101 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 121 122 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 Groups and geometry, ROGER C LYNDON Surveys in combinatorics 1985, I ANDERSON (ed) Elliptic structures on 3-manifolds, C.B THOMAS A local spectral theory for closed operators, L ERDELYI & WANG SHENGWANG Syzygies, E.G EVANS & P GRIFFITH Compactification of Siegel moduli schemes, C-L CHAI Some topics in graph theory, H.P YAP Diophantine Analysis, LOXTON & A VAN DER POORTEN (eds) An introduction to surreal numbers, H GONSHOR Analytical and geometric aspects of hyperbolic space, D.B.A.EPSTEIN (ed) Low-dimensional topology and Kleinian groups, D.B.A EPSTEIN (ed) Lectures on the asymptotic theory of ideals, D REES Lectures on Bochner-Riesz means, K.M DAVIS & Y-C CHANG An introduction to independence for analysts, H.G DALES & W.H WOODIN Representations of algebras, P.J WEBB (ed) Homotopy theory, E REES & J.D.S JONES (eds) Skew linear groups, M SHIRVANI & B WEHRFRITZ Triangulated categories in the representation theory of finite-dimensional algebras, D HAPPEL Proceedings of Groups - St Andrews 1985, E ROBERTSON & C CAMPBELL (eds) Non-classical continuum mechanics, R.J KNOPS & A.A LACEY (oos) Lie groupoids and Lie algebroids in differential geometry, K MACKENZIE Commutator theory for congruence modular varieties, R FREESE & R MCKENZIE Van der Corpufs method for exponential sums, S.W GRAHAM & O KOLESNIK New directions in dynamical systems, T.1 BEDFORD & J.W SWIFf (eds) Descriptive set theory and the structure of sets of uniqueness, A.S KECHRIS & A LOUVEAU The subgroup structure of the finite classical groups, P.B KLEIDMAN & M.W.LIEBECK Model theory and modules, M PREST Algebraic, e~tremal & metric combinatorics, M-M DEZA, P FRANKL & LO ROSENBERG (eds) Whitehead groups of fmite groups, ROBERT OLIVER Linear algebraic monoids, MOHAN S PUTCHA Number theory and dynamical systems, M DODSON & J VICKERS (eds) Operator algebras and applications, 1, D EVANS & M TAKESAKI (eds) Operator algebras and applications, 2, D EVANS & M TAKESAKI (eds) Analysis at Urbana, I, E BERKSON, T PECK, & J UHL (eds) Analysis at Urbana, II, E BERKSON, T PECK, & UHL (eds) Advances in homotopy theory, S SALAMON, B STEER & W SUTHERLAND (eds) Geometric aspects of Banach spaces, E.M PEINADOR and A RODES (eds) Surveys in combinatorics 1989, J SIEMONS (ed) The geometry of jet bundles, D.J SAUNDERS The ergodic theory of discrete groups, PETER J NICHOLLS Introduction to unifonn spaces, I.M JAMES Homological questions in local algebra, JAN R STROOKER Cohen-Macaulay modules over Cohen-Macaulay rings, Y YOSHINO Continuous and discrete modules, S.H MOHAMED & BJ MOLLER Helices and vector bundles, A.N RUDAKOV et al Solitons, nonlinear evolution equations and inverse scattering, M.A ABLOWITZ & P.A CLARKSON Geometry of low-dimensional manifolds 1, S.K DONALDSON & C.B THOMAS (eds) Geometry of lOW-dimensional manifolds 2, S.K.:DONALDSON & C.B THOMAS (eds) Oligomorphic permutation groups, P.J CAMERON L-functions in Arithmetic, J COATES & M.J TAYLOR Number theory and cryptography, J LOXTON (ed) Classification theories of polarized varieties, TAKAO FUJITA Twistors in mathematics and physics, T.N BAILEY & RJ BASTON (eds) 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 print and sell (1/1 nl4InRtr oj books 10 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 © 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 521 39978 CONTENTS vi ( ·c'Illents of Volume ( 'c.ntributors Nnlnes of Participants IUlroduction Acknowledgments I vii viii ix xiv JlAKrr 1: FOUR-MANIFOlDS AND ALGEBRAIC SURFACES ! Ynng-Mills invariants of four-manifolds S K Donaldson I ( )n the topology of algebraic surfaces l~c)hCI1 E Gompf 41 'Ithe topology of algebraic surfaces with q = Pg = I )Icler Kotsehick ( )n the homeomorphism classification of smooth knotted surfaces in the 4-sphere Mnllhias Kreck It'lnt algebraic manifolds II,A.E Johnson 55 ItAI~T 73 93 97 2: FLOER'S INSTANTON HOMOLOGY GROUPS I stunton homology, surgery and knots Andreas Floer hununton hontology Andreas Floer, notes by Dieter Kotschick Invariants for homology 3-spheres I~()nuld Fintushel and Ronald J Stem ()n the Floer homology of Seifert fibered homology 3-spheres ( ~hristian Okonek 115 125 149 liu-invariant SU(2) instantons over the four-sphere Mikio Furuta PARrr 3: DIFFERENTIAL GEO:METRY AND MATHEMATICAL PHYSICS Skynne fields and instantons N.S Manton I~cpresentations of braid groups and operators coupled to monopoles :' '~ I(nlph E Cohen and John D.S Jones ~ I ~xlremal immersions and the extended frame bundle 63 161 175 179 191 207 1>.11 Hartley and R.W Tucker Minimal surfaces in quatemionic symmetric spaces I", E Burstall 231 "three-dimensional Einstein-Weyl geometry 237 K.P Too Ilannonic Morphisms, confonnal foliations and Seifert fibre spaces John C Wood 247 CONTENTS OF VOLUME Contents of Volume Contributors Names of Participants Introduction Acknowledgements PART 1: SYMPLECIlCGEOMElRY Introduction Rational and ruled symplectic 4-manifolds DusaMcDuff Symplectic capacities H Hofer The nonlinear Maslov index A.B Givental 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) Robion Kirby and Paul Melvin Representations of braid groups M.F Atiyah, notes by S.K Donaldson PART 3: THREE-DIMENSIONAL MANIFOlDS Introduction An introduction to polyhedral metrics of non-positive curvature on 3-manifolds I.R Aitchison and J.H Rubinstein Finite groups of hyperbolic isometries C.B Thomas Pin structures on low-dimensional manifolds R.C Kirby and L.R Taylor vi~ I XJ xi~ 1 10 11 12 1 1: 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 o H Setting A(X) maps x x ' Itdcpx( V) !I at points where Ae Theorem 2.1 (Fuglede, A smooth map qJ:M"' ~ = A(x)1I vii dtp x ¢ for (called ? A"" Note that a horizontally weakly qJ is just a Riemannian submersion We have (b) is a harmonic morphism if and only if it is horizontally weakly conformal We remark that in the case tp:fR ~ x' H H VE =0 M"' xE x = VxJ n T x M , that is, for all conformal map with any harmonic mor- M = JR3 , N= 1R =