18 (1983) 269-316

[Fl] Floer, A. Morse Theory 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-holomorphic 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 spberes d'homologie

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

151-164 )

[SU] Sacks, J. and Uhlenbeck, K.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

[Tal Taubes, C.R. Self-dual connections on non-self-dual four 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 Theory 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 Council for their generous suppor˜

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

FOUR-MANIFOLDS AND ALGEBRAIC

SURFACES

3

The last few years have seen important advances in our understanding of 4-

Inanifolds: their topology, differential topology and geometry. On the topological

side there is a good picture of the full classification, through Freedman's h-cobordism

n.nd restricted s-cobordism theorems. In the differential topological category we are

now well-aquainted with the special features of 4-manifold theory which are detected

hy the instanton solutions of the Yang-Mills equations, but the general classification

iH, for the moment, a matter of speculation. The 4-manifolds underlying complex

n.lgebraic surfaces have always provided a particularly interesting stock of examples,

IUld the fascinating problems of understanding the interaction between the complex

Ht,ructure and the differential topology lie at the forefront of current research. One

..n.n obtain a good idea of the present position of the subject, and of the progress

thnt has been made in recent years, by reading the two survey articles [M], [FM].

'('he five articles in this section cover many facets of the subject. The paper of

I)onaldson contains a general account of the use of Yang-Mills moduli spaces to de-

1111(˜ 4-manifold invariants, and some discussion of geometrical apects of the theory.

III particular it gives a brief summary of the link between Yang-Mills theory over

C'oillplex surfaces and stable holomorphic bundles, which in large measure accounts

fC)I' the prominence of algebraic surfaces in the results. The paper of Gompf sur-

vc'ys the general picture of smooth 4-manifolds, especially algebraic surfaces, and

pn'H(˜nts partial classification results. It also contains wonderfully explicit "Kirby

f'".Ietllus" descriptions of some distinct differentiable structures on a family of open

,1 Ilumifolds, and ties these in to the ideas of Floer homology which we consider at

1'.1'('u.t,er length in the next section. The paper of Kotschick takes a more algebro-

Kt'olllet,ric stance, and surveys what is known about the differential topology of a

˜I.tÂ·(Â·in.l, but very important, class of complex surfaces. This class includes the "Dol-

K",c"lu\v surfaces", which provided some of the first applications of the new techniques

I'nll .. Yang-Mills theory and which are also the starting point for Gompf's examples.

'I'lar' interaction between the complex geometry and the topology is particularly ap-

p......Â·llt in Kotschick's paper, and leading open problems, of detecting rationality,

"tlU hc˜ traced back to early work on algebraic surfaces.

'1111f' Dolgachev surfaces are also the starting point for the work described in the

"l'ti("I<, of Kreck; the general setting is the relative theory, of 2-dimensional surfaces

i,. 4 Ilutnifolds, and the Dolgachev manifolds appear as branched covers. Kreck's

I ttl I)('1' Kives us an example of the application of the topological s-cobordism theorem,

t'0IT.('f,h(˜r with surgery theory, to a very concrete problem.

'l'la" pn.per of Johnson deals with a rather different facet of the topology of algebraic

\"'" I("t.i("s; the structure of the fundamental group. There has been a good deal of

'H't.iVit,y in the last few years on the problems of describing what groups can occur

".. t.lu" fundamental groups of Kahler manifolds or of complex projective manifolds,

with work of Johnson and Rees, Gromov, Toledo, Corlette, Goldman and Millson

nud ()t.Il<˜rH. A wide variety of techniques have been used, ranging from algebra to

.laJrf"rc'lltill.l geometry and analysis. These qnestions arc, at least vaguely, related

to the techniques applied in defining differentiable invariants of complex swfaces,

since the moduli spaces of unitary representations of the fundamental group of a

compact Kahler manifold can be interpreted as moduli spaces of stable holomorphic

vector bundles (compare, for example, the contribution of Okonek below).

[FM] Friedman, R. and Morgan, J.W. Algebraic surfaces and 4-manifolds: some

conjectures and speculations Bull. Amer. Math. Soc. (New Series) 18 (1988)

1-18

[M] Mandelbaum,R. Four-dimensional topology: an introduction Bull. Amer.

Math. Soc. (New Series) 2 (1980) 1-159

Yang-Mills Invariants of Four-manifolds

S.K.DONALDSON

'Itl}(˜ Mathematical Institute, Oxford.

'l'hiH article is based on three lectures given at the Symposium in Durham. In

1.1." first section we review the well-known analogies between Yang-Mills instantons

OVf'1' 4-manifolds and pseudo-holomorphic curves in almost-Kahler manifolds. The

NC'c'oIHl section contains a rapid summary of the definition of invariants for smooth

,1 IIIH.nifolds using Yang-Mills moduli spaces, and of their main properties. In the

tl,il'(l section we outline an extension of this theory, defining new invariants which

WI' hope will have applications to connected sums of complex algebraic surfaces.

l-'iulLlly, in the fourth section, we take the opportunity to make some observations

nil pH(˜udo-holomorphiccurves and discuss the possibility of using linear analysis to

.'C allHtruct symplectic submanifolds, 'in analogy with the Kodaira embedding theorem

f"˜'lll e()mplex geometry.

1, ELLIPTIC TECHNIQUES IN TOPOLOGICAL PROBLEMS

SECTION

tl'lac' II\.Ht ten years have seen the development and application of new techniques in

1.1 .., t.wo fields of 4-manifold topology and symplectic geometry. There are striking

tU,,Â·u.llc˜ls between these developments, both in detail and in general methodology.

III tJu' first case one is interested primarily in smooth, oriented 4-manifolds, and

tit.' prohlems of classification up to diffeomorphism. In the second case one is inter-

t'Rl.t'cl in, for example, problems of existence and uniqueness of symplectic structures

(t'IOf-U'd, nowhere degenerate, 2-forms). In each case the structure considered is 10-

fully st.andard : the only questions are global ones and it is reasonable to describe

IUlt.1a suhjects as "topological" in an extended sense of the word.

'1'I.f' IIC'W developments which we have in mind bring methods of geometry and anal-

yuiN t.o hear on these topological questions. One introduces, as an auxiliary tool,

"f.II â€¢â€¢ , ".I>propriate geometrical structure, which will have local invariants like curva-

'.1." ,....<1 torsion. In the case of symplectic manifolds this structure is a Riemannian

tll .. t..,ic adapted to the symplectic form or, equivalently, a compatible almost com-

1'1.Â·)( tit.rlleture. Such a metric can appropriately be called almost-Ka_hler. In the

â€¢â€¢11"Â°,, case one considers Riemannian metrics on 4-manifolds. With this structure

II )(.Â·d W(' study associated geometric objects: in the first case these are the pseudo-

1,,,l,,u,.tl1'1)hic curves in an almost complex manifold V (i.e. maps f : ˜ -+ V from a

Donaldson: Yang-Mills invariants of four-manifolds

6

Riemann surface ˜ with complex-linear derivative) ; in the other case the objects

are the Yang-Mills instantons over a 4-manifold X (i.e. connections A on a princi-

pal bundle P -+ X with anti-self-dual curvature). In either case the objects can be

viewed as the solutions of certain non-linear, elliptic, differential equations. Infor-

mation about the original topological problem is extracted from properties of th˜

solutions of these equations. In the symplectic case this strategy was first employed

by Gromov [115], and the developments in both fields are instances of the use of

"hard" techniques, in the terminology of Gromov [16].

The detailed analogies between these two set-ups are wide ranging. Among the

most important are

(1) In each theory there is a "classical" or "integrable" case. On the one hand

we can consider Kahler metrics on complex manifolds V, and their associated

symplectic forms. Then the pseudo-holomorphic curves are the holomorphic

curves in the ordinary sense. On the other hand we can consider the 4-

manifolds obtained from complex projective surfaces, with Kahler metrics.

Then, as we shall describe in Section 2 (c) below, the Yang-Mills instantons

can be identified with certain holomorphic bundles over the complex surface.

So in either theory our differential geometric objects can be described in

algebro-geometric terms in these important cases.

(2) There is a fundamental integral formula in each case. The area of a compact

pseudo-holomorphic curve equals its topological "degree" ( the pairing of its

fundamental class with the cohomology class of the symplectic form) ; and

the Yang-Mills energy (mean- square of the curvature) of an instanton over

a compact base manifold is a topological characteristic number of the bundle

carrying the conection.

(3) Both theories are conformally invariant; with regard to the structures on E

and X respectively.

(4) The non-linear elliptic differential equations which arise in the two cases can

have non-zero Fredhohn indices. Thus the solutions are typically not isolated

but are parametrised by moduli manifolds.

(5) Both theories enjoy strong links with Mathematical Physics ( q - models and

gauge theories). A unified treatment of these developments from the point:

of view of q˜antum field theory has been given by Witten [22]. '

(6) Both theories exploit exploit special "low-dimensional" features - they areJ

tied to the 2-dimensionality of E and the 4-dimensionality of X respectivelY.l

There are many other points of contact between the theories. Notable among these˜

are the developments in the two fields brought about through the magnificent work'

of Floor ( see [10], and the articles on Floor's work in these Proceedings). Many of'

the developments in the two fields bear strongly on the representation variety W ot

conjugacy classes of r˜presentationsof the fundamental group of a closed Riemann':

surface, which has a natural Kahler structure. For example the Casson invariant of:

a 3-manifold can be obtained from the intersection number of a pair of Lagrangian

7

Donaldson: Yang-Mills invariants of four-manifolds

Ittll>manifolds in W. In a different setting we will encounter the space W in Section

2 (c) below, in our discussion of instantons over complex algebraic surfaces. It is

,ut,riguing that these representation spaces have also come to the fore recently in the

.Iones/Witten theory of invariants for knots and 3-manifolds (see the contributions

â€¢˜r Atiyah, Hitchin, Kirby and Witten in the accompanying volume), and it seems

.lllite likely that this points the way towards the possibility of obtaining some unified

IltHlerstanding of these different developments in Low-Dimensional Topology and

( :(.( Hnetry.

2,

SECTION YANG-MILLS INVARIANTS

(n) Definition. We will now describe how the Yang-Mills instantons yield invari-

.\111.1-1 of certain smooth 4-manifolds. For more details see [8] or [9]. For brevity we

will (˜onfine our discussion here to the gauge group SU(2), so we fix a.ttention on a

pl'ill(˜ipa1 SU(2) bundle P over a compact, oriented Riemannian 4-manifold X. We

will 11180 assume that X is simply connected. The bundle P is determined up to

bUHllorphism by the integer k =< C2(P), [Xl >, and if P is to support any anti-self-

dunl eonnection k must be non-negative, by the integral formula mentioned in (2)

t.r St,(˜tion 1. For each k 2: 0 we have a moduli 8pace Mk of anti-self-dual connec-

UOIIM OIl P modulo equivalence, and M o consists of a single point, representing the

ltHuluet connection on the trivial bundle.

I,t,t. Ao be a solution of the instanton equations, i.e. F* (A o) = 0, where F+ =

( 1/2)( F + *F) denotes the self-dual part of the curvature. The curvature of another

n"lllt˜(˜tion A o + a can be written

= F(Ao) + dAoa + a 1\ a,

F(A o + a)

wlu'J'c d Ao is the coupled exterior derivative. Taking the self-dual part we get, in

"t.fuiclard notation,

F+(Ao +a) = d10a + (a 1\ a)+.

Il'lu' uloduli space is obtained by dividing the solutions of this equation by the

hl,t.ic'll of the "gauge group" g = Aut P. For small deformations a this division can

tlf' ""I)laced by imposing the Coulomb gauge condition (provided the connection A o

,˜ lIl'('ducible )

dAoa = 0 ,

which defines a local transversal slice for the action of g. Thus ( assuming irre-

d".Â·ihility ) a neighbourhood of the point [A o] in the moduli space is given by the

n.. lul.ioliH of the differential equations

=0

dAoa

d10a +(a 1\ a)+ = O.

Donaldson: Yang-Mi11s invariants of four-manifolds

8

These are non-linear, first order, equations; the non-linearity coming from the

quadratic term (a A a)+. The linearisation about a = 0 can be written DAoa = 0,

d1

where DAo = d Ao E9 0 is a elliptic operator which plays the role in this four-

dimensional situation of the Cauchy-Riemann operator in the theory of pseudo-

holomorphic curves. The Fredholm index s = ind DAo of this operator is given b˜

the formula:

in which b+(X) is the dimension of a maximal positive subspace for the inter-

section form on H2(X). The nlllllber s is the "virtual dimension" of the moduli

space; more precisely, according to a theorem of Freed and UWenbeck [11], [9],

for a generic Riemannian metric on X the part of the moduli space consisting of

irreducible connections will be a smooth manifold of dimension s.

Let us now assume that b+(X) is strictly positive. Then it can be shown that for

generic metrics and all k ;:::: 1 every instanton is irreducible. It is easy to see why b+

enters here. A reducible anti-self-dual connection on P corresponds to an element

c of H 2 (X;R) which is in the intersection of the integer lattice and the subspace

H- C H 2 consisting of classes represented by anti-self dual forms. The codimension

of H- is b+, so if b+ > 0 and H- is in general position there are no non-zero classes

in the intersection. On the same lines one can show that if b+ > 1 then for generic 1

-parameter families of Riemannian metrics on X we do not encounter any non-trivial

reducible connections.

We can now indicate how to define differential topological invariants of the under-

lying 4-manifold X. We introduce the space 8* of all irreducible connections on

P, modulo equivalence. It is an infinite dimensional manifold and, under our as-

sumptions the moduli space MA: is a submanifold of 8*, for generic metrics on X.

Roughly, the invariants we define are the pairings of the fundamental homology class

of the moduli space with the cohomology of B*. To see that this is a reasonable,

strategy we have to consider the dependence of the definition on the Riemannian

metric on X. The moduli space itself certainly depends on the choice of metric,

so let us temporarily write MA;(g) for the the moduli space defined with respect

to a metric g. Suppose go , gl are two generic metrics on X. We join them by a

smooth path gt; t E [0,1] of metrics. If b+ > 1 then, as explained above, we do not

encounter any reducible connections so we can define

J

.˜

[0, 1] I [A]

N = { ([A], t) B* Mk(gt)}.

E E

X

˜

For a generic path gt the space N is a manifold- with- boundary, the boundary ˜

consisting of the disjoint union of Mk(gO) and Mk(gl). Using the obvious projection.

from N to 8* , we can regard N as giving a "homology" between the two moduli.:

spaces. .

This idea needs to be amplified in a number of ways. First we need to show that:

the moduli space is orientable ( and to fix signs one must find a rule for choosing ˜

Donaldson: Yang-Mills invariants of four-manifolds 9

n. definite orientation). Second we need to construct cohomology classes on 8*.

,I'his second step is an exercise in algebraic topology. Fix a base point in X and

Ic,t, B be the 80(3) bundle over B* whose points represent equivalence classes of

cÂ·ounections on a bundle which is trivialised over the base point. The space B is

wc-n.k-homotopyequivalent to the space Maps(X,BG) of based maps (of "degree"k)

f.'cnIl X to the classifying space BG (which can be identified with HpcxÂ» of the

nt.l'tlcture group SU(2). One can show then that the rational cohomology of B is a

IH )lynomial algebra on 2-dimensional cohomology classes labelled by a basis for the

:˜ clilnensional homology of X. That is, the cohomology is generated by the image

c˜f n. natural map

H 2 (8; Z),

jJ : H 2 (X; Z) --+

which is just the slant product in Maps(X,BG) x X with the 4-dimensional

.Â·IU.HH pulled back from the generator of H 4 (BG) under the evaluation pairing

AlalJs(X,BG) X X --+ BG. One can show further that this map jJ descends to

lunp

It.

H 2 (B*; Z),

IJ : H2 (X; Z) --+

.uld that the rational cohomology of B* is freely generated as a ring by the image

.)f' I.hiH map and by a 4-dimensional class ( the Pontryagin class of the fibration

l˜ â€¢ B*). The upshot of this algebro-topological excursion is that the rational

"OhOlllOlogy classes of B* are labelled by polynomials in the homology of X.

'1'11(' t,hird and most important step required to define invariants is to understand

tI,f" ("olnpactness properties of the moduli space. If the moduli spaces were compact

tl1t they would carry fundamental homology classes in the usual way and there

4

"

wcndd be little extra to say. However in practice the moduli spaces are scarcely ever

â€¢Â·â€¢â€¢IIIIUl.et, but they do have natural compactifications. The compactification M k of

,.,AÂ· iH n, subset of

s2(X) U ....

U Mk-l X X U Mk-2 X

Mk

'1'1 ..â€¢ topology is defined by a notion of convergence of the following kind. If

(.1'1," ., Xl) is a point in the symmetric product s'(X), a sequence [An] in Mk

â€¢Â·..uvC'rp;es to a limit ([A], (Xl, ... Xl)) E Mk-l X s'(X) if the connections converge

(lip t.o equivalence) away from Xl, â€¢â€¢â€¢ ,X" and the energy densities IF(A n )1 2 converge

,

nit IlIC'nsures to

L6

2 2

+ 81r

IF(A)1 Xi â€¢

i=l

'1'Iu' Ht.n.tement that the closure M k of M k in this topology is compact is essentially

ft handy formulation of analytical results of Uhlenbeck on Yang-Mills fields. This

t I,,'ory enters into our discussion of invariants because it can be used to show that

Donaldson: Yang-Mills invariants of four-manifolds

10

if the moduli space has even dimension, s = 2d say, then for k such that 4k >

(3b+(X) +3) there is a natural pairing between the moduli space Mk and a product

of cohomology classes /t(al) '-' /t(a2) '-' ... '-' /t(ad), for any al, ... , ad in H2(X).

We will refer to this range of values of as the "stable range" for k.

The cleanest conceptual definition of these pairings proceeds by extending the')

cohomology classes to the compactified space. For I > 0 and c E H2(X) let

s'(c) E H 2 (s'(XÂ» be the natural "symmetric sum " of copies of c. Then for a

in H2(X) we let a(l) be the class

where c is the Poincare dual of a. One then shows that, for any k, there is an

extension Ji(a) of lL(a) to H 2 (Mk), which agrees with a(l) on Mk,l == M k n (Mk- 1x

s'(XÂ». Consequently, for any al, â€¢â€¢ . ,ad there is a class

Granted this we can define a pairing < II, [M kl > so long as the compactified

space carries a fundamental homology class, and this fact follows from standard

homology theory provided that the "strata" Mk,' making up M k have codimension

2 or more, for I > O. But the dimension of Mk" is certainly bounded by that of

M k - I x s'(X) which is :

(1) dim Mk-I + 41 = dim M k - 81 + 41 = dim M k - 41, if 1 < k ;

(2) dim Sk(X) =4k, if I = k.

Since b+ is odd the condition for Mk,k to have codimension 2 is that 8k - 3(1 +

b+(XÂ» > 4k, which is just the stable range condition stated above.

The disadvantage with this approach is that the only definition of the classes p(a)

known to the author is rather complicated (the main points in the definition are

given in Chapter 7 of [9]). However the same pairing can be defined by a much

more elementary, although less perspicuous, procedure. For a generic surface ˜ in"

X the restriction of any ireducible anti-self-dual connection over X to E is again

irreducible, so we get restriction maps :

J

1

8;,

r: Mj -+

where BE is the space of irreducible connections over E, modulo equivalence. Ifi

a is the fundamental class of E in H 2 (X) the cohomology class p,(a) is pulled back

from BE by the restriction map. We choose a generic codimension 2 submanifold l

in this target space which represents the cohomology class, and let VE be the pre- :,

image of this in the moduli space. By abuse of notation we use the same symbol "

to denote subsets of all the different moduli spaces M j (since they are all pulled

back from the same representative over E). Let now E 1 , â€¢â€¢â€¢ ,Ed be sudaces in X,

11

Donaldson: Yang-Mills invariants of four-manifolds

Il:eneral position, and write Vi for representatives VE., as above. The crux of the

ill

IUIl.t,t,er is to show that, for k in the stable range, the intersection

iH compact. We can then define the pairing to be the corresponding algebraic

lut.'l'scction number; the number of points, counted with signs. The argument to

.,t4t....hlish this compactness is elementary, given two basic facts. First we can choose

tlu' ˜ so that all intersections in all the moduli spaces are transverse (and the

product connection is not in the closure of the Vi). Second, if [An] is a sequence in

\'. (. M k which converges to ([A],Xl' ... 'X,) in the sense considered above, and if

uolU' of the points Xj lies in ˜i then the limit [A] is in Vi C M j . One then goes on

t,4. H}U)W that this intersection number is independent of the choice of Riemannian

....˜t,l'ic on X by intersecting N with the Vi. Similar arguments show that the

IlIt,f-I'H(˜ction number is independent of the choice of Vi, and of the surfaces Ei,

within their homology classes.

I.. ""Ill, we have found new invariants of 4-manifolds which are multi-linear functions

til t.h(' homology. We introduce the notation

Sym˜,R

ru" t.lu˜ Bet of d-linear, symmetric, functions on H 2 (X;Z) with values in a ring R.

W(˜

'l'h."11 have

1. Let X be a smooth, oriented, compact and simply connected 4-

'l'IIII;OIlEM

,uulJi(old with b+(X) = 2a + 1 for a 2: 1. For each k with 4k > (3b+(X) + 3)

tl." IIULp :

= qk,X : ([Ell,Â·Â·Â·, [˜d]) .-+ n(Vi nÂ·Â·Â· n Vd n M k )

qk

f/ftlllU˜S an element of Sym˜,z , where d = 4k - 3(1 + a), which is (up to sign) a

.IU''''''t'utinl-topological invariant ofX, natural with respect to orientation -preserving

","i.,J"' t'c)rphisms.

W.' lut.(trpose a few remarks here. First, if b+ = lone can still define invariants,

I.. tt, have a more complicated form; see the article by Kotschick in these

t.lu'He

1Â·lucÂ·(˜(Â·dings. Second, it should be possible to extend the range of values of k

râ€¢â€¢â€¢ wlli("h invariants are defined. In a simple model case (where b+ = k = 1)

t'llf˜ 1<llC.WS how to introduce a boundary term to compensate for a codimension-l

",'.ut,11I1I Mt,l, then one obtains the "r-invariant" of a 4-manifold. This approach

I&un I˜f'C'Jl (˜xtended in the Oxford D.Phil. thesis of K.C.Mong, and can probably be

"ppli"e1 quite generally, although this has yet to be worked out in detail. A simpler

I'''H'('dur(˜ has been developed by J.W. Morgan, using components of the invariants

-2

'tt' It. counected sum XI rep, to define qk,X for values of k below the "stable

nll˜"'"

t

Donaldson: Yang-Mills invariants of four-manifolds

12

As a third remark; it would be good to have a definition of the invariants which

was both elementary and conceptually clear. To do this one would need to fully

understand the interaction between the topology used to define the compactification

and the homotopy theory of the spaces of connections. It is worth emphasising that

the anti-self dual equation itself plays no essential part in this discussion. Let B˜

denote the space of irreducible connections modulo equivalence on a bundle of Chern

class k. We can define a topology on the union:

˜= 8A: U Bi-l x X U 8 k- 2

s2(X) U ...

X

in much the same way as before, decreeing that a sequence [An] converges to

([A], (x 1 , â€¢.. , xI) if

(1) The connections converge away from the Xi.

(2) The self-dual parts IF+(A n )1 2 of the energy densities are uniformly bounded.

(3) The Chern-Wei! integrands Tr(F(A n )2) converge as measures to the limit

Tr(F(A))2 + 87r 2 2: 6x â€¢â€¢

It would be interesting to identify the homotopy type of˜. Similar questions can

be posed for the spaces of maps from a Riemann surface, which are relevant to the

analogous "weak" convergence encountered in the theory of harmonic maps and

holomorphic curves.

(b) Connected sums. One of the main features of the invariants constructed

above is that they vanish for a large class of connected sums. We have

2. Let X be a 4-manifold which satisfies the conditions of Theorem 1.

THEOREM

H X can be written as a smooth, oriented, connected sum X = X 1 UX2 and each of

the numbers b+(Xi) is strictly positive, then qk,X is identically zero for all k.

This strong statement reflects the fact that one can give a rather detailed description

of the moduli spaces over a connected sum, in terms of data on each factor. This uses

analytical techniques, which go back to work of Taubes [18 ], for "glueing" together

anti-self- dual solutions, and the ideas lead on to Floor's instanton homology groups

(which appear in the context of "generalised conected sums" across a homology 3-

sphere). We will now indicate the kind of analytical techniques involved, and sketch

how they lead to Theorem 2.

Let A 1 ,A2 be instantons on bundles Pl,P2 over the manifolds X 1 ,X2 respectively.

Assume that the connections are irreducible and that the operators <11. appearing˜

in the linearisation of the anti-self-dual equations are surjective (which is true for.˜

generic metrics on Xi). We also suppose that the metrics on the Xi are flat in ˜

small neighbourhoods of points Xi. We introduce a parameter .A > 0 and consider

a conformal structure on the connected sum based on the "glueing" map given, in

local Euclidean co-ordinates about these points, by

13

Donaldson: Yang-Mills invariants of four-manifolds

eis

e.

where ....-+ a reflection. A suitable metric on the conformal class represents aÂ·

c'onnected sum with a "neck" of diameter 0(,\1/2) ( another, conformally equivalent,

IIlodel is a connected sum joined by a tube of radius 0(1) and length O(exp(tX-1 ))0

We want to construct an instanton on X, for a small parameter '\, which is close to

Ai away from the neck region in the connected sum. As an approximation to what

W(˜ want we fix an identification of the fibres :

We construct a connection A o on a bundle P over X by flattening the connections

A. near Xi, and glueing together the bundles using the identification p, spread out

t.v(˜r balls around the Xi using the flat structures. We want to find an anti-self-dual

4'CHluection A o + a near to A oâ€¢ This is rather similar to our discussion above of the

Ic)(Â°n.l behaviour of the moduli space about a solution, the difference is that now A o

iN Iiot itself a solution. We want to solve the equation

= -F+(Ao) + (a 1\ a)+,

d10a

with a small. Suppose that S is a right inverse to d˜o i.e. d10Sw = w, and that

4

WC' have a uniform bound on the operator norm of S, mapping from L2 to L , that

iflt

with a constant CÂ· independent of A (which should be regarded as a parameter

t.broughout the discussion). Note that the L 2 norm on 2-forms and L 4 norm on

I f()rrns are conformally invariant, so we need only specify the conformal structure

flU ..Y". We will come back to the construction of S in a moment, but first we show

how it leads to a solution of our problem. We seek a solution in the form a = S(w),

,˜() t.he equation becomes:

w = -(Sew) 1\ S(w))+ - F+(Ao).

We use the Cauchy- Schwartz inequality to estimate the quadratic term, or rather

1.llfÂ· corresponding bilinear form:

'I'his means that we can write our equation in the fo˜ w = T(w), where T(w) =

(S'(w) 1\ S(w))+ - F+(A o ), and ˜e have:

Donaldson: Yang-Mills invariants of four-manifolds

14

say, if the L2_norms of W1,W2 are smaller than some fixed constant, independent of

A. On the other hand

and it is easy to see that this can be made arbitrarily small by making A small (since

one neeeds to flatten the connections over corespondingly small neighbourhoods or

the points Xi ). It follows easily then from the contraction mapping principle that,

for small A, there is a solution to our problem in the form

TR(O).

w = lim

n-+oo

, We now come back to explain how to construct the right inverse 5, obeying the

c'rucial uniform estimate. By standard elliptic theory there are right inverses Si

d1, over the compact manifolds Xi which are bounded as maps

to the operators

4 â€¢ To save notation (and an additional, unimportant, term in the

from L2 to L

estimates) let us at this stage ignore the distinction between Ai and the slightly

flattened connection over Xi used to form Ao- Let <P1' <P2 be cut-off functions on

X whose derivatives are supported in the neck region, with tPi equal to 1 on the

"Xi side" and to 0 on the other side, and with tP˜ + tP˜ = 1 on X. The function

tPi can be regarded in an obvious way as a function on Xi, and we can choose the

functions so that ( for small A ) the L 4 norm of dtPi is as small as we please. (This

is essentially the failure of the Sobolev embedding Lr --+ Co, when p = 4.) By the

conformal invariance it does not matter whether we measure this L4 norm in X or

in Xi-

Now, as a first approximation to the desired inverse 5, we set

for any self-dual 2-form w over X. The cut-off functions allow us to make sense˜

of this formula, over X, even though the Si are defined ov˜r Xi, using obvious:˜

identifications. Moreover we have

This gives, much as before, that

When Ais small we can choose 4>i with derivative small in L 4 , then this inequality,;

says that at0 0we Â·

A N - 1 is a contraction ; hence at N is invertible and can put ;

A

S = No (d1oN)-l.

15

Donaldson: Yang-Mills invariants of four-manifolds

I'Ia is completes our brief excursion into the analytical aspects of the theory. Taking

I

tolu' ideas further one shows that, with Ai fixed and ,\ small, one constructs a family

of solutions parametrised by a copy of SO(3), the choice of gluing parameter p.

I,.,tt.ing the Ai vary we construct open sets in the moduli space Mk,X which are

nbl"(˜ bundles over open sets in Mkt,X t X Mk"X2' for k = k1 + k 2 and each ki > o.

d1

WI)(˜n one of the ki is zero, say k2 , the picture is different, since the operator then

2

h".H n. cokemel of dimension 3b+(X2 ). One obtains another open set in Mk,x which

1˜ Illodelled on a subset Z of Mk,X t , this being the zero set of a section of a vector

IHIIHlle E of rank 3b+(X2 ) over Mk,X t â€¢ The bundle E is the direct sum of b+(X2 )

'fopi('s of a canonical 3-plane bundle over the moduli space: the vector bundle

n˜Ho("in.ted by the adjoint representation to the principal 80(3) bundle B --+ B*

lIu'lltioned in (a) above. In particular the rational Euler class of E is zero.

J'(˜lationbetween these different open sets in the manifold Mk,x, and their de-

'I'IIC'

p'˜ll(l(˜llCe on the parameter '\, is rather complicated but to sketch the ideas involved

I,. f.la(˜ I)roof of the "vanishing theorem" (Theorem 2) we can proced by imagining

U'h,f, the moduli space Mk,X is actually decomposed into compact components in

t.laiH wny, labelled by (k 1 ,k2 ). We then invoke two mechanisms. First, for com-

' .... UÂ·llt.S with neither ki equal to zero, the 80(3) fibre in the description of the

"lIluponent fibering over Mkt,X 1 X Mk 2 ,X2 is trivial as far as the cohomology classes

1'( f ˜) u.r(˜ concerned. These classes are all lifted up from the base in the fibration (

tIdnk of restricting to surfaces in Xl, X 2 ) and so their cup-product must obviously

vt\'Iiula ()Il the fundamental class. The second mechanism applies when one of the

AI, i14 1,('ro, k 2 say. We can then think ( under our unrealistic hypotheses) of the

.Â·tlil f'Hponding component of Mk,X as being identified with the zero set Z. Under

'111˜ ic I.'I Itification the cohomology classes p,(a) are all obtained by restricting the

':"1 n˜Mpouding classes over Mk,x t â€¢ On the other hand the fundamental class of Z in

tllf' hOlllology of Mk,X t is Poincare dual to the Euler class of E, and hence is zero

,,, I nt.iollu.l homology, so the contribution from this component to all the homology

"nh III˜H p;ives zero.

'˜lllphasise

W.' again that all we have tried to do here is to give the main ideas in

,1"1 proof of Theorem 2, since we will take up these ideas again in Section 3 ; the

.If\toI1H."d proof is long and complicated and we refer to [8] for this.

('1) I UHC,nntons and holomorphic bundles. We will now consider the "integrable

"""f˜ 1IJ('llt,ioned in Section 1. We suppose that our base manifold is endowed with

u

.. I .. u.pu.t,ihle complex structure: then we will see that any instanton naturally

.It,llu,oH u. holomorphic bundle. In this discussion it is simplest to work with vector

t.tltldl.-B, :-10 we identify our connections with covariant derivatives on the complex

'-f'. 1..1' htllldle associated to the fundamental representation of SU(2). The relation

wit I. J.olorllorphic structures can been seen most simply if we consider first the

I-nm' WIt.'1l t,he base space is C 2 , with the standard flat metric, and choose complex

Donaldson: Yang-Mills invariants of four-manifolds

16

= Xl + iX2, W = X3 + iX4.

co-ordinates z A covariant derivative has components

and its' curvature has components

Fij = [Vi, Vj].

The anti-self-dual condition becomes the three equations :

l

F12 +F34 = 0 l

F13 +F42 = 0

1

F14 +F23 = o. j

1

Now write D z = (1/2)(V1 + iV2 ), D w = (1/2)(V 3 + iV.) i these are the coupled j

Cauchy-Riemann operators in the two complex directions. Then the second and ,

third of the three anti-self-dual equations can be expressed in the tidy form .

[Dz,D w ] = o.

This is the integrability condition which is necessary and sufficient for the exis- ,

tence of a map 9 from C 2 to GL(2, C) such that:

8 8

-1 -1

oz ' = OWÂ·

gDzg gD w 9

=

So, in the presence of the complex structure on the base, we can write our three

anti-self-dual equations as the integrability condition plus the remaining equation,

which can be written :

[D z , D:l + [D w , D:,] = O.

For a global formulation of this we suppose X is a complex Kahler surface andÂ·

w is the metric 2-form on X. The anti-self dual forms are just the "primitive" (1,1).

forms: the forms of type (1,1) which are orthogonal to the ˜ahler form. The;

covariant derivative of a connection over X can be decomposed into (1,0) and (0,1)

parts:

VA = V˜ +V˜

We extend the operators to coupled exterior derivatives OA , 8A on bundle-.

valued forms, equal to VA, VA respectively on the O-forms. If the curvature hasÂ·;'

type (1,'1) then a˜ = 0 and the connection defines a holomorphic bundle, whose'

17

Donaldson: Yang-Mills invariants of four-manifolds

= o.

....".1 holomorphic sections are the solutions of the equation BAS So an anti-

1

˜t˜lf (Inal connection defines a holomorphic bundle. Conversely, given a holomorphic

l"IIUllc, we get an anti-self dual connection from any compatible unitary connection

which satisfies the remaining equation F(A).w = O. This relation can be taken

1I11I.. h further.It has been shown ( [6], [19] ) that it induces a (1,1) correspondence

'If˜t.w(˜en:

( I) The equivalence classes of irreducible anti-self-dual connections ( with struc-

ture group SU(2) in the present discussion).

(2) Equivalence classes of holomorphic 5L(2, C) bundles E over X which satisfy

the condition of "stability" with respect to the polarisation [w] E H 2 (X).

This stability condition is the requirement that for every line bundle bundle

L which admits a holomorphic map to E we must have cl(L) '-' [w] < 0 .

'1'11'1 :-;ubstance of this assertion is an existence theorem: for any stable bundle

We' ("lUi find a compatible connection A which satisfies the differential equation

III( A ).w = o. The effect is that, as far as discussion of moduli questions go (and

Itt pu.rticular for the purposes of defining invariants), we can shift our focus from

lIu' differential geometry of anti-self-dual connections to the algebraic geometry of

"â€¢â€¢1. )I11orphic bundles.

11'1 ..,s(' ideas have been used in two ways. On the one hand we can, in favourable

.. "m's, apply algebraic teclmiques to describe the moduli spaces explicitly and then

,'"I.'ulate invariants. Two standard techniques are available for constructing rank-2

lanlolllorphic bundles over surfaces. In one we consider a bundle V of rank 2, with a

1...lolllOrphic section s which vanishes on a set of points {Xi} in X, with multiplicity

".H' ",t, each point. Then we have an exact sequence

0--+ 0 --+ V AÂ®I --+ 0,

--+

wll˜'rc A is the line bundle

A 2 V and I is the ideal sheaf of functions vanishingÂ·pn. '

t IH\ points Xi. These extensions are classified by a group Ext == Ext 1 (A Â® I, 0),

which fits into an exact sequence:

H 1 (A*) HO(Kx Â®A)*,

EB(Kx Â® A):.

Ext

--+ --+ --+

i

wllc're the last map is the transpose of the evaluation map at the points Xi. SO

read off complete information about these extensions if we have sufficient

W'" cnll

'\Il˜)wl(˜dge of the cohomology groups of the line bundles over X. In principle this

.. ppronch can be used to describe all bundles over X since, if E is any rank-2 bundle

Wf' c'nn always find a line bundle L such that V = E Â® L has a section vanishing

nf UII isolated set of points (for a complete theory one needs also to consider zeros

waUl hip;her multiplicity).

"'or the second construction techriique we consider a double branched-cover 7r :

˜ "t". If J is a line bundle over X the direct image 1["*(J) is a rank-2 vector bundle

\

Donaldson: Yang-Mills invariants of four-manifolds

18

(locally free sheaf) over X. Conversely, starting from a rank-2 bundle E over X, r

we have a trace-free section s of the bundle EndE Â® L for some line bundle Love'

X, which has distinct eigenvalues at the generic point of X, and whose determinan

vanishes with multiplicity one on a curve C in X then we can construct a doubl

cover X ˜ X, branched over C. The points of X represent choices of eigenvalue 0

s. The associated eigenspace defines a line bundle J' over X, and E is the diret

image of J = J' Â® [C], where C is regarded as a divisor in X. This theory can b_

extended to cases where X is singular and J is a rank-l sheaf of a suitable kind. '

The other application of these ideas is more general. The Yang-Mills invariants giv

strong information about the differential topology of complex surfaces even in cas

where one cannot, at present, calculate the invariants explicitly. This comes ahou ˜

through a general positivity property of the invariants. Let a E H 2 (X) be Poincar

dual to the Kahler class [w] over a surface X, as above. Then we have :

3.

THEOREM

> O.

For all large enough k the invariant qk,X satisfies qk,X (a, a, ... , a)

For the proof of this one considers the restriction of holomorphic bundles over X t :

a hyperplane section ˜ - a complex curve representing Q. We have a moduli spac

WE of stable bundles over E, just as considered in Witten's interpretation of th,

Jones invariants.(See the account of Witten's lectures in these Proceedings). Le'

llS suppose for simplicity that stable bundles over X remain stable when restricte'

to C ( the technical difficultyies that arise here can be overcome by replacing Q b

pa for p >> 0, and considering restriction to a finite collection of curves.) Then vi

have a restriction map

r:Mk˜WE.

Over WE we have a basic holomorphic line bundle Â£, again just as considere

in Witten's theory. It is easy enough to show that p(a) is the pull-back by r .

the first Chern class of .e. On the other hand Â£, is an ample line bundle over WE

for large N the sections of .eN define a holomorphic embedding j : WE -+ cpm:

Furthermore one can easily see that r is an embedding, so the composite j 0 r give'

a projective embedding of Mk, and N IJ(a) is the restriction of the hyperplane cl '

over projective space. In this way one shows that,under one important hypothesi'

the pairing Ndqk,X(Q, â€¢â€¢â€¢ ,a) is the degree of the closure of the image of Mk in CP ,

(a projective variety). The degree of a non-empty projective variety is positive, an:

this gives the result. :

The vital hypothesis we require for this argument to work is the condition that,

least over a dense set of points in Mk, the Kahler metric w behaves like a generi

Riemannian metric, i.e. that for a dense set of anti-self-dual connections A th

d1

cokemel of the operator is zero. In algebra-geometric terms we require that fo',

a dense set of stable bundles E the cohomology group H2 (EndoE) should be zer

(Here Endo denotes the trace -free endomorphisms).

Donaldson: Yang-Mills invariants of four-manifolds 19

'1\ â€¢â€¢'olnplete the proof of (3) then we must show that this hypothesis is satisfied,

t.his is where the condition that k be large enters. What one proves is that the

I ..el

IUIt"t't, Sic of the moduli space representing bundles E with H2(EndoE) non-zero

h... cÂ·cHoplex dimension bounded by

(4)

rtl., ,.uJue constants A,B. This grows more more slowly than the virtual (complex)

,IhuruHion d = 4k - (3/2)(1 + b+(XÂ» of the moduli space M" , and it follows that

AI, \ a˜k is dense in M Ic, for large k. To establish the bound (4) one uses the fact that

N'I( I'J'7I.doE) is Serre- dual to HO(EndoE f8J Kx). So if E represents a point in Sic

U."rr iH a non-trivial section s of the bundle EndoEÂ®Kx. Two cases arise according

'u wlu˜ther the determinant of s is identically zero or not. If the determinant is zero

Ull' l((˜rnel of s defines a line bundle L* and a section of E Â® L. Then we can fit

h,tI4' th(˜ first construction described above and estimate the nmnber of parameters

IYAlllthle in the group Ext in terms of k. H the determinant is non-zero we fit into

.ht:' "f˜(Â·ond construction, using a branched cover, and we again estimate the number

uf ,.n.nuneters which determine the branched cover X and rank-1 sheaf J.

(tI) Itemarks.

c') ...˜ ("lUi roughly summarise the first results which are obtained from these Yang-

MIIiN invariants by saying that they show that there are at least two distinct classes

It' ,.,,('11 manifolds (up to diffeomorphism), which are not detected by classical meth-

lub.. ()n the one hand we have the connected sums of elementary building blocks,

r.,1' f˜x,unple the manifolds:

2 2

(U C P2# ... #CP 2 ) UCP U UCp

.â€¢â€¢

Xo,fJ =

vt '

, y ",

a copies fJ copies

r..1' which the invariants are trivial. Any (simply connected) 4-manifold X with

luld illt.('rsection form is homotopy equivalent to one of the Xo,fJ . On the other hand

w" lIn.v(˜ complex algebraic surfaces, where the invariants are non-trivial. (There is

.tt.Il.- overlap between these classes in the case when a = 1.) For a more extensive

t U˜(Â·uHHion of this general picture see the article by Gompf in these proceedings. More

'˜"flf˜(1 results show that the second class of 4-manifolds itself contains many distinct

'tl"uifolds with the same classical invariants (that is, homotopy equivalent but non-

tllll'c'olllorphic, simply connected smooth 4-manifolds.) The strongest results of this

˜il,.lltave been obtained by Friedman and Morgan in their work on elliptic surfaces.

1(.",,,,11 that a K3 surface is a compact, simply connected complex surface with

tII vinl canonical bundle. All K3 surfaces are diffeomorphic, but not necessarily bi-

t.,.lc)IIH)l-phicallyequivalent. Some K3 surfaces are "elliptic surfaces", that is they

ndfuit a holomorphic map 1r : S ˜. Cpl whose generic fibre is an elliptic curve

(:' clilllcnsional torus). Starting with such a K3 surface one can construct a family

Donaldson: Yang-Mills invariants of four-manifolds

20

of complex surfaces Sp,q p, q 2: 1 by performing logarithmic transformations to :

pair of fibres of 11", with multiplicities p and q. From a differentiable point of vie

a logarithmic transform of multiplicity r can be effected by removing a tubul !,

neighbourhood of a fibre, with boundary a 3-dimensional torus, and glueing it bac

ñòð. 1 |