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Bordered Heegaard Floer homology
About this Title
Robert Lipshitz, Department of Mathematics, University of Oregon, Eugene, Oregon 97403, Peter S. Ozsváth, Department of Mathematics, Princeton University, Princeton, New Jersey 08544 and Dylan P. Thurston, Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Publication: Memoirs of the American Mathematical Society
Publication Year:
2018; Volume 254, Number 1216
ISBNs: 978-1-4704-2888-4 (print); 978-1-4704-4748-9 (online)
DOI: https://doi.org/10.1090/memo/1216
Published electronically: June 13, 2018
Keywords: Three-manifold topology,
low-dimensional topology,
Heegaard Floer homology,
holomorphic curves,
extended topological field theory
MSC: Primary 57R58, 57M27; Secondary 53D40, 57R57
Table of Contents
Chapters
- 1. Introduction
- 2. $\mathcal {A}_{\infty }$ structures
- 3. The algebra associated to a pointed matched circle
- 4. Bordered Heegaard diagrams
- 5. Moduli spaces
- 6. Type $D$ modules
- 7. Type $A$ modules
- 8. Pairing theorem via nice diagrams
- 9. Pairing theorem via time dilation
- 10. Gradings
- 11. Bordered manifolds with torus boundary
- A. Bimodules and change of framing
Abstract
We construct Heegaard Floer theory for 3-manifolds with connected boundary. 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 $\mathcal A_\infty$ module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the $\mathcal A_\infty$ tensor product of the type $D$ module of one piece and the type $A$ module from the other piece is $\widehat {\textit {HF}}$ of the glued manifold.
As a special case of the construction, we specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for $\widehat {\textit {HF}}$. We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.
- Casim Abbas, Pseudoholomorphic strips in symplectisations. I. Asymptotic behavior, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 2, 139–185 (English, with English and French summaries). MR 2047354, DOI 10.1016/S0294-1449(03)00038-6
- Casim Abbas, An introduction to compactness results in symplectic field theory, Springer, Heidelberg, 2014. MR 3157146
- Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
- F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799–888. MR 2026549, DOI 10.2140/gt.2003.7.799
- A. I. Bondal and M. M. Kapranov, Framed triangulated categories, Mat. Sb. 181 (1990), no. 5, 669–683 (Russian); English transl., Math. USSR-Sb. 70 (1991), no. 1, 93–107. MR 1055981, DOI 10.1070/SM1991v070n01ABEH001253
- F. Bourgeois, A Morse-Bott approach to contact homology, Ph.D. thesis, Stanford University, 2006.
- S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726
- Eaman Eftekhary, Longitude Floer homology and the Whitehead double, Algebr. Geom. Topol. 5 (2005), 1389–1418. MR 2171814, DOI 10.2140/agt.2005.5.1389
- E. Eftekhary, Floer homology and splicing knot complements, 2008, arXiv:0802.2874.
- Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume (2000), 560–673. GAFA 2000 (Tel Aviv, 1999). MR 1826267, DOI 10.1007/978-3-0346-0425-3_{4}
- Julius Farkas, Theorie der einfachen Ungleichungen, J. Reine Angew. Math. 124 (1902), 1–27 (German). MR 1580578, DOI 10.1515/crll.1902.124.1
- Andreas Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 6, 775–813. MR 948771, DOI 10.1002/cpa.3160410603
- Andreas Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), no. 4, 575–611. MR 987770
- A. Floer, Instanton homology and Dehn surgery, The Floer memorial volume, Progr. Math., no. 133, Birkhäuser, 1995, pp. 77–97.
- Sergey Fomin and Richard P. Stanley, Schubert polynomials and the nil-Coxeter algebra, Adv. Math. 103 (1994), no. 2, 196–207. MR 1265793, DOI 10.1006/aima.1994.1009
- Kenji Fukaya, Morse homotopy, $A^\infty$-category, and Floer homologies, Proceedings of GARC Workshop on Geometry and Topology ’93 (Seoul, 1993) Lecture Notes Ser., vol. 18, Seoul Nat. Univ., Seoul, 1993, pp. 1–102. MR 1270931
- M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
- M. Hedden, On knot Floer homology and cabling, Ph.D. thesis, Columbia University, 2005.
- Matthew Hedden, On knot Floer homology and cabling, Algebr. Geom. Topol. 5 (2005), 1197–1222. MR 2171808, DOI 10.2140/agt.2005.5.1197
- Matthew Hedden, Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007), 2277–2338. MR 2372849, DOI 10.2140/gt.2007.11.2277
- Helmut Hofer, Véronique Lizan, and Jean-Claude Sikorav, On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J. Geom. Anal. 7 (1997), no. 1, 149–159. MR 1630789, DOI 10.1007/BF02921708
- H. Hofer, A general Fredholm theory and applications, Current developments in mathematics, 2004, Int. Press, Somerville, MA, 2006, pp. 1–71, arXiv:math/0509366v1.
- András Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006), 1429–1457. MR 2253454, DOI 10.2140/agt.2006.6.1429
- A. Juhész, Floer homology and surface decompositions, Geom. Topol. 12 (2008), no. 1, 299–350, arXiv:math/0609779.
- B. Keller, Koszul duality and coderived categories (after K. Lefévre), http://people.math.jussieu.fr/~keller/publ/kdc.dvi.
- Bernhard Keller, Introduction to $A$-infinity algebras and modules, Homology Homotopy Appl. 3 (2001), no. 1, 1–35. MR 1854636, DOI 10.4310/hha.2001.v3.n1.a1
- Mikhail Khovanov, Nilcoxeter algebras categorify the Weyl algebra, Comm. Algebra 29 (2001), no. 11, 5033–5052. MR 1856929, DOI 10.1081/AGB-100106800
- M. Khovanov, How to categorify one-half of quantum $\mathfrak {gl}(1\mid 2)$, 2010, \eprint{arXiv:1007.3517}.
- Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR 2388043
- P. Kronheimer, T. Mrowka, P. Ozsváth, and Z. Szabó, Monopoles and lens space surgeries, Ann. of Math. (2) 165 (2007), no. 2, 457–546. MR 2299739, DOI 10.4007/annals.2007.165.457
- Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120–139. MR 1403918
- Adam Simon Levine, Knot doubling operators and bordered Heegaard Floer homology, J. Topol. 5 (2012), no. 3, 651–712. MR 2971610, DOI 10.1112/jtopol/jts021
- Adam Simon Levine, Slicing mixed Bing-Whitehead doubles, J. Topol. 5 (2012), no. 3, 713–726. MR 2971611, DOI 10.1112/jtopol/jts019
- K. Lefèvre-Hasegawa, Sur les $A_{\infty }$-catégories, Ph.D. thesis, Université Denis Diderot–Paris 7, 2003, arXiv:math.CT/0310337.
- Robert Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10 (2006), 955–1096. [Paging previously given as 955–1097]. MR 2240908, DOI 10.2140/gt.2006.10.955
- R. Lipshitz, A Heegaard-Floer invariant of bordered 3-manifolds, Ph.D. thesis, Stanford University, Palo Alto, CA, 2006.
- Robert Lipshitz, Ciprian Manolescu, and Jiajun Wang, Combinatorial cobordism maps in hat Heegaard Floer theory, Duke Math. J. 145 (2008), no. 2, 207–247. MR 2449946, DOI 10.1215/00127094-2008-050
- R. Lipshitz, P. S. Ozsváth, and D. P. Thurston, Computing cobordism maps with bordered Floer homology, in preparation.
- R. Lipshitz, P. S. Ozsváth, and D. P. Thurston, Slicing planar grid diagrams: a gentle introduction to bordered Heegaard Floer homology, Proceedings of Gökova Geometry-Topology Conference 2008, Gökova Geometry/Topology Conference (GGT), Gökova, 2009, pp. 91–119, \eprint{arXiv:0810.0695}.
- Robert Lipshitz, Peter S. Ozsváth, and Dylan P. Thurston, Heegaard Floer homology as morphism spaces, Quantum Topol. 2 (2011), no. 4, 381–449. MR 2844535, DOI 10.4171/qt/25
- Robert Lipshitz, Peter S. Ozsváth, and Dylan P. Thurston, A faithful linear-categorical action of the mapping class group of a surface with boundary, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1279–1307. MR 3055762, DOI 10.4171/JEMS/392
- Robert Lipshitz, Peter S. Ozsváth, and Dylan P. Thurston, Bordered Floer homology and the spectral sequence of a branched double cover I, J. Topol. 7 (2014), no. 4, 1155–1199. MR 3286900, DOI 10.1112/jtopol/jtu012
- Robert Lipshitz, Peter S. Ozsváth, and Dylan P. Thurston, Bordered Floer homology and the spectral sequence of a branched double cover I, J. Topol. 7 (2014), no. 4, 1155–1199. MR 3286900, DOI 10.1112/jtopol/jtu012
- Robert Lipshitz, Peter S. Ozsváth, and Dylan P. Thurston, Computing $\widehat {HF}$ by factoring mapping classes, Geom. Topol. 18 (2014), no. 5, 2547–2681. MR 3285222, DOI 10.2140/gt.2014.18.2547
- Robert Lipshitz, Peter S. Ozsváth, and Dylan P. Thurston, Bimodules in bordered Heegaard Floer homology, Geom. Topol. 19 (2015), no. 2, 525–724. MR 3336273, DOI 10.2140/gt.2015.19.525
- Dusa McDuff, Singularities and positivity of intersections of $J$-holomorphic curves, Holomorphic curves in symplectic geometry, Progr. Math., vol. 117, Birkhäuser, Basel, 1994, pp. 191–215. With an appendix by Gang Liu. MR 1274930, DOI 10.1007/978-3-0348-8508-9_{7}
- Dusa McDuff and Dietmar Salamon, $J$-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2004. MR 2045629
- Mario J. Micallef and Brian White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves, Ann. of Math. (2) 141 (1995), no. 1, 35–85. MR 1314031, DOI 10.2307/2118627
- Y. Ni, Non-separating spheres and twisted Heegaard Floer homology, 2009, \eprint{arXiv:0902.4034}.
- Peter Ozsváth and Zoltán Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615–639. MR 2026543, DOI 10.2140/gt.2003.7.615
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004), 311–334. MR 2023281, DOI 10.2140/gt.2004.8.311
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58–116. MR 2065507, DOI 10.1016/j.aim.2003.05.001
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2) 159 (2004), no. 3, 1159–1245. MR 2113020, DOI 10.4007/annals.2004.159.1159
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019, DOI 10.4007/annals.2004.159.1027
- Peter Ozsváth and Zoltán Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006), no. 2, 326–400. MR 2222356, DOI 10.1016/j.aim.2005.03.014
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks, link invariants and the multi-variable Alexander polynomial, Algebr. Geom. Topol. 8 (2008), no. 2, 615–692. MR 2443092, DOI 10.2140/agt.2008.8.615
- Peter Ozsváth, Zoltán Szabó, and Dylan Thurston, Legendrian knots, transverse knots and combinatorial Floer homology, Geom. Topol. 12 (2008), no. 2, 941–980. MR 2403802, DOI 10.2140/gt.2008.12.941
- John Pardon, An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves, Geom. Topol. 20 (2016), no. 2, 779–1034. MR 3493097, DOI 10.2140/gt.2016.20.779
- J. Pardon, Contact homology and virtual fundamental cycles, 2015, \eprint{arXiv:1508.03873}.
- I. Petkova, Cables of thin knots and bordered Heegaard Floer homology, 2009, \eprint{arXiv:0911.2679}.
- Jacob Andrew Rasmussen, Floer homology and knot complements, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–Harvard University. MR 2704683
- Sucharit Sarkar, Maslov index formulas for Whitney $n$-gons, J. Symplectic Geom. 9 (2011), no. 2, 251–270. MR 2811652
- Paul Seidel, Fukaya categories and deformations, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 351–360. MR 1957046
- P. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008.
- P. Seidel, Lectures on four-dimensional Dehn twists, Symplectic 4-manifolds and algebraic surfaces, Lecture Notes in Math., vol. 1938, Springer, Berlin, 2008, pp. 231–267, \eprint{arXiv:math/0309012}.
- S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866. MR 185604, DOI 10.2307/2373250
- James Dillon Stasheff, Homotopy associativity of $H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293–312. MR 0158400, DOI 10.1090/S0002-9947-1963-0158400-5
- Sucharit Sarkar and Jiajun Wang, An algorithm for computing some Heegaard Floer homologies, Ann. of Math. (2) 171 (2010), no. 2, 1213–1236. MR 2630063, DOI 10.4007/annals.2010.171.1213
- Clifford Henry Taubes, Self-dual Yang-Mills connections on non-self-dual $4$-manifolds, J. Differential Geometry 17 (1982), no. 1, 139–170. MR 658473
- Clifford Henry Taubes, Self-dual connections on $4$-manifolds with indefinite intersection matrix, J. Differential Geom. 19 (1984), no. 2, 517–560. MR 755237
- Vladimir Turaev, Torsion invariants of $\textrm {Spin}^c$-structures on $3$-manifolds, Math. Res. Lett. 4 (1997), no. 5, 679–695. MR 1484699, DOI 10.4310/MRL.1997.v4.n5.a6
- Edward Witten, Topological quantum field theory, Comm. Math. Phys. 117 (1988), no. 3, 353–386. MR 953828
- Edward Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), no. 6, 769–796. MR 1306021, DOI 10.4310/MRL.1994.v1.n6.a13
- R. Zarev, Bordered Floer homology for sutured manifolds, 2009, \eprint{arXiv:0908.1106}.
- R. Zarev, Joining and gluing sutured Floer homology, 2010, \eprint{arXiv:1010.3496}.