×

zbMATH — the first resource for mathematics

Symplectic Floer homology of area-preserving surface diffeomorphisms. (English) Zbl 1179.37077
An algorithm to compute \(HF_*(h)\) on surfaces of a negative Euler characteristic is given. In addition, an algorithm to compute \(HF_*(h)\) for all mapping classes on surfaces with boundary is defined and given.
In Section 2 symplectic Floer homology is reviewed and a detailed discussion of when it is well-defined is given. The invariance of the \(H_*(\Sigma)\)-module structure using an algebraic interpretation of the \(H_1(\Sigma)\)-reaction in terms of the twisted Floer homology complex is shown.
In Section 3 is shown that every map in a pseudo-Anosov mapping class is weakly monotone. Also, there is described the canonical singular representative of a pseudo-Anosov mapping class \(\varphi_{\text{sing}}\) in the closed case, and a Hamiltonian perturbation supported near its singularities which results in a smooth symplectomorphism \(\varphi_{sm}\) is given.
In Section 4, the topology of the space of weakly monotone maps in arbitrary mapping classes in order to prove a stronger invariance result for \(HF_*\) is studied. The standard form for reducible maps and behavior of pseudo-Anosov maps near the boundary are described. Is showed that \(HF_*(\varphi_{sm})= HF_*(h)\).
In Section 5 the theory of train tracks is reviewed and a combinatorial formula is given computing \(HF_*(\varphi_{sm})\) for pseudo-Anosov mapping classes from the action of \(\phi_{sm}\) on an invariant train track.
The results obtained in Section 5 are applied in Section 6: a large collection of examples of pseudo-Anosov maps due to Penner and an explicit formula for the rank of \(HF_*(h)\) are obtained.

MSC:
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
53D40 Symplectic aspects of Floer homology and cohomology
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] M Bestvina, M Handel, Train-tracks for surface homeomorphisms, Topology 34 (1995) 109 · Zbl 0837.57010 · doi:10.1016/0040-9383(94)E0009-9
[2] J S Birman, M E Kidwell, Fixed points of pseudo-Anosov diffeomorphisms of surfaces, Adv. in Math. 46 (1982) 217 · Zbl 0508.55001 · doi:10.1016/0001-8708(82)90024-X
[3] F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799 · Zbl 1131.53312 · doi:10.2140/gt.2003.7.799 · emis:journals/UW/gt/GTVol7/paper25.abs.html · eudml:128488 · arxiv:math/0308183
[4] A J Casson, S A Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Math. Soc. Student Texts 9, Cambridge Univ. Press (1988) · Zbl 0649.57008
[5] K Cieliebak, K Mohnke, Compactness for punctured holomorphic curves, J. Symplectic Geom. 3 (2005) 589 · Zbl 1113.53053 · doi:10.4310/JSG.2005.v3.n4.a5 · euclid:jsg/1154467631 · www.intlpress.com
[6] V Colin, K Honda, Reeb vector fields and open book decompositions · Zbl 1266.57013 · arxiv:0809.5088
[7] A Cotton-Clay, A sharp bound on fixed points of area-preserving surface diffeomorphisms, in preparation · Zbl 0862.58006 · doi:10.1090/S0002-9947-96-01502-4
[8] S Dostoglou, D A Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. \((2)\) 139 (1994) 581 · Zbl 0812.58031 · doi:10.2307/2118573
[9] E Eftekhary, Floer homology of certain pseudo-Anosov maps, J. Symplectic Geom. 2 (2004) 357 · Zbl 1081.53075 · doi:10.4310/JSG.2004.v2.n3.a3 · euclid:jsg/1118755325
[10] Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. (2000) 560 · Zbl 0989.81114
[11] O Fabert, Contact homology of Hamiltonian mapping tori · Zbl 1188.53102 · doi:10.4171/CMH/193 · www.ems-ph.org
[12] A Fathi, F Laudenbach, V Poenaru, editors, Travaux de Thurston sur les surfaces, Astérisque 66, Soc. Math. France (1991) 286
[13] A L Fel\(^{\prime}\)shtyn, Floer homology, Nielsen theory, and symplectic zeta functions, Tr. Mat. Inst. Steklova 246 (2004) 283 · Zbl 1105.53069
[14] A Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988) 513 · Zbl 0674.57027 · euclid:jdg/1214442477
[15] A Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989) 575 · Zbl 0755.58022 · doi:10.1007/BF01260388
[16] A Floer, H Hofer, D Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math. J. 80 (1995) 251 · Zbl 0846.58025 · doi:10.1215/S0012-7094-95-08010-7
[17] R Gautschi, Floer homology of algebraically finite mapping classes, J. Symplectic Geom. 1 (2003) 715 · Zbl 1084.53075 · doi:10.4310/JSG.2001.v1.n4.a4 · euclid:jsg/1092749567
[18] M Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc. \((\)JEMS\()\) 4 (2002) 313 · Zbl 1017.58005 · doi:10.1007/s100970100041
[19] M Hutchings, Reidemeister torsion in generalized Morse theory, Forum Math. 14 (2002) 209 · Zbl 0990.57011 · doi:10.1515/form.2002.010
[20] M Hutchings, Floer homology of families. I, Algebr. Geom. Topol. 8 (2008) 435 · Zbl 1170.57025 · doi:10.2140/agt.2008.8.435 · arxiv:math/0308115
[21] M Hutchings, M Sullivan, The periodic Floer homology of a Dehn twist, Algebr. Geom. Topol. 5 (2005) 301 · Zbl 1089.57021 · doi:10.2140/agt.2005.5.301 · emis:journals/UW/agt/AGTVol5/agt-5-14.abs.html · eudml:126089 · arxiv:math/0410059
[22] M Hutchings, M Sullivan, Rounding corners of polygons and the embedded contact homology of \(T^3\), Geom. Topol. 10 (2006) 169 · Zbl 1101.53053 · doi:10.2140/gt.2006.10.169 · eudml:126791 · arxiv:math/0410061
[23] S Jabuka, T Mark, Heegaard Floer homology of certain mapping tori, Algebr. Geom. Topol. 4 (2004) 685 · Zbl 1052.57046 · doi:10.2140/agt.2004.4.685 · emis:journals/UW/agt/AGTVol4/agt-4-31.abs.html · eudml:124002 · arxiv:math/0405314
[24] B J Jiang, J H Guo, Fixed points of surface diffeomorphisms, Pacific J. Math. 160 (1993) 67 · Zbl 0829.55001 · doi:10.2140/pjm.1993.160.67
[25] P Kronheimer, T Mrowka, Monopoles and three-manifolds, New Math. Monogr. 10, Cambridge Univ. Press (2007) · Zbl 1158.57002
[26] E Lebow, Embedded contact homology of \(2\)-torus bundles over the circle, PhD thesis, University of California, Berkeley (2007)
[27] Y J Lee, Heegaard splittings and Seiberg-Witten monopoles (editors U Boden Hans, I Hambleton, A J Nicas, B D Park), Fields Inst. Commun. 47, Amer. Math. Soc. (2005) 173 · Zbl 1094.57030
[28] Y J Lee, Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori. I, J. Symplectic Geom. 3 (2005) 221 · Zbl 1093.53092 · doi:10.4310/JSG.2005.v3.n3.a4 · euclid:jsg/1144954879
[29] Y J Lee, Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori. II, J. Symplectic Geom. 3 (2005) 385 · Zbl 1093.53092 · doi:10.4310/JSG.2005.v3.n3.a4 · euclid:jsg/1144954879
[30] Y J Lee, C H Taubes, Periodic Floer homology and Seiberg-Witten Floer cohomology · Zbl 1280.57029 · doi:10.4310/JSG.2012.v10.n1.a4 · euclid:jsg/1332853050
[31] G Liu, G Tian, On the equivalence of multiplicative structures in Floer homology and quantum homology, Acta Math. Sin. \((\)Engl. Ser.\()\) 15 (1999) 53 · Zbl 0928.53041 · doi:10.1007/s10114-999-0060-x
[32] D McDuff, D Salamon, Introduction to symplectic topology, Oxford Math. Monogr., The Clarendon Press, Oxford Univ. Press (1998) · Zbl 0844.58029
[33] L Mosher, Train track expansions of measured foliations, Preprint (2003)
[34] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. \((2)\) 159 (2004) 1027 · Zbl 1073.57009 · doi:10.4007/annals.2004.159.1027
[35] R C Penner, A construction of pseudo-Anosov homeomorphisms, Trans. Amer. Math. Soc. 310 (1988) 179 · Zbl 0706.57008 · doi:10.2307/2001116
[36] R C Penner, J L Harer, Combinatorics of train tracks, Annals of Math. Studies 125, Princeton Univ. Press (1992) · Zbl 0765.57001
[37] T Perutz, A hypercube for fixed-point Floer homology, Lecture (2008) · Zbl 1193.30059
[38] T Perutz, Lagrangian matching invariants for fibred four-manifolds. II, Geom. Topol. 12 (2008) 1461 · Zbl 1144.53104 · doi:10.2140/gt.2008.12.1461 · arxiv:math/0606062
[39] M Pozniak, Floer homology, Novikov rings and clean intersections, PhD thesis, University of Warwick (1994)
[40] J Robbin, D Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995) 1 · Zbl 0859.58025 · doi:10.1112/blms/27.1.1
[41] D Salamon, E Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992) 1303 · Zbl 0766.58023 · doi:10.1002/cpa.3160451004
[42] M Schwarz, Cohomology operations from \(S^1\)-cobordisms in Floer homology, PhD thesis, ETH Zürich (1995)
[43] P Seidel, The symplectic Floer homology of a Dehn twist, Math. Res. Lett. 3 (1996) 829 · Zbl 0876.57022 · doi:10.4310/MRL.1996.v3.n6.a10
[44] P Seidel, More about vanishing cycles and mutation (editors K Fukaya, Y G Oh, K Ono, G Tian), World Sci. Publ. (2001) 429 · Zbl 1079.14529
[45] P Seidel, Symplectic Floer homology and the mapping class group, Pacific J. Math. 206 (2002) 219 · Zbl 1061.53065 · doi:10.2140/pjm.2002.206.219
[46] C H Taubes, Seiberg Witten and Gromov invariants for symplectic \(4\)-manifolds, (R Wentworth, editor), First Intl. Press Lecture Ser. 2, Intl. Press (2000) · Zbl 0967.57001
[47] C H Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007) 2117 · Zbl 1135.57015 · doi:10.2140/gt.2007.11.2117 · arxiv:math/0611007
[48] C H Taubes, Embedded contact homology and Seiberg-Witten Floer cohomology I-IV, Preprint (2008) · Zbl 1276.57026 · doi:10.2140/gt.2010.14.2819
[49] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. \((\)N.S.\()\) 19 (1988) 417 · Zbl 0674.57008 · doi:10.1090/S0273-0979-1988-15685-6
[50] M Usher, Vortices and a TQFT for Lefschetz fibrations on \(4\)-manifolds, Algebr. Geom. Topol. 6 (2006) 1677 · Zbl 1131.57031 · doi:10.2140/agt.2006.6.1677 · arxiv:math/0603128
[51] M L Yau, A holomorphic \(0\)-surgery model for open books, Int. Math. Res. Not. (2007) · Zbl 1175.57021 · doi:10.1093/imrn/rnm041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.