Floer Homology, Gauge Theory, and Low Dimensional Topology: Proceedings of the Clay Mathematics Institute 2004 Summer School, Alfred Renyi Institute of Mathematics, Budapest, Hungary, June 5-26, 2004 (Clay Mathematics Proceedings, Vol. 5)

314 Pages·2006·3.12 MB·English

by  David A. Ellwood

Clay Mathematics Proceedings
5
Volume 5
Mathematical gauge theory studies connections on principal bundles,
aF
or, more precisely, the solution spaces of certain partial differential nl
o
equations for such connections. Historically, these equations have de

come from mathematical physics, and play an important role in the Lr

oH
description of the electro-weak and strong nuclear forces. The use
w
o
of gauge theory as a tool for studying topological properties of
-m
D
four-manifolds was pioneered by the fundamental work of Simon
io
Donaldson in the early 1980s, and was revolutionized by the ml
o
introduction of the Seiberg–Witten equations in the mid-1990s. e
g
n
Since the birth of the subject, it has retained its close connection y
s,
with symplectic topology. The analogy between these two fields of io G
study was further underscored by Andreas Floer’s construction of na
an infinite-dimensional variant of Morse theory that applies in two au
lg
a priori different contexts: either to define symplectic invariants for
Te
pairs of Lagrangian submanifolds of a symplectic manifold, or to define o
T
p
topological invariants for three-manifolds, which fit into a framework for h
o
calculating invariants for smooth four-manifolds. “Heegaard Floer homology”, le Proceedings of the
oo
the recently-discovered invariant for three- and four-manifolds, comes from an
gr Clay Mathematics Institute
application of Lagrangian Floer homology to spaces associated to Heegaard yy
, 2004 Summer School
diagrams. Although this theory is conjecturally isomorphic to Seiberg–
Witten theory, it is more topological and combinatorial in flavor and thus
Alfréd Rényi Institute of Mathematics
easier to work with in certain contexts. The interaction between gauge
E
theory, low-dimensional topology, and symplectic geometry has led l Budapest, Hungary
l
w
to a number of striking new developments in these fields. The aim
o
of this volume is to introduce graduate students and researchers in o June 5–26, 2004
d
other fields to some of these exciting developments, with a special ,

O
emphasis on the very fruitful interplay between disciplines.
z
s David A. Ellwood
This volume is based on lecture courses and advanced seminars v
á Peter S. Ozsváth
given at the 2004 Clay Mathematics Institute Summer School at t
h
the Alfréd Rényi Institute of Mathematics in Budapest, Hungary. , András I. Stipsicz

S
Several of the authors have added a considerable amount of t Zoltán Szabó
i
additional material to that presented at the school, and the resulting p
s Editors
volume provides a state-of-the-art introduction to current research, i
c
covering material from Heegaard Floer homology, contact geometry, z
,

smooth four-manifold topology, and symplectic four-manifolds. a
n
d

S
z
a
b
ó
,

E
d
it
o
r
s
www.ams.org American Mathematical Society
CMIP/5 AMS
www.claymath.org CMI Clay Mathematics Institute
4 color process 312 pages pages on 50 lb stock • 9/16 inch spine
Floer Homology,
Gauge Theory, and
Low-Dimensional
Topology

Clay Mathematics Proceedings
Volume 5
Floer Homology,
Gauge Theory, and
Low-Dimensional
Topology
Proceedings of the
Clay Mathematics Institute
2004 Summer School
Alfréd Rényi Institute of Mathematics
Budapest, Hungary
June 5–26, 2004
David A. Ellwood
Peter S. Ozsváth
András I. Stipsicz
Zoltán Szabó
Editors
American Mathematical Society
Clay Mathematics Institute
2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05,
53D40, 57M27,14J26.
ThecoverillustratesaKinoshita-Terasakaknot(aknotwithtrivialAlexanderpolyno-
mial),andtwoKauffmanstates. ThesestatesrepresentthetwogeneratorsoftheHeegaard
Floer homology of the knot in its topmost filtration level. The fact that these elements
are homologically non-trivial can be used to show that the Seifert genus of this knot is
two, a result first proved by David Gabai.
Library of Congress Cataloging-in-Publication Data
ClayMathematicsInstitute. SummerSchool(2004: Budapest,Hungary)
Floerhomology,gaugetheory,andlow-dimensionaltopology: proceedingsoftheClayMathe-
maticsInstitute2004SummerSchool,Alfr´edR´enyiInstituteofMathematics,Budapest,Hungary,
June5–26,2004/DavidA.Ellwood...[etal.],editors.
p.cm. —(Claymathematicsproceedings,ISSN1534-6455;v.5)
ISBN0-8218-3845-8(alk.paper)
1.Low-dimensionaltopology—Congresses. 2.Symplecticgeometry—Congresses. 3.Homol-
ogytheory—Congresses. 4.Gaugefields(Physics)—Congresses. I.Ellwood,D.(David),1966–
II.Title. III.Series.
QA612.14.C55 2004
514(cid:1).22—dc22 2006042815
Copyingandreprinting. Materialinthisbookmaybereproducedbyanymeansforeduca-
tionalandscientificpurposeswithoutfeeorpermissionwiththeexceptionofreproductionbyser-
vicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledgment
ofthesourceisgiven. Thisconsentdoesnotextendtootherkindsofcopyingforgeneraldistribu-
tion,foradvertisingorpromotionalpurposes,orforresale. Requestsforpermissionforcommercial
use of material should be addressed to the Clay Mathematics Institute, One Bow Street, Cam-
bridge,MA02138,[email protected].
Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In
suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s).
(Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof
eacharticle.)
(cid:1)c 2006bytheClayMathematicsInstitute. Allrightsreserved.
PublishedbytheAmericanMathematicalSociety,Providence,RI,
fortheClayMathematicsInstitute,Cambridge,MA.
PrintedintheUnitedStatesofAmerica.
TheClayMathematicsInstituteretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
(cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
VisittheClayMathematicsInstitutehomepageathttp://www.claymath.org/
10987654321 111009080706
Contents
Introduction vii
Heegaard Floer Homology and Knot Theory 1
An Introduction to Heegaard Floer Homology 3
Peter Ozsva´th and Zolta´n Szabo´
Lectures on Heegaard Floer Homology 29
Peter Ozsva´th and Zolta´n Szabo´
Circle Valued Morse Theory for Knots and Links 71
Hiroshi Goda
Floer Homologies and Contact Structures 101
Lectures on Open Book Decompositions and Contact Structures 103
John B. Etnyre
Contact Surgery and Heegaard Floer Theory 143
Andra´s I Stipsicz
Ozsv´ath-Szabo´ Invariants and Contact Surgery 171
Paolo Lisca and Andra´s I Stipsicz
Double Points of Exact Lagrangian Immersions and Legendrian Contact
Homology 181
Tobias Ekholm
Symplectic 4–manifolds and Seiberg–Witten Invariants 193
Knot Surgery Revisited 195
Ronald Fintushel
Will We Ever Classify Simply-Connected Smooth 4-manifolds? 225
Ronald J. Stern
A Note on Symplectic 4-manifolds with b+ =1 and K2 ≥0 241
2
Jongil Park
The Kodaira Dimension of Symplectic 4-manifolds 249
Tian-Jun Li
Symplectic 4-manifolds, Singular Plane Curves, and Isotopy Problems 263
Denis Auroux
Monodromy, Vanishing Cycles, Knots and the Adjoint Quotient 277
Ivan Smith
List of Participants 293
v

Introduction
The Clay Mathematical Institute hosted its 2004 Summer School on Floer ho-
mology, gauge theory, and low–dimensional topology at the Alfr´ed R´enyi Institute
ofMathematicsinBudapest,Hungary. Theaimofthisschoolwastobringtogether
students and researchers in the rapidly developing crossroads of gauge theory and
low–dimensional topology. In part, the hope was to foster dialogue across closely
related disciplines, some of which were developing in relative isolation until fairly
recently. The lectures centered on several topics, including Heegaard Floer theory,
knot theory, symplectic and contact topology, and Seiberg–Witten theory. This
volume is based on lecture notes from the school, some of which were written in
closecollaborationwithassignedteaching assistants. Thelectureshave revisedthe
choiceofmaterialsomewhatfromthatpresentedattheschool,andthetopicshave
been organizedto fit together in logicalcategories. Each course consistedof two to
five lectures, and some had associated problem sessions in the afternoons.
Mathematical gauge theory studies connections on principal bundles, or, more
precisely,thesolutionspacesofcertainpartialdifferentialequationsforsuchconnec-
tions. Historically, these equations have come from mathematical physics. Gauge
theoryasatoolforstudyingtopologicalpropertiesoffour–manifoldswaspioneered
by the fundamental work of Simon Donaldson in the early 1980’s. Since the birth
of the subject, it has retained its close connection with symplectic topology, a
subject whose intricate structure was illuminated by Mikhail Gromov’s introduc-
tion of pseudo–holomorphic curve techniques, also introduced in the early 1980’s.
The analogybetween these two fields of study was further underscored by Andreas
Floer’sconstructionofaninfinite–dimensionalvariantofMorsetheorythatapplies
in two a priori different contexts: either to define symplectic invariants for pairs of
Lagrangian submanifolds of a symplectic manifold (the so–called Lagrangian Floer
homology), providing obstructions to disjoining the submanifolds through Hamil-
tonianisotopies,ortogivetopologicalinvariantsforthree–manifolds(theso–called
instanton Floer homology), which fit into a framework for calculating Donaldson’s
invariants for smooth four–manifolds.
In the mid–1990’s, gauge–theoretic invariants for four–manifolds underwent a
dramaticchange withthe introductionof a new set of partialdifferential equations
introduced by Nathan Seiberg and Edward Witten in their study of string theory.
Very closely connected with the underlying geometry of the four–manifolds over
whichtheyaredefined, theSeiberg–Wittenequationsleadtofour–manifoldinvari-
ants which are in many ways much easier to work with than the anti–self–dual
Yang–Mills equations which Donaldson had studied. The introduction of the new
invariants led to a revolution in the field of smooth four–manifold topology.
vii
viii INTRODUCTION
Highlights in four–manifold topology from this period include the deep theo-
remsofCliffordTaubesaboutthedifferentialtopologyofsymplecticfour–manifolds.
These give an interpretation of some of Gromov’s invariants for symplectic man-
ifolds in terms of the Seiberg–Witten invariants of the underlying smooth four–
manifold. Another striking consequence of the new invariants was a quick, elegant
proof by Kronheimer and Mrowka of a conjecture by Thom, stating that the alge-
braiccurvesinthecomplexprojectiveplaneminimizegenusintheirhomologyclass.
The invariants were also used particularly effectively in work of Ron Fintushel and
Ron Stern, who discovered several operations on smooth four–manifolds, for which
theSeiberg–Witteninvariantstransforminapredictablemanner. Theseoperations
include rational blow–downs, where the neighborhood of a certain chain of spheres
is replaced by a space with vanishing second homology, and also knot surgery, for
whichtheAlexanderpolynomialofaknotisreflectedintheSeiberg–Witteninvari-
ants of a corresponding four–manifold. These operations can be used to construct
a number of smooth four–manifolds with interesting properties.
In an attempt to better understand the somewhat elusive gauge theoretic in-
variants, a different construction was given by Peter Ozsva´th and Zolta´n Szabo´.
They formulated an invariant for three– and four–manifolds which takes as its
starting point a Lagrangian Floer homology associated to Heegaard diagrams for
three–manifolds. The resulting “Heegaard Floer homology” theory is conjecturally
isomorphictoSeiberg–Wittentheory,butmoretopologicalandcombinatorialinits
flavor and correspondingly easier to work with in certain contexts. Moreover, this
theory has benefitted a great deal from an array of contemporary results rendering
variousanalyticalandgeometricstructuresinamoretopologicalandcombinatorial
form,suchasDonaldson’sintroductionof“Lefschetzpencils”inthesymplecticcat-
egoryandGiroux’scorrespondencebetweenopenbookdecompositionsandcontact
structures.
The two lecture series of Ozsv´ath and Szabo´ in the first section of this volume
provide a leisurely introduction to Heegaard Floer theory. The first lecture series
(thelecturesgivenbySzab´oattheSummerSchool)startwiththebasicnotions,and
moveontotheconstructionsoftheprimaryvariantsofFloerhomologygroupsand
maps between them. These lectures also cover basics of a corresponding Heegaard
Floer homology invariant for knots. The second lecture series (given by Ozsva´th)
givesarapidproofofoneofthebasiccalculationaltoolsofthesubject,thesurgery
exacttriangle,anditsimmediateapplications. SpecialemphasisisplacedonaDehn
surgery characterization of the unknot, a result whose proof is outlined in these
lectures. Section 1 concludes with the lecture notes from Goda’s course. Whereas
Heegaard diagrams correspond to real–valued Morse theory in three dimensions,
in these lectures, Goda considers circle–valued Morse theory for link complements.
He uses this theory to give obstructions to a knot being fibered.
The main theme in Section 2 is contact geometry and its interplay with Floer
homology. The lectures of John Etnyre give a detailed account of open book de-
compositions and contact structures, and the Giroux correspondence. The proof
of the Giroux correspondence is followed by some applications of this theory, in-
cluding an embedding theorem for weak symplectic fillings, which turned out to
be a crucial step in many of the recent developments of the subject, including the
INTRODUCTION ix
verification of Property P by Kronheimer and Mrowka. The definition of the con-
tact invariant in Heegaard Floer theory (resting on the above mentioned Giroux
correspondence) is discussed in the lecture notes of Andra´s Stipsicz, together with
ashortdiscussiononcontactsurgeries. Resultsregardingexistenceoftightcontact
structures on various 3–manifolds and their fillability properties are also given. A
similarapplicationofthecontactinvariantsisdescribedinthepaperofPaoloLisca
and Andra´s Stipsicz, with the use of minimum machinery required in the proof.
A different type of Floer homology (called contact homology) is studied in Tobias
Ekholm’s paper. A classical result of Gromov states that any exact Lagrangian
immersion into Cn has at least one double–point. Ekholm generalizes this result,
using Floer homology to give estimates on the minimum number of double–points
of an exact Legendrian immersion into some Euclidean space.
Section 3 discusses symplectic geometry and Seiberg–Witten invariants. Ron
Fintushel’slecturesgiveanintroductiontoSeiberg–Witteninvariantsandtheknot
surgery construction. The lectures give a thorough discussion of how the Seiberg–
Witteninvariantstransformundertheknotsurgeryoperation. Applicationsinclude
exotic embeddings of surfaces in smooth four–manifolds. Ron Stern’s contribution
describes the current state of art in the classification of smooth 4–manifolds, and
collects a number of intriguing questions and problems which can motivate further
resultsinthesubject. ThepaperofJongilParkprovidesnewapplicationsofthera-
tional blow–down construction, which led him to discover symplectic 4–manifolds
homeomorphic but not diffeomorphic to rational surfaces with small Euler char-
acteristic. Tian–Jun Li studies symplectic 4–manifolds systematically using the
generalization of the notion of the holomorphic Kodaira dimension κ to this cat-
egory. After the discussion of the κ = −∞ case, the state of the art for κ = 0
is described, where a reasonably nice classification scheme is expected. The con-
tribution of Denis Auroux also addresses the problem of understanding symplectic
4–manifolds, but from a completely different point of view. In this case the mani-
foldsarepresentedasbranchedcoversofthecomplexprojectiveplanealongcertain
curves,andthediscussioncentersonthepossibilityofgettingsymplecticinvariants
from topological properties of these branch sets. The volume concludes with Ivan
Smith’s contribution, where the author reviews basics about symplectic fibrations,
leading him (in a joint project with Paul Seidel) to knot invariants defined using
symplectic topology and Floer homology, conjecturally recapturing the celebrated
knot invariants of Khovanov.
It is hoped that this volume will give the reader a sampling of these many new
and exciting developments in low–dimensional topology and symplectic geometry.
Beforecommencingwiththemathematics,wewouldliketopausetothanksomeof
the many people who have contributed in one way or another to this volume. We
would like to thank Arthur Greenspoon for a meticulous proofreading of this text.
We would like to thank the Clay Mathematical Institute for making this program
possible,throughboththeirfinancialsupportandtheirenthusiasm;specialmention
goestoVidaSalahiforhercarefulanddiligentworkinbringingthisvolumetoprint.
Next, we thank the staff at the R´enyi Institute for helping to create a conducive
environment for the Summer School. We would like to thank the lecturers for
giving clear, accessible accounts of their research, and we are also grateful to their
course assistants, who helped make these courses run smoothly. Finally, we thank