The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I

Vincent Colin; Paolo Ghiggini; Ko Honda

doi:10.1007/s10240-024-00145-x

Article

The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I

Vincent Colin ¹; Paolo Ghiggini ¹; Ko Honda ¹

¹

Publications Mathématiques de l'IHÉS, Volume 139 (2024), pp. 13-187

Abstract

Given an open book decomposition $(S, 𝔥)$ adapted to a closed, oriented 3-manifold $M$ , we define a chain map $Φ$ from a certain Heegaard Floer chain complex associated to $(S, 𝔥)$ to a certain embedded contact homology chain complex associated to $(S, 𝔥)$ , as defined in (Colin et al. in Geom. Topol., 2024), and prove that it induces an isomorphism on the level of homology. This implies the isomorphism between the hat version of Heegaard Floer homology of $- M$ and the hat version of embedded contact homology of $M$ .

Received: 2012-08-07
Accepted: 2024-02-28
Online First: 2024-04-02
Published online: 2024-06-07

MR Zbl

DOI: 10.1007/s10240-024-00145-x

Author's affiliations:

Vincent Colin ¹; Paolo Ghiggini ¹; Ko Honda ¹

¹

@article{PMIHES_2024__139__13_0,
     author = {Vincent Colin and Paolo Ghiggini and Ko Honda},
     title = {The equivalence of {Heegaard} {Floer} homology and embedded contact homology via open book decompositions {I}},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {13--187},
     year = {2024},
     publisher = {Springer International Publishing},
     address = {Cham},
     volume = {139},
     doi = {10.1007/s10240-024-00145-x},
     mrnumber = {4750569},
     zbl = {1557.57017},
     language = {en},
     url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-024-00145-x/}
}

TY  - JOUR
AU  - Vincent Colin
AU  - Paolo Ghiggini
AU  - Ko Honda
TI  - The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I
JO  - Publications Mathématiques de l'IHÉS
PY  - 2024
SP  - 13
EP  - 187
VL  - 139
PB  - Springer International Publishing
PP  - Cham
UR  - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-024-00145-x/
DO  - 10.1007/s10240-024-00145-x
LA  - en
ID  - PMIHES_2024__139__13_0
ER  -

%0 Journal Article
%A Vincent Colin
%A Paolo Ghiggini
%A Ko Honda
%T The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I
%J Publications Mathématiques de l'IHÉS
%D 2024
%P 13-187
%V 139
%I Springer International Publishing
%C Cham
%U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-024-00145-x/
%R 10.1007/s10240-024-00145-x
%G en
%F PMIHES_2024__139__13_0

Vincent Colin; Paolo Ghiggini; Ko Honda. The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I. Publications Mathématiques de l'IHÉS, Volume 139 (2024), pp. 13-187. doi: 10.1007/s10240-024-00145-x

References
Cited by

[0.] V. Colin, P. Ghiggini and K. Honda, Embedded contact homology and open book decompositions, Geom. Topol., to appear.

[I.] V. Colin, P. Ghiggini and K. Honda, The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I. Publ. Math. Inst. Hautes Études Sci., (2024), this paper. | DOI | MR | Zbl | Numdam

[II.] V. Colin, P. Ghiggini and K. Honda, The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions II, Publ. Math. Inst. Hautes Études Sci., (2024). | DOI | MR | Zbl | Numdam

[III.] V. Colin, P. Ghiggini and K. Honda, The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus, Publ. Math. Inst. Hautes Études Sci., (2024). | DOI | MR | Numdam

[Abb1.] C. Abbas Finite energy surfaces and the chord problem, Duke Math. J., Volume 96 (1999), pp. 241-316 | MR | Zbl | DOI

[Abb2.] C. Abbas An Introduction to Compactness Results in Symplectic Field Theory, Springer, Heidelberg, 2014 | MR | Zbl | DOI

[Abo.] M. Abouzaid A geometric criterion for generating the Fukaya category, Publ. Math. Inst. Hautes Études Sci., Volume 112 (2010), pp. 191-240 | MR | Numdam | Zbl | DOI

[BEHWZ.] F. Bourgeois; Y. Eliashberg; H. Hofer; K. Wysocki; E. Zehnder Compactness results in symplectic field theory, Geom. Topol., Volume 7 (2003), pp. 799-888 | MR | Zbl | DOI

[CV.] K. Cieliebak; E. Volkov First steps in stable Hamiltonian topology, J. Eur. Math. Soc. (JEMS), Volume 17 (2015), pp. 321-404 | MR | Zbl | DOI

[CGH1.] V. Colin; P. Ghiggini; K. Honda Equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions, Proc. Natl. Acad. Sci., Volume 108 (2011), pp. 8101-8105 | MR | Zbl | DOI

[CHL.] V. Colin; K. Honda; F. Laudenbach On the flux of pseudo-Anosov homeomorphisms, Algebr. Geom. Topol., Volume 8 (2008), pp. 2147-2160 | MR | Zbl | DOI

[DS.] S. Donaldson; I. Smith Lefschetz pencils and the canonical class for symplectic four-manifolds, Topology, Volume 42 (2003), pp. 743-785 | MR | Zbl | DOI

[Dr.] D. Dragnev Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations, Comm. Pure Appl. Math., Volume 57 (2004), pp. 726-763 | MR | Zbl | DOI

[EGH.] Y. Eliashberg, A. Givental and H. Hofer, Introduction to symplectic field theory, GAFA 2000, Geom. Funct. Anal., Special Volume, Part II (2000), 560–673. | MR

[Frau.] U. Frauenfelder, Floer homology of symplectic quotients and the Arnold-Givental conjecture, PhD Thesis, ETH, Zürich, 2003. | MR

[Gh.] P. Ghiggini Knot Floer homology detects genus-one fibred knots, Amer. J. Math., Volume 130 (2008), pp. 1151-1169 | MR | Zbl | DOI

[Gi.] E. Giroux Géométrie de contact: de la dimension trois vers les dimensions supérieures, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 405-414 | MR | Zbl

[Ho.] H. Hofer Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math., Volume 114 (1993), pp. 515-563 | MR | Zbl | DOI

[HLS.] H. Hofer; V. Lizan; J.-C. Sikorav On genericity for holomorphic curves in four-dimensional almost complex manifolds, J. Geom. Anal., Volume 7 (1997), pp. 149-159 | MR | DOI | Zbl

[HWZ1.] H. Hofer; K. Wysocki; E. Zehnder Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 13 (1996), pp. 337-379 | MR | Numdam | Zbl | DOI

[HWZ2.] H. Hofer; K. Wysocki; E. Zehnder Properties of pseudoholomorphic curves in symplectisation. IV. Asymptotics with degeneracies, Contact and Symplectic Geometry, Volume 8 (1996), pp. 78-117 | MR | Zbl

[HKM.] K. Honda; W. Kazez; G. Matić On the contact class in Heegaard Floer homology, J. Differential Geom., Volume 83 (2009), pp. 289-311 | MR | Zbl | DOI

[Hu1.] M. Hutchings An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc. (JEMS), Volume 4 (2002), pp. 313-361 | MR | Zbl | DOI

[Hu2.] M. Hutchings The embedded contact homology index revisited, new perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes, 49, Am. Math. Soc., Providence, 2009, pp. 263-297 | MR | Zbl | DOI

[Hu3.] M. Hutchings Lecture notes on embedded contact homology, Contact and Symplectic Topology, 26, 2004 | Zbl

[HS1.] M. Hutchings; M. Sullivan The periodic Floer homology of a Dehn twist, Algebr. Geom. Topol., Volume 5 (2005), pp. 301-354 | MR | Zbl | DOI

[HS2.] M. Hutchings; M. Sullivan Rounding corners of polygons and the embedded contact homology of $T^{3}$ , Geom. Topol., Volume 10 (2006), pp. 169-266 (electronic) | MR | Zbl | DOI

[HT1.] M. Hutchings; C. Taubes Gluing pseudoholomorphic curves along branched covered cylinders I, J. Symplectic Geom., Volume 5 (2007), pp. 43-137 | MR | Zbl | DOI

[HT2.] M. Hutchings; C. Taubes Gluing pseudoholomorphic curves along branched covered cylinders II, J. Symplectic Geom., Volume 7 (2009), pp. 29-133 | MR | DOI | Zbl

[HT3.] M. Hutchings; C. Taubes Proof of the Arnold chord conjecture in three dimensions I, Math. Res. Lett., Volume 18 (2011), pp. 295-313 | MR | Zbl | DOI

[HT4.] M. Hutchings; C. Taubes Proof of the Arnold chord conjecture in three dimensions II, Geom. Topol., Volume 17 (2013), pp. 2601-2688 | MR | Zbl | DOI

[IP.] E. Ionel; T. Parker Relative Gromov-Witten invariants, Ann. of Math. (2), Volume 157 (2003), pp. 45-96 | MR | Zbl | DOI

[KrM.] P. Kronheimer; T. Mrowka Monopoles and Three-Manifolds, 10, Cambridge University Press, Cambridge, 2007 | MR | Zbl | DOI

[KMOS.] P. Kronheimer; T. Mrowka; P. Ozsváth; Z. Szabó Monopoles and lens space surgeries, Ann. of Math. (2), Volume 165 (2007), pp. 457-546 | MR | Zbl | DOI

[KLT1.] C. Kutluhan; Y. Lee; C. Taubes HF=HM I: Heegaard Floer homology and Seiberg-Witten Floer homology, Geom. Topol., Volume 24 (2020), pp. 2829-2854 | MR | Zbl | DOI

[KLT2.] C. Kutluhan; Y. Lee; C. Taubes HF=HM II: Reeb orbits and holomorphic curves for the ech/Heegaard-Floer correspondence, Geom. Topol., Volume 24 (2020), pp. 2855-3012 | MR | Zbl | DOI

[KLT3.] C. Kutluhan; Y. Lee; C. Taubes HF=HM III: holomorphic curves and the differential for the ech/Heegaard Floer correspondence, Geom. Topol., Volume 24 (2020), pp. 3013-3218 | MR | Zbl | DOI

[KLT4.] C. Kutluhan; Y. Lee; C. Taubes HF=HM IV: the Seiberg-Witten Floer homology and ech correspondence, Geom. Topol., Volume 24 (2020), pp. 3219-3469 | MR | Zbl | DOI

[KLT5.] C. Kutluhan; Y. Lee; C. Taubes HF=HM V: Seiberg-Witten-Floer homology and handle addition, Geom. Topol., Volume 24 (2020), pp. 3471-3748 | MR | Zbl | DOI

[LT.] Y. Lee; C. Taubes Periodic Floer homology and Seiberg-Witten Floer cohomology, J. Symplectic Geom., Volume 10 (2012), pp. 81-164 | MR | Zbl | DOI

[Li.] R. Lipshitz A cylindrical reformulation of Heegaard Floer homology, Geom. Topol., Volume 10 (2006), pp. 955-1097 | MR | DOI | Zbl

[Li2.] R. Lipshitz Correction to the article: a cylindrical reformulation of Heegaard Floer homology, Geom. Topol., Volume 18 (2014), pp. 17-30 | MR | Zbl | DOI

[M.] D. McDuff Singularities and positivity of intersections of $J$ -holomorphic curves, Holomorphic Curves in Symplectic Geometry, 117, Birkhäuser, Basel, 1994, pp. 191-215 (With an appendix by Gang Liu) | MR | DOI

[MS.] D. McDuff; D. Salamon $J$ -Holomorphic Curves and Symplectic Topology, 52, Am. Math. Soc., Providence, 2004 | MR | Zbl

[MW.] M. Micallef; B. White The structure of branch points in minimal surfaces and in pseudoholomorphic curves, Ann. of Math. (2), Volume 141 (1995), pp. 35-85 | MR | Zbl | DOI

[Ni.] Y. Ni Knot Floer homology detects fibred knots, Invent. Math., Volume 170 (2007), pp. 577-608 | MR | Zbl | DOI

[OSz1.] P. Ozsváth; Z. Szabó Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2), Volume 159 (2004), pp. 1027-1158 | MR | Zbl | DOI

[OSz2.] P. Ozsváth; Z. Szabó Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2), Volume 159 (2004), pp. 1159-1245 | MR | Zbl | DOI

[Po.] M. Poźniak Floer homology, Novikov rings and clean intersections, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, Volume 196 (1999), pp. 119-181 | MR | Zbl

[Ra.] J. Rasmussen, Floer homology and knot complements, Ph.D. thesis, 2003, | arXiv | MR

[Ru.] W. Rudin Real and Complex Analysis, McGraw-Hill, New York, 1987 | MR | Zbl

[SW.] S. Sarkar; J. Wang An algorithm for computing some Heegaard Floer homologies, Ann. of Math. (2), Volume 171 (2010), pp. 1213-1236 | MR | Zbl | DOI

[Schm.] F. Schmäschke, Floer homology of Lagrangians in clean intersection, preprint (2016), | arXiv

[Se1.] P. Seidel A long exact sequence for symplectic Floer cohomology, Topology, Volume 42 (2003), pp. 1003-1063 | MR | DOI | Zbl

[Se2.] P. Seidel Fukaya Categories and Picard-Lefschetz Theory, European Mathematical Society (EMS), Zürich, 2008 | DOI | Zbl

[T1.] C. Taubes The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol., Volume 11 (2007), pp. 2117-2202 | MR | Zbl | DOI

[T2.] C. Taubes Embedded contact homology and Seiberg-Witten Floer cohomology I–V, Geom. Topol., Volume 14 (2010), pp. 2497-3000 | MR | Zbl | DOI

[T3.] C. Taubes The structure of pseudoholomorphic subvarieties for a degenerate almost complex structure and symplectic form on $S^{1} \times B^{3}$ , Geom. Topol., Volume 2 (1998), pp. 221-332 | MR | DOI | Zbl

[TW.] W. P. Thurston; H. E. Winkelnkemper On the existence of contact forms, Proc. Amer. Math. Soc., Volume 52 (1975), pp. 345-347 | MR | DOI | Zbl

[Wa1.] S. Warschawski On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc., Volume 38 (1935), pp. 310-340 | MR | DOI | Zbl

[Wa2.] S. Warschawski On differentiability at the boundary in conformal mapping, Proc. Amer. Math. Soc., Volume 12 (1961), pp. 614-620 | MR | DOI | Zbl

[We1.] C. Wendl Finite energy foliations on overtwisted contact manifolds, Geom. Topol., Volume 12 (2008), pp. 531-616 | MR | DOI | Zbl

[We2.] C. Wendl Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, Comment. Math. Helv., Volume 85 (2010), pp. 347-407 | MR | Zbl | DOI

[We3.] C. Wendl, Punctured holomorphic curves with boundary on 3-manifolds: Fredholm theory and embeddedness, to appear.

Cited by Sources: