Strand algebras and contact categories

Mathews, Daniel¹

¹ School of Mathematical Sciences, Monash University, Clayton, VIC, Australia

Geometry & topology, Tome 23 (2019) no. 2, pp. 637-683.

Résumé

We demonstrate an isomorphism between the homology of the strand algebra of bordered Floer homology, and the category algebra of the contact category introduced by Honda. This isomorphism provides a direct correspondence between various notions of Floer homology and arc diagrams, on the one hand, and contact geometry and topology on the other. In particular, arc diagrams correspond to quadrangulated surfaces, idempotents correspond to certain basic dividing sets, strand diagrams correspond to contact structures, and multiplication of strand diagrams corresponds to stacking of contact structures. The contact structures considered are cubulated, and the cubes are shown to behave equivalently to local fragments of strand diagrams.

Détail

DOI : 10.2140/gt.2019.23.637

Classification : 53D10, 57R17, 57R58, 18F99
Keywords: strand algebra, contact category, contact structures, bordered Floer homology

Bibliographie

[1] I Agol, Virtual properties of 3–manifolds, from: "Proceedings of the International Congress of Mathematicians" (editors S Y Jang, Y R Kim, D W Lee, I Ye), Kyung Moon Sa (2014) 141

[2] D Bennequin, Entrelacements et équations de Pfaff, from: "Troisième rencontre de géométrie du Schnepfenried, I", Astérisque 107, Soc. Math. France (1983) 87

[3] N Bergeron, D T Wise, A boundary criterion for cubulation, Amer. J. Math. 134 (2012) 843 | DOI

[4] D Calegari, Notes on Agol’s virtual Haken theorem, course notes (2013)

[5] V Colin, E Giroux, K Honda, On the coarse classification of tight contact structures, from: "Topology and geometry of manifolds" (editors G Matić, C McCrory), Proc. Sympos. Pure Math. 71, Amer. Math. Soc. (2003) 109 | DOI

[6] V Colin, E Giroux, K Honda, Finitude homotopique et isotopique des structures de contact tendues, Publ. Math. Inst. Hautes Études Sci. 109 (2009) 245 | DOI

[7] B Cooper, Formal contact categories, preprint (2015)

[8] Y Eliashberg, Classification of overtwisted contact structures on 3–manifolds, Invent. Math. 98 (1989) 623 | DOI

[9] Y Eliashberg, Contact 3–manifolds twenty years since J Martinet’s work, Ann. Inst. Fourier (Grenoble) 42 (1992) 165 | DOI

[10] J B Etnyre, D S Vela-Vick, R Zarev, Sutured Floer homology and invariants of Legendrian and transverse knots, Geom. Topol. 21 (2017) 1469 | DOI

[11] D Gabai, Foliations and the topology of 3–manifolds, Bull. Amer. Math. Soc. 8 (1983) 77 | DOI

[12] E Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991) 637 | DOI

[13] K Honda, Contact structures, Heegaard Floer homology and triangulated categories, in preparation

[14] K Honda, On the classification of tight contact structures, I, Geom. Topol. 4 (2000) 309 | DOI

[15] K Honda, Gluing tight contact structures, Duke Math. J. 115 (2002) 435 | DOI

[16] K Honda, W H Kazez, G Matić, Contact structures, sutured Floer homology and TQFT, preprint (2008)

[17] K Honda, W H Kazez, G Matić, The contact invariant in sutured Floer homology, Invent. Math. 176 (2009) 637 | DOI

[18] K Honda, W H Kazez, G Matić, On the contact class in Heegaard Floer homology, J. Differential Geom. 83 (2009) 289 | DOI

[19] K Honda, Y Tian, Contact categories of disks, preprint (2016)

[20] A Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006) 1429 | DOI

[21] T Kálmán, D V Mathews, Tight contact structures on Seifert surface complements, preprint (2017)

[22] L H Kauffman, Formal knot theory, 30, Princeton Univ. Press (1983)

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

[24] R Lipshitz, P S Ozsvath, D P Thurston, Bordered Heegaard Floer homology, 1216, Amer. Math. Soc. (2018) | DOI

[25] D V Mathews, On contact categories, in preparation

[26] D Mathews, Chord diagrams, contact-topological quantum field theory and contact categories, Algebr. Geom. Topol. 10 (2010) 2091 | DOI

[27] D V Mathews, Sutured Floer homology, sutured TQFT and noncommutative QFT, Algebr. Geom. Topol. 11 (2011) 2681 | DOI

[28] D V Mathews, Contact topology and holomorphic invariants via elementary combinatorics, Expo. Math. 32 (2014) 121 | DOI

[29] D V Mathews, Itsy bitsy topological field theory, Ann. Henri Poincaré 15 (2014) 1801 | DOI

[30] D V Mathews, Twisty itsy bitsy topological field theory, Internat. J. Math. 25 (2014) | DOI

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

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

[33] P Ozsváth, Z Szabó, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005) 39 | DOI

[34] P Ozsváth, Z Szabó, Heegaard diagrams and Floer homology, from: "International Congress of Mathematicians" (editors M Sanz-Solé, J Soria, J L Varona, J Verdera), Eur. Math. Soc. (2006) 1083

[35] P Webb, An introduction to the representations and cohomology of categories, from: "Group representation theory" (editors M Geck, D Testerman, J Thévenaz), EPFL (2007) 149

[36] D T Wise, From riches to raags : 3–manifolds, right-angled Artin groups, and cubical geometry, 117, Amer. Math. Soc. (2012) | DOI

[37] R Zarev, Bordered Floer homology for sutured manifolds, preprint (2009)

[38] R Zarev, Joining and gluing sutured Floer homology, preprint (2010)