The Floer Memorial Volume

687 Pages·1995·17.944 MB·English

by  Ian McAllister, Peter Braam (auth.), Helmut Hofer, Clifford H. Taubes, Alan Weinstein, Eduard Zehnder (eds.)

Progress in Mathematics
Volume 133
Series Editors
H. Bass
J. Oesterh!
A. Weinstein
The Floer Memorial
Volume
Helmut Hofer
Clifford H. Taubes
Alan Weinstein
Eduard Zehnder
Editors
Birkhauser Verlag
Basel . Boston· Berlin
Editors:
Helmut Hofer Clifford H. Taubes
Dept. of Mathematics Dept. of Mathematics
ETH-Zentrum Harvard University
8092 Zurich Cambridge, MA 02138
Switzerland USA
Alan Weinstein Eduard Zehnder
Dept. of Mathematics Dept. of Mathematics
University of California ETH-Zentrum
Berkeley, CA 94720 8092 Zurich
USA Switzerland
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
The Floer memorial volume / Helmut Hofer ... ed. - Basel
Boston ; Berlin : Birkhauser, 1995
(Progress in mathematics ; Vol. 133)
NE: Hofer, Helmut [Hrsg.]; Floer, Andreas: Festschrift; GT
This work is subject to copyright. All rights are reserved, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, re-use
of illustrations, broadcasting, reproduction on microfilms or in other ways, and
storage in data banks. For any kind of use permission of the copyright owner must
be obtained.
© 1995 Birkhauser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland
Softcover reprint of the hardcover 1st edition 1995
Printed on acid-free paper produced of chlorine-free pulp
00
ISBN-13: 978-3-0348-9948-2 e-ISBN-13: 978-3-0348-9217-9
DOl: 10.1007/978-3-0348-9217-9
987654321
Contents
Preface .............................................................. Vll
Andreas Floer 1956-1991 ............................................. ix
Publications of Andreas Floer Xl
McAllister, I. and Braam, P.
Floer's work on monopoles .......................................... .
Floer, A.
Monopoles on asymptotically flat manifolds ............................ 3
Floer, A.
The configuration space of Yang-Mills-Higgs theory on asymptotically
flat manifolds ...................................................... 43
Floer, A.
Instanton homology and Dehn surgery ................................. 77
Arnold, VI.
Some remarks on symplectic monodromy of Milnor fibrations ........... 99
Atiyah, M.
Floer homology ...................................................... 105
Audin, M.
Topologie des systemes de Moser en dimension quatre 109
Austin, D.M. and Braam, P.J.
Morse-Bott theory and equivariant cohomology. . . . . . . . . . . . . . . . . . . . . . . .. 123
Bates, S.M.
Some simple continuity properties of symplectic capacities .............. 185
Braam, PJ . and Donaldson, S.K.
Floer's work on instanton homology, knots and surgery ................. 195
Braam, P.I. and Donaldson, SK.
Fukaya-Floer homology and gluing formulae for polynomial invariants 257
Chaperon, M.
On generating families ............................................... 283
Cohen, R.L., lones, ID.S, and Segal, G.B.
Floer's infinite dimensional Morse theory and homotopy theory .......... 297
Dell'Antonio, G., D'Onofrio, B., and Ekeland, I.
Periodic solutions of elliptic type for strongly nonlinear
Hamiltonian systems ............................................... 327
Eliashberg, Y
Topology of 2-knots in ~4 and symplectic geometry .................... 335
Ernst, KD.
The ends of the monopole moduli space over ~3#,
(homology sphere): Part I .......................................... 355
vi Contents
Ernst, KD.
The ends of the monopole moduli space over 1R3#,
(homology sphere): Part II .......................................... 409
Fintushel, R. and Stern, R.
Using Floer's exact triangle to compute Donaldson invariants. . . . . . . . . . .. 435
Givental, AB.
A symplectic fixed point theorem for toric manifolds .................... 445
Hofer, H. and Salamon, D.A.
Floer homology and Novikov rings .................................... 483
Hofe/~ H. and Zehnder, E.
Symplectic invariants and Hamiltonian dynamics ....................... 525
McDuff, D.
An irrational ruled 4-manifold .............. . . . . . . . . . . . . . . . . . . . . . . . . . .. 545
Oh, Y.-G.
Floer cohomology of Lagrangian intersections and pseudoholomorphic
discs, III: Arnold-Givental conjecture ................................ 555
Polterovich, L.
An obstacle to non-Lagrangian intersections ............................ 575
Taubes, CH.
A Mayer-Vietoris model for Donaldson-Floer theory .................... 587
Viterbo, C.
The cup-product on the Thom-Smale-Witten complex, and
Floer cohomology ................................................. 609
Weinstein, A.
The symplectic structure on moduli space .............................. 627
Witten, E.
Chern-Simons gauge theory as a string theory .......................... 637
Index............................................................... 679
Preface
The death of Andreas Floer in Spring 1991 has left us shaken and sad. He was
an exceptionally gifted mathematician, who during his short life has significantly
influenced the development of mathematics. His untimely death was a personal
tragedy and a great loss for mathematics.
Mathematics was one of the focal points in Andreas' life, giving him pleasure
and satisfaction. We decided to collect essays for a memorial volume in his honour
related to his fields of interest: gauge theory, dynamical systems, symplectic geo
metry and topology. Mathematicians with a special relationship to Andreas Floer,
to his work and his ideas were invited to contribute. We did not set any conditions
as to style and contents and welcomed new research results as well as surveys,
speculations and personal reminiscences. The response was overwhelming, bearing
testimony to the esteem and appreciation of this colleague.
We would like to thank all the contributors. We also owe thanks to Peter
Braam for preparing two unfinished manuscripts by Andreas Floer for inclusion in
this volume and adding a short commentary. Finally, our thanks go to Birkhauser
Verlag for support and professional help.
H. Hofer
C. Taubes
A. Weinstein
E. Zehnder
Andreas Floer 1956-1991
Photo courtesy Prof. George Bergmann, Berkeley
Andreas Floer was born on August 23, 1956 in Duisburg, Germany. He studied
at the Ruhr-University Bochum, where he specialized in algebraic topology and
dynamical systems and received his Diploma in mathematics in 1982. He then
spent a year and a half at Berkeley, working with C. Taubes and A. Weinstein.
After his return to Bochum, he received his doctoral degree under E. Zehnder in
1984 with a thesis on V.I. Arnold's fixed point problem for global Hamiltonian
mappings. There followed research positions at SUNY at Stony Brook (1985-
1986) and at the Courant Institute of NYU (1986-1988) and a period as assistant
professor in Berkeley. In the fall of 1990 he was appointed to the chair of Analysis
and Geometry at the Ruhr-University in Bochum. He died on May 15, 1991 in
tragic circumstances.
Andreas Floer worked in the fields of dynamical systems, symplectic geom
etry, Yang-Mills theory and low dimensional topology. Motivated by the global
x Andreas Floer 1956-1991
existence problem of periodic solutions for Hamiltonian systems and starting from
ideas of Conley, Gromov and Witten, he developed his Floer homology. Floer's
homology theory is based on the combinatorial study of solutions of certain el
liptic partial differential equations on manifolds. They occur as connecting orbits
in variational principles like the action principle of classical mechanics and that
for the Chern-Simons functional in 3-dimensional topology. Applied to the highly
degenerate action functional on the loop space, his construction recovers the ho
mology of the underlying compact symplectic manifold and leads to a solution
v.I.
of the Arnold conjecture in many cases. For other functionals, it produces
previously unknown invariants of the underlying structures. The best known ex
ample is the Floer homology of the homology 3-sphere, which refines the Casson
invariant and is related to the Donaldson invariants for 4-manifolds. In joint work
with H. Hofer, the combination of the Floer homology and the symplectic capacity
theory led to a symplectic homology theory and to new symplectic invariants. In
his last work, Andreas Floer defined Yang-Mills type invariants for knots and gave
an axiomatic characterization of his 3-manifold- and knot invariants.
Andreas Floer's work was tragically interrupted, but his visions and deep con
tributions have provided new, powerful methods which can be applied to problems
inaccessible only a few years ago. Andreas Floer was a brilliant mathematician,
an exceptionally gifted and a very sensitive man, and a friend. We miss him.
Publications of Andreas Floer
[1] Integrality of the monopole number in SU(2) Yang-Mills-Higgs theories on
~3. Comm. Math. Phys., 93:367-378, 1984.
[2] Fixed point results for symplectic maps related to the Arnold conjecture (with
E. Zehnder). In: Dynamical Systems and Bifurcation, Proc. Groningen 1984,
Lecture Notes in Math. 1125, Springer 1985, 47-63.
[3] Proof of the Arnold conjecture for surfaces and generalizations to certain
Kahler manifolds. Duke math. 1. 53 (1986), 1-32.
[4] A refinement of the Conley index and an application to the stability of hy
perbolic invariant sets. Ergod. Th. and Dynam. Sys. (1987), 7, 93-103.
[5] The equivariant Conley index and bifurcations of periodic solutions of Hamil
tonian systems (with E. Zehnder). Charles Conley Memorial Vol., Ergod. Th.
Dynamical Systems 8* (1988), 87-98.
[6] Nonspreading wave packets for the cubic SchrOdinger equation with a bound
ed potential (with A. Weinstein). J. Funct. Anal. 69 (1986), 397--408.
[7] Monopoles on asymptotically Euclidean 3-manifolds. Bull. Amer. Math. Soc.
16 (1987), 125-127.
[8] A topological persistance theorem for normally hyperbolic manifolds via the
Conley index. Transact. Amer. Math. Soc. 321 (1990) No.2, 647-657.
[9] Viterbo's index and the Morse index for the symplectic action, in: Periodic
Solutions of Hamiltonian Systems and related topics, edited by A. Ambrosetti,
1. Ekeland, P. Rabinowitz and E. Zehnder, NATO ASI Series C, Vol. 209,
(1987) 147-152.
[10] Holomorphic curves and a Morse theory for exact symplectomorphisms. In:
Aspect dynamiques et topologiques des group infinis de transformation de la
mecanique classique, Traveaux en Cours 25, 49-61.
[11] Morse theory for fixed points of symplectic diffeomorphisms. Bull. Amer.
Math. Soc. 16 (1987), 279-281.
[12] Morse theory for Lagrangian intersections. J. Diff. Geom. 28, (1988) 513-
547.
[13] A relative Morse index for the symplectic action. Comm. Pure Appl. Math.,
Vol. XLI (1988), 393--407.
[14] The unregularized gradient flow of the symplectic action. Comm. Pure Appl.
Math., Vol. XLI (1988), 775-813.
[15] Witten's complex and infinite dimensional Morse theory. Journal Differential
Geom. 30 (1989), 207-221.
[16] Cuplength estimates for Lagrangian intersections. Comm. Pure Appl. Math.,
Vol. XLII (1989), 335-356.
[17] Selfdual conformal structures on lCP2. Journal Differential Geometry 33
(1991),551-573.