Combinatorial proofs in bordered Heegaard Floer homology

Zhan, Bohua¹

¹Department of Mathematics, Massachusetts Institute of Technology, Room 2-231B, 77 Massachusetts Avenue, Cambridge, MA 02139, United States

Algebraic and Geometric Topology, Tome 16 (2016) no. 5, pp. 2571-2636

Résumé

Using bordered Floer theory, we give a combinatorial construction and proof of invariance for the hat version of Heegaard Floer homology. As part of the proof, we also establish combinatorially the invariance of the linear-categorical representation of the strongly based mapping class groupoid given by the same theory.

Détail

DOI : 10.2140/agt.2016.16.2571

Classification : 57R58, 57R56
Keywords: bordered Heegaard Floer homology

Bibliographie

[1] J E Andersen, A J Bene, R C Penner, Groupoid extensions of mapping class representations for bordered surfaces, Topology Appl. 156 (2009) 2713 | DOI

[2] A J Bene, A chord diagrammatic presentation of the mapping class group of a once bordered surface, Geom. Dedicata 144 (2010) 171 | DOI

[3] S Cautis, J Kamnitzer, Knot homology via derived categories of coherent sheaves, I : the sl(2)–case, Duke Math. J. 142 (2008) 511 | DOI

[4] B Farb, D Margalit, A primer on mapping class groups, 49, Princeton Univ. Press (2012)

[5] M Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002) 665 | DOI

[6] M Khovanov, P Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002) 203 | DOI

[7] M Khovanov, R Thomas, Braid cobordisms, triangulated categories, and flag varieties, Homology Homotopy Appl. 9 (2007) 19

[8] A S Levine, Knot doubling operators and bordered Heegaard Floer homology, J. Topol. 5 (2012) 651 | DOI

[9] R Lipshitz, P S Ozsváth, D P Thurston, A faithful linear-categorical action of the mapping class group of a surface with boundary, J. Eur. Math. Soc. 15 (2013) 1279 | DOI

[10] R Lipshitz, P Ozsváth, D P Thurston, Bordered Heegaard Floer homology: invariance and pairing, preprint (2014)

[11] R Lipshitz, P S Ozsváth, D P Thurston, Computing HF by factoring mapping classes, Geom. Topol. 18 (2014) 2547 | DOI

[12] R Lipshitz, P S Ozsváth, D P Thurston, Bimodules in bordered Heegaard Floer homology, Geom. Topol. 19 (2015) 525 | DOI

[13] P Ozsváth, A I Stipsicz, Z Szabó, Combinatorial Heegaard Floer homology and nice Heegaard diagrams, Adv. Math. 231 (2012) 102 | DOI

[14] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159 | DOI

[15] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027 | DOI

[16] S Sarkar, J Wang, An algorithm for computing some Heegaard Floer homologies, Ann. of Math. 171 (2010) 1213 | DOI

[17] P Seidel, I Smith, A link invariant from the symplectic geometry of nilpotent slices, Duke Math. J. 134 (2006) 453 | DOI

[18] P Seidel, R Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001) 37 | DOI

[19] K Siegel, A geometric proof of a faithful linear-categorical surface mapping class group action, (2011)

[20] S Suzuki, On homeomorphisms of a 3–dimensional handlebody, Canad. J. Math. 29 (1977) 111

[21] B Zhan, Explicit Koszul-dualizing bimodules in bordered Heegaard Floer homology, Algebr. Geom. Topol. 16 (2016) 231 | DOI