Bordered Heegaard Floer Homology

294 Pages·2018·5.418 MB·English

by  Robert Lipshitz, Peter S. Ozsvath, Dylan P. Thurston

Number 1216
Bordered Heegaard Floer Homology
Robert Lipshitz
Peter S. Ozsvath
Dylan P. Thurston
July 2018 • Volume 254 • Number 1216 (fourth of 5 numbers)
Number 1216
Bordered Heegaard Floer Homology
Robert Lipshitz
Peter S. Ozsvath
Dylan P. Thurston
July 2018 • Volume 254 • Number 1216 (fourth of 5 numbers)
Library of Congress Cataloging-in-Publication Data
Names: Lipshitz,R.(Robert),author. |Ozsv´ath,PeterSteven,1967–author. |Thurston,Dylan
P.,1972–author.
Title: BorderedHeegaardFloerhomology/RobertLipshitz,PeterS.Ozsv´ath,DylanP.Thurston.
Description: Providence, RI : American Mathematical Society, [2018] | Series: Memoirs of the
American Mathematical Society, ISSN 0065-9266 ; volume 254, number 1216 | Includes biblio-
graphicalreferencesandindex.
Identifiers: LCCN2018029366|ISBN9781470428884(alk. paper)
Subjects: LCSH:Floer homology. |Three-manifolds(Topology)|Topologicalmanifolds. |Sym-
plecticgeometry.
Classification: LCCQA665.L572018|DDC516.3/6–dc23
LCrecordavailableathttps://lccn.loc.gov/2018029366
DOI:https://doi.org/10.1090/memo/1216
Memoirs of the American Mathematical Society
Thisjournalisdevotedentirelytoresearchinpureandappliedmathematics.
Subscription information. Beginning with the January 2010 issue, Memoirs is accessible
from www.ams.org/journals. The 2018 subscription begins with volume 251 and consists of six
mailings, each containing one or more numbers. Subscription prices for 2018 are as follows: for
paperdelivery,US$999list,US$799.20institutionalmember;forelectronicdelivery,US$879list,
US$703.20institutional member. Uponrequest, subscribers topaper delivery ofthis journalare
also entitled to receive electronic delivery. If ordering the paper version, add US$20 for delivery
withintheUnitedStates;US$80foroutsidetheUnitedStates. Subscriptionrenewalsaresubject
tolatefees. Seewww.ams.org/help-faqformorejournalsubscriptioninformation. Eachnumber
maybeorderedseparately;please specifynumber whenorderinganindividualnumber.
Back number information. Forbackissuesseewww.ams.org/backvols.
Subscriptions and orders should be addressed to the American Mathematical Society, P.O.
Box 845904, Boston, MA 02284-5904USA. All orders must be accompanied by payment. Other
correspondenceshouldbeaddressedto201CharlesStreet,Providence,RI02904-2213USA.
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of the material, such as to copy select pages for
useinteachingorresearch. Permissionisgrantedtoquotebriefpassagesfromthispublicationin
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
ispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Requestsforpermission
toreuseportionsofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. For
moreinformation,pleasevisitwww.ams.org/publications/pubpermissions.
Sendreq[e-mail].
Excludedfromtheseprovisionsismaterialforwhichtheauthorholdscopyright. Insuchcases,
requestsforpermissiontoreuseorreprintmaterialshouldbeaddresseddirectlytotheauthor(s).
Copyrightownershipisindicatedonthecopyrightpage,oronthelowerright-handcornerofthe
firstpageofeacharticlewithinproceedingsvolumes.
MemoirsoftheAmericanMathematicalSociety (ISSN0065-9266(print);1947-6221(online))
ispublishedbimonthly(eachvolumeconsistingusuallyofmorethanonenumber)bytheAmerican
MathematicalSocietyat201CharlesStreet,Providence,RI02904-2213USA.Periodicalspostage
paid at Providence, RI.Postmaster: Send address changes to Memoirs, AmericanMathematical
Society,201CharlesStreet,Providence,RI02904-2213USA.
(cid:2)c 2018bytheAmericanMathematicalSociety. Allrightsreserved.
ThispublicationisindexedinMathematicalReviews(cid:2),Zentralblatt MATH,ScienceCitation
Index(cid:2),ScienceCitation IndexTM-Expanded,ISI Alerting ServicesSM,SciSearch(cid:2),Research
Alert(cid:2),CompuMathCitationIndex(cid:2),CurrentContents(cid:2)/Physical,Chemical&EarthSciences.
ThispublicationisarchivedinPortico andCLOCKSS.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttps://www.ams.org/
10987654321 232221201918
Contents
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. The bordered Floer homology package . . . . . . . . . . . . . . . . . 3
1.3. On gradings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4. The case of three-manifolds with torus boundary . . . . . . . . . . . 6
1.5. Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6. Further developments . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7. Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Chapter 2. A∞ structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1. A∞ algebras and modules . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2. A∞ tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3. Type D structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4. Another model for the A∞ tensor product . . . . . . . . . . . . . . 22
2.5. Gradings by non-commutative groups . . . . . . . . . . . . . . . . . 26
Chapter 3. The algebra associated to a pointed matched circle . . . . . . . 29
3.1. The strands algebra A(n,k) . . . . . . . . . . . . . . . . . . . . . . . 29
3.2. Matched circles and their algebras . . . . . . . . . . . . . . . . . . . 34
3.3. Gradings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Chapter 4. Bordered Heegaard diagrams . . . . . . . . . . . . . . . . . . . 45
4.1. Bordered Heegaard diagrams: definition, existence, and uniqueness . 45
4.2. Examples of bordered Heegaard diagrams . . . . . . . . . . . . . . . 51
4.3. Generators, homology classes and spinc structures . . . . . . . . . . 53
4.4. Admissibility criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5. Closed diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Chapter 5. Moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1. Overview of the moduli spaces . . . . . . . . . . . . . . . . . . . . . 61
5.2. Holomorphic curves in Σ×[0,1]×R . . . . . . . . . . . . . . . . . . 64
5.3. Holomorphic curves in R×Z×[0,1]×R . . . . . . . . . . . . . . . 70
5.4. Compactifications via holomorphic combs . . . . . . . . . . . . . . . 73
5.5. Gluing results for holomorphic combs . . . . . . . . . . . . . . . . . 82
5.6. Degenerations of holomorphic curves . . . . . . . . . . . . . . . . . . 89
5.7. More on expected dimensions . . . . . . . . . . . . . . . . . . . . . . 99
Chapter 6. Type D modules . . . . . . . . . . . . . . . . . . . . . . . . . . 109
iii
iv CONTENTS
6.1. Definition of the type D module . . . . . . . . . . . . . . . . . . . . 109
6.2. ∂2 =0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3. Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4. Twisted coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Chapter 7. Type A modules . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.1. Definition of the type A module . . . . . . . . . . . . . . . . . . . . 145
7.2. Compatibility with algebra . . . . . . . . . . . . . . . . . . . . . . . 147
7.3. Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.4. Twisted coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Chapter 8. Pairing theorem via nice diagrams . . . . . . . . . . . . . . . . 157
Chapter 9. Pairing theorem via time dilation . . . . . . . . . . . . . . . . . 161
9.1. Moduli of matched pairs . . . . . . . . . . . . . . . . . . . . . . . . 162
9.2. Dilating time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
9.3. Dilating to infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9.4. Completion of the proof of the pairing theorem . . . . . . . . . . . . 183
9.5. A twisted pairing theorem . . . . . . . . . . . . . . . . . . . . . . . 184
9.6. An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Chapter 10. Gradings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
10.1. Algebra review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
10.2. Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
10.3. Type A structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.4. Type D structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.5. Refined gradings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
10.6. Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Chapter 11. Bordered manifolds with torus boundary . . . . . . . . . . . . 205
11.1. Torus algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
11.2. Surgery exact triangle . . . . . . . . . . . . . . . . . . . . . . . . . 208
11.3. Preliminaries on knot Floer homology . . . . . . . . . . . . . . . . 209
(cid:2) −
11.4. From CFD to HFK . . . . . . . . . . . . . . . . . . . . . . . . . . 212
− (cid:2)
11.5. From CFK to CFD: Statement of results . . . . . . . . . . . . . 216
11.6. Generalized coefficient maps and boundary degenerations . . . . . 220
− (cid:2)
11.7. From CFK to CFD: Basis-free version . . . . . . . . . . . . . . . 225
11.8. Proof of Theorem 11.26 . . . . . . . . . . . . . . . . . . . . . . . . 244
11.9. Satellites revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Appendix A. Bimodules and change of framing . . . . . . . . . . . . . . . . 255
A.1. Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
A.2. Sketch of the construction . . . . . . . . . . . . . . . . . . . . . . . 256
A.3. Computations for 3-manifolds with torus boundary . . . . . . . . . 261
(cid:2)
A.4. From HFK to CFD for arbitrary integral framings . . . . . . . . . 265
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Index of Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Abstract
We construct Heegaard Floer theory for 3-manifolds with connected bound-
ary. The theory associates to an oriented, parametrized two-manifold a differential
graded algebra. For a three-manifold with parametrized boundary, the invariant
comes in two different versions, one of which (type D) is a module over the algebra
and the other of which (type A) is an A∞ module. Both are well-defined up to
chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces,
the A∞ tensor product of the type D module of one piece and the type A module
(cid:2)
from the other piece is HF of the glued manifold.
Asaspecialcaseoftheconstruction,wespecializetothecaseofthree-manifolds
with torus boundary. This case can be used to give another proof of the surgery
(cid:2)
exact triangle for HF. We relate the bordered Floer homology of a three-manifold
with torus boundary with the knot Floer homology of a filling.
Received by the editor August 10, 2014 and, in revised form, August 20, 2015, October 23,
2015,andJanuary4,2016.
ArticleelectronicallypublishedonJune13,2018.
DOI:https://doi.org/10.1090/memo/1216
2010MathematicsSubjectClassification. Primary57R58,57M27;Secondary53D40,57R57.
Keywordsandphrases. Three-manifoldtopology,low-dimensionaltopology,HeegaardFloer
homology,holomorphiccurves,extendedtopologicalfieldtheory.
The first author was supported by an NSF Mathematical Sciences Postdoctoral Research
Fellowship,NSFgrantsDMS-0905796andDMS-1149800andaSloanResearchFellowship.
The first author is affiliated with the Department of Mathematics, University of Oregon,
Eugene,Oregon97403. Email: [email protected].
The second author was supported by NSF grants DMS-0505811, DMS-0804121, and DMS-
1105810.
The second author is affiliated with the Department of Mathematics, Princeton University,
Princeton,NewJersey08544. Email: [email protected].
ThethirdauthorwassupportedbyNSFgrantsDMS-1008049andDMS-1358638andaSloan
ResearchFellowship.
ThethirdauthorisaffiliatedwiththeDepartmentofMathematics,IndianaUniversity,Bloom-
ington,Indiana47405. Email: [email protected].
(cid:2)c2018 American Mathematical Society
v

List of Figures
1.1 Example of cutting a Heegaard diagram . . . . . . . . . . . . . . . . 3
3.1 Splittings and shuffles of Reeb chords . . . . . . . . . . . . . . . . . . 34
3.2 Pointed matched circles for the genus 2 surface . . . . . . . . . . . . 35
3.3 Orientation of a pointed matched circle . . . . . . . . . . . . . . . . . 35
4.1 3-manifold represented by a bordered Heegaard diagram . . . . . . . 48
4.2 Bordered Heegaard diagram for the trefoil complement . . . . . . . . 52
4.3 Heegaard diagrams for handlebodies . . . . . . . . . . . . . . . . . . 52
4.4 Another Heegaard diagram for a genus 2 handlebody . . . . . . . . . 53
5.1 Examples of three kinds of codimension 1 degenerations . . . . . . . 63
5.2 A capping operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Sources of curves at east ∞ . . . . . . . . . . . . . . . . . . . . . . . 72
5.4 Visualizing shuffle curves, I . . . . . . . . . . . . . . . . . . . . . . . 72
5.5 Visualizing shuffle curves, II . . . . . . . . . . . . . . . . . . . . . . . 73
5.6 Schematic of a height 2 holomorphic comb . . . . . . . . . . . . . . . 75
5.7 MB(x,y;S(cid:2);P) may be a proper subset of MB(x,y;S(cid:2);P), I . . . 77
5.8 MB(x,y;S(cid:2);P) may be a proper subset of MB(x,y;S(cid:2);P), II . . . 78
5.9 1-parameter family of maps v from an odd shuffle component . . . . 88
5.10 Annulus at east ∞ that does not exist . . . . . . . . . . . . . . . . . 91
5.11 Expected and unexpected behavior of M(x,y;S(cid:2);P(cid:3))⊂M(x,y;S(cid:2)) 96
5.12 Degenerations discussed in Theorem 5.61 . . . . . . . . . . . . . . . . 97
(cid:3)
5.13 Domains of model curves used to compute inv . . . . . . . . . . . . 104
5.14 Jittered intersection number . . . . . . . . . . . . . . . . . . . . . . . 105
5.15 Parts of the proof of Lemma 5.73 (index additivity) . . . . . . . . . . 106
(cid:3)
6.1 Local illustrations of possible contributions to ∂2 in CFD . . . . . . 116
6.2 Labeling of triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3 Curves α and αH, before and after a handleslide . . . . . . . . . . . 121
6.4 A triangle is a strip with a boundary puncture . . . . . . . . . . . . 124
vii
viii LISTOFFIGURES
6.5 A degeneration at v . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
12
6.6 Straightening α-arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.7 Illustration of a boundary bigon degenerating, in proof of invariance 136
6.8 Curves α, αH and α(cid:3) in Σ . . . . . . . . . . . . . . . . . . . . . . . . 139
6.9 Two compositions of triangles, as degenerations of a rectangle . . . . 141
7.1 Local illustrations of terms occurring the A∞ relation for C(cid:3)FA . . . 148
7.2 Domain where m is necessary for associativity . . . . . . . . . . . . 149
3
8.1 First step in making a bordered Heegaard diagram nice . . . . . . . 158
9.1 Schematic of an ideal-matched comb . . . . . . . . . . . . . . . . . . 172
9.2 Illustration of the fibered product in the proof of Proposition 9.40 . . 182
9.3 Degenerating a hexagon . . . . . . . . . . . . . . . . . . . . . . . . . 185
11.1 Bordered Heegaard diagrams with torus boundary . . . . . . . . . . 205
11.2 Labeling of regions around ∂Σ (torus case) . . . . . . . . . . . . . . . 206
11.3 From bordered diagrams to doubly-pointed diagrams . . . . . . . . . 214
11.4 Diagram for the longitude of a solid torus . . . . . . . . . . . . . . . 216
(cid:3)
11.5 Examples of reconstructing CFD from CFK . . . . . . . . . . . . . . 221
11.6 Boundary degeneration as end of a one-dimensional moduli space . . 222
11.7 Sketch of some of the maps from Theorem 11.36 . . . . . . . . . . . . 228
11.8 Twisting, in proof of Theorem 11.36 . . . . . . . . . . . . . . . . . . 229
11.9 Pinching off the winding region . . . . . . . . . . . . . . . . . . . . . 234
11.10 Homology classes in the winding region . . . . . . . . . . . . . . . . . 234
11.11 Pinching Ω along C . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
2
11.12 Illustration of the proof of Theorem 11.26 . . . . . . . . . . . . . . . 248
11.13 Diagram for the (2,1) cable . . . . . . . . . . . . . . . . . . . . . . . 250
11.14 Cable of the trefoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
A.1 Bordered Heegaard diagrams for diffeomorphisms . . . . . . . . . . . 258
A.2 Heegaard diagrams for Dehn twists of the torus . . . . . . . . . . . . 262
A.3 DA bimodules for generators of the genus-one mapping class group . 264
A.4 Isotoped Heegaard diagram for identity map of torus . . . . . . . . . 265
A.5 The bimodule C(cid:3)FAA(I,0) for the torus . . . . . . . . . . . . . . . . . 266
A.6 The bimodule CF(cid:3)DD(I,0) for the torus . . . . . . . . . . . . . . . . 266
CHAPTER 1
Introduction
1.1. Background
Since the pioneering work of Simon Donaldson, techniques from gauge theory
have taken a central role in the study of smooth four-manifold topology [DK90].
His numerical invariants, associated to closed, smooth four-manifolds with b+ >1,
2
have shed much light on our understanding of differential topology in dimension
four. Moreover,theseinvariants,andthesubsequentclosely-relatedSeiberg-Witten
invariants [Wit94] and Heegaard Floer invariants [OSz04d] all fit into a formal
frameworkreminiscentofthe“topologicalquantumfieldtheories” proposedbyWit-
ten[Wit88]. Crudelyspeaking, thesetheorieshave thefollowingform. Toaclosed
three-manifoldY oneassociatesa(suitablygraded)abeliangroup,theFloerhomol-
ogy of Y, and, to a four-manifold W with boundary identified with Y, a homology
class in the Floer homology of the boundary. If a closed, smooth four-manifold X
decomposes along Y into a union of two four-manifolds with boundary X and X ,
1 2
thenthenumerical(Donaldson,Seiberg-Witten,orHeegaardFloer)invariantofthe
closedfour-manifoldisobtainedasasuitablepairingbetweentherelativeinvariants
coming from X and X in a corresponding version of Floer homology of Y.
1 2
As the name suggests, the first such construction was proposed by Andreas
Floer (for a restricted class of three-manifolds) as a tool for studying Donaldson’s
theory. A complete construction of the corresponding three-dimensional Floer the-
ory for Seiberg-Witten invariants was given by Kronheimer and Mrowka [KM07].
Heegaard Floer homology was defined by Zoltán Szabó and the second author.
The aim of the present work is to perform a corresponding construction one
dimension lower. Specifically, we produce an invariant which, loosely speaking,
associatestoaparametrized,closed,orientedsurfaceF adifferentialgradedalgebra
A(F) and associates to a three-manifold whose boundary is identified with F a
differential graded module over A(F). When a closed, oriented three-manifold can
be decomposed along F into two pieces Y and Y , a suitable variant of Floer
1 2
homology is gotten as a pairing of the differential graded modules associated to Y
1
and Y .
2
Wegivenow aslightly moredetailedversionofthispicture, startingwithsome
more remarks about Heegaard Floer homology, and then a more precise sketch of
the invariants constructed in the present book.
Recall that there are several variants of Heegaard Floer homology stemming
from the fact that, in its most basic form, the Heegaard Floer homology of a three-
manifold is the homology of a chain complex defined over a polynomial ring in an
indeterminantU. Thefulltheorycanbepromotedtoconstructinvariantsforclosed,
1