Combinatorial Floer homology

de Silva, Vin; Robbin, Joel; Salamon, Dietmar

doi:10.1090/memo/1080

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Combinatorial Floer homology

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Vin de Silva, Joel W. Robbin and Dietmar A. Salamon

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 230, Number 1080
ISBNs: 978-0-8218-9886-4 (print); 978-1-4704-1670-6 (online)
DOI: http://dx.doi.org/10.1090/memo/1080
Published electronically: December 10, 2013

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Chapters

Chapter 1. Introduction
Chapter 2. Chains and Traces
Chapter 3. The Maslov Index
Chapter 4. The Simply Connected Case
Chapter 5. The Non Simply Connected Case
Chapter 6. Lunes and Traces
Chapter 7. Arcs
Chapter 8. Combinatorial Lunes
Chapter 9. Combinatorial Floer Homology
Chapter 10. Hearts
Chapter 11. Invariance under Isotopy
Chapter 12. Lunes and Holomorphic Strips
Chapter 13. Further Developments
Appendix A. The Space of Paths
Appendix B. Diffeomorphisms of the Half Disc
Appendix C. Homological Algebra
Appendix D. Asymptotic behavior of holomorphic strips

Abstract

We define combinatorial Floer homology of a transverse pair of noncontractible nonisotopic embedded loops in an oriented $2$ -manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original Lagrangian Floer homology. Our proof uses a formula for the Viterbo-Maslov index for a smooth lune in a $2$ -manifold.

[1] Mohammed Abouzaid, On the Fukaya categories of higher genus surfaces, Adv. Math. 217 (2008), no. 3, 1192–1235. MR 2383898, 10.1016/j.aim.2007.08.011
[2] Mohammed Abouzaid, Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov, and Dmitri Orlov, Homological mirror symmetry for punctured spheres, J. Amer. Math. Soc. 26 (2013), no. 4, 1051–1083. MR 3073884, 10.1090/S0894-0347-2013-00770-5
[3] Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197
[4] Samuel J. Blank, Extending immersions of the circle, Dissertation, Brandise University, Waltham, Mass., 1967. (Cited on page [[pageref]]FRANCIS2,TITUS,BLANK,POENARU.)
[5] Paul Biran and Octav Cornea, A Lagrangian quantum homology, New perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes, vol. 49, Amer. Math. Soc., Providence, RI, 2009, pp. 1–44. MR 2555932
[6] Vin De Silva, Products in the symplectic Floer homology of Lagrangian intersections, PhD thesis, Oxford, 1999. (Cited on pages [[pageref]]DESILVA0, [[pageref]]DESILVA1, [[pageref]]DESILVA2.)
[7] Clifford J. Earle and James Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43. MR 0276999
[8] D. B. A. Epstein, Curves on $2$-manifolds and isotopies, Acta Math. 115 (1966), 83–107. MR 0214087
[9] David Epstein, private communication, 6 April 2000. (Cited on pages [[pageref]]DBAE1, [[pageref]]DBAE2.)
[10] Andreas Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 6, 775–813. MR 948771, 10.1002/cpa.3160410603
[11] Andreas Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR 965228
[12] George K. Francis, Spherical curves that bound immersed discs, Proc. Amer. Math. Soc. 41 (1973), 87–93. MR 0321112, 10.1090/S0002-9939-1973-0321112-0
[13] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction. AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009. MR2310372 (2008d:53033) (Cited on page [[pageref]]FOOO1, [[pageref]]FOOO2.)
[14] Ralf Gautschi, Joel W. Robbin, and Dietmar A. Salamon, Heegaard splittings and Morse-Smale flows, Int. J. Math. Math. Sci. 56 (2003), 3539–3572. MR 2020083, 10.1155/S0161171203210115
[15] Michael Hutchings and Yi-Jen Lee, Circle-valued Morse theory and Reidemeister torsion, Geom. Topol. 3 (1999), 369–396. MR 1716272, 10.2140/gt.1999.3.369
[16] Michael Hutchings and Yi-Jen Lee, Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of $3$-manifolds, Topology 38 (1999), no. 4, 861–888. MR 1679802, 10.1016/S0040-9383(98)00044-5
[17] Michael Khanevsky, Hofer’s metric on the space of diameters, J. Topol. Anal. 1 (2009), no. 4, 407–416. MR 2597651, 10.1142/S1793525309000187
[18] 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
[19] John W. Milnor, Topology from the differentiable viewpoint, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Based on notes by David W. Weaver; Revised reprint of the 1965 original. MR 1487640
[20] 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
[21] Yong-Geun Oh, Floer cohomology of Lagrangian intersections and pseudoholomorphic discs I, II. Comm Pure Appl. Math., 46 (1993), 949–94, 995–1012. III. Arnold–Givental conjecture. In The Floer Memorial Volume, edited by Hofer, Taubes, Weinstein, and Zehnder, Birkhäuser, Basel, 1995, pp. 555–574. (Cited on pages [[pageref]]OHa, [[pageref]]OHb.)
[22] A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz numbers, and matrix models, Algebraic geometry—Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 325–414. MR 2483941, 10.1090/pspum/080.1/2483941
[23] A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2) 163 (2006), no. 2, 517–560. MR 2199225, 10.4007/annals.2006.163.517
[24] A. Okounkov and R. Pandharipande, The equivariant Gromov-Witten theory of ${\bf P}^1$, Ann. of Math. (2) 163 (2006), no. 2, 561–605. MR 2199226, 10.4007/annals.2006.163.561
[25] A. Okounkov and R. Pandharipande, Virasoro constraints for target curves, Invent. Math. 163 (2006), no. 1, 47–108. MR 2208418, 10.1007/s00222-005-0455-y
[26] 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, 10.4007/annals.2004.159.1027
[27] 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, 10.4007/annals.2004.159.1159
[28] Valentin Poenaru, Extensions des immersions en codimension 1 (d'apres Samuel Blank), Sem. Bourbaki, 1967/68, Exposé 342. (Cited on page [[pageref]]FRANCIS2,TITUS,BLANK,POENARU.)
[29] Marcin Pozniak, Floer homology, Novikov rings, and clean intersections, PhD thesis, University of Warwick, 1994. (Cited on page [[pageref]]POZNIAK.)
[30] Joel Robbin and Dietmar Salamon, The Maslov index for paths, Topology 32 (1993), no. 4, 827–844. MR 1241874, 10.1016/0040-9383(93)90052-W
[31] Joel Robbin and Dietmar Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), no. 1, 1–33. MR 1331677, 10.1112/blms/27.1.1
[32] Joel W. Robbin and Dietmar A. Salamon, Asymptotic behaviour of holomorphic strips, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), no. 5, 573–612 (English, with English and French summaries). MR 1849689, 10.1016/S0294-1449(00)00066-4
[33] Paul Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000), no. 1, 103–149 (English, with English and French summaries). MR 1765826
[34] Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2441780
[35] Paul Seidel, Homological mirror symmetry for the genus two curve, J. Algebraic Geom. 20 (2011), no. 4, 727–769. MR 2819674, 10.1090/S1056-3911-10-00550-3
[36] Tatjana Simcevic, A Hardy space approach to Floer gluing. PhD thesis, ETH Zürich, in preparation. (Cited on page [[pageref]]SIMCEVIC.)
[37] Stephen Smale, Diffeomorphisms of the $2$-sphere, Proc. Amer. Math. Soc. 10 (1959), 621–626. MR 0112149, 10.1090/S0002-9939-1959-0112149-8
[38] Charles J. Titus, The combinatorial topology of analytic functions on the boundary of a disk, Acta Math. 106 (1961), 45–64. MR 0166375
[39] Claude Viterbo, Intersection de sous-variétés lagrangiennes, fonctionnelles d’action et indice des systèmes hamiltoniens, Bull. Soc. Math. France 115 (1987), no. 3, 361–390 (French, with English summary). MR 926533

References

[1] Mohammed Abouzaid, On the Fukaya categories of higher genus surfaces, Adv. Math. 217 (2008), no. 3, 1192–1235. MR 2383898 (2008k:53198), http://dx.doi.org/10.1016/j.aim.2007.08.011
[2] Mohammed Abouzaid, Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov, and Dmitri Orlov, Homological mirror symmetry for punctured spheres, J. Amer. Math. Soc. 26 (2013), no. 4, 1051–1083. MR 3073884, http://dx.doi.org/10.1090/S0894-0347-2013-00770-5
[3] Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197 (80c:30001)
[4] Samuel J. Blank, Extending immersions of the circle, Dissertation, Brandise University, Waltham, Mass., 1967. (Cited on page [[pageref]]FRANCIS2,TITUS,BLANK,POENARU.)
[5] Paul Biran and Octav Cornea, A Lagrangian quantum homology, New perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes, vol. 49, Amer. Math. Soc., Providence, RI, 2009, pp. 1–44. MR 2555932 (2010m:53132)
[6] Vin De Silva, Products in the symplectic Floer homology of Lagrangian intersections, PhD thesis, Oxford, 1999. (Cited on pages [[pageref]]DESILVA0, [[pageref]]DESILVA1, [[pageref]]DESILVA2.)
[7] Clifford J. Earle and James Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43. MR 0276999 (43 \#2737a)
[8] D. B. A. Epstein, Curves on $2$ -manifolds and isotopies, Acta Math. 115 (1966), 83–107. MR 0214087 (35 \#4938)
[9] David Epstein, private communication, 6 April 2000. (Cited on pages [[pageref]]DBAE1, [[pageref]]DBAE2.)
[10] Andreas Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 6, 775–813. MR 948771 (89g:58065), http://dx.doi.org/10.1002/cpa.3160410603
[11] Andreas Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR 965228 (90f:58058)
[12] George K. Francis, Spherical curves that bound immersed discs, Proc. Amer. Math. Soc. 41 (1973), 87–93. MR 0321112 (47 \#9645)
[13] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction. AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009. MR2310372 (2008d:53033) (Cited on page [[pageref]]FOOO1, [[pageref]]FOOO2.)
[14] Ralf Gautschi, Joel W. Robbin, and Dietmar A. Salamon, Heegaard splittings and Morse-Smale flows, Int. J. Math. Math. Sci. 56 (2003), 3539–3572. MR 2020083 (2004m:57065), http://dx.doi.org/10.1155/S0161171203210115
[15] Michael Hutchings and Yi-Jen Lee, Circle-valued Morse theory and Reidemeister torsion, Geom. Topol. 3 (1999), 369–396. MR 1716272 (2000h:57063), http://dx.doi.org/10.2140/gt.1999.3.369
[16] Michael Hutchings and Yi-Jen Lee, Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of $3$ -manifolds, Topology 38 (1999), no. 4, 861–888. MR 1679802 (2000i:57055), http://dx.doi.org/10.1016/S0040-9383(98)00044-5
[17] Michael Khanevsky, Hofer's metric on the space of diameters, J. Topol. Anal. 1 (2009), no. 4, 407–416. MR 2597651 (2011c:53195), http://dx.doi.org/10.1142/S1793525309000187
[18] Maxim Kontsevich, Homological algebra of mirror symmetry, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120–139. MR 1403918 (97f:32040)
[19] John W. Milnor, Topology from the differentiable viewpoint, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Based on notes by David W. Weaver; Revised reprint of the 1965 original. MR 1487640 (98h:57051)
[20] 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 (2004m:53154)
[21] Yong-Geun Oh, Floer cohomology of Lagrangian intersections and pseudoholomorphic discs I, II. Comm Pure Appl. Math., 46 (1993), 949–94, 995–1012. III. Arnold–Givental conjecture. In The Floer Memorial Volume, edited by Hofer, Taubes, Weinstein, and Zehnder, Birkhäuser, Basel, 1995, pp. 555–574. (Cited on pages [[pageref]]OHa, [[pageref]]OHb.)
[22] A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz numbers, and matrix models, Algebraic geometry—Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 325–414. MR 2483941 (2009k:14111)
[23] A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2) 163 (2006), no. 2, 517–560. MR 2199225 (2007b:14123), http://dx.doi.org/10.4007/annals.2006.163.517
[24] A. Okounkov and R. Pandharipande, The equivariant Gromov-Witten theory of ${\bf P}^1$ , Ann. of Math. (2) 163 (2006), no. 2, 561–605. MR 2199226 (2006j:14075), http://dx.doi.org/10.4007/annals.2006.163.561
[25] A. Okounkov and R. Pandharipande, Virasoro constraints for target curves, Invent. Math. 163 (2006), no. 1, 47–108. MR 2208418 (2006m:14077), http://dx.doi.org/10.1007/s00222-005-0455-y
[26] 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 (2006b:57016), http://dx.doi.org/10.4007/annals.2004.159.1027
[27] 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 (2006b:57017), http://dx.doi.org/10.4007/annals.2004.159.1159
[28] Valentin Poenaru, Extensions des immersions en codimension 1 (d'apres Samuel Blank), Sem. Bourbaki, 1967/68, Exposé 342. (Cited on page [[pageref]]FRANCIS2,TITUS,BLANK,POENARU.)
[29] Marcin Pozniak, Floer homology, Novikov rings, and clean intersections, PhD thesis, University of Warwick, 1994. (Cited on page [[pageref]]POZNIAK.)
[30] Joel Robbin and Dietmar Salamon, The Maslov index for paths, Topology 32 (1993), no. 4, 827–844. MR 1241874 (94i:58071), http://dx.doi.org/10.1016/0040-9383(93)90052-W
[31] Joel Robbin and Dietmar Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), no. 1, 1–33. MR 1331677 (96d:58021), http://dx.doi.org/10.1112/blms/27.1.1
[32] Joel W. Robbin and Dietmar A. Salamon, Asymptotic behaviour of holomorphic strips, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), no. 5, 573–612 (English, with English and French summaries). MR 1849689 (2002e:53135), http://dx.doi.org/10.1016/S0294-1449(00)00066-4
[33] Paul Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000), no. 1, 103–149 (English, with English and French summaries). MR 1765826 (2001c:53114)
[34] Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2441780 (2009f:53143)
[35] Paul Seidel, Homological mirror symmetry for the genus two curve, J. Algebraic Geom. 20 (2011), no. 4, 727–769. MR 2819674 (2012f:53186), http://dx.doi.org/10.1090/S1056-3911-10-00550-3
[36] Tatjana Simcevic, A Hardy space approach to Floer gluing. PhD thesis, ETH Zürich, in preparation. (Cited on page [[pageref]]SIMCEVIC.)
[37] Stephen Smale, Diffeomorphisms of the $2$ -sphere, Proc. Amer. Math. Soc. 10 (1959), 621–626. MR 0112149 (22 \#3004)
[38] Charles J. Titus, The combinatorial topology of analytic functions on the boundary of a disk, Acta Math. 106 (1961), 45–64. MR 0166375 (29 \#3652)
[39] Claude Viterbo, Intersection de sous-variétés lagrangiennes, fonctionnelles d'action et indice des systèmes hamiltoniens, Bull. Soc. Math. France 115 (1987), no. 3, 361–390 (French, with English summary). MR 926533 (89b:58081)

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Combinatorial Floer homology

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Combinatorial Floer homology