Algebraic & Geometric Topology

Dimensionally reduced sutured Floer homology as a string homology

Daniel V Mathews and Eric Schoenfeld

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Abstract

We show that the sutured Floer homology of a sutured 3–manifold of the form (D2 × S1,F × S1) can be expressed as the homology of a string-type complex, generated by certain sets of curves on (D2,F) and with a differential given by resolving crossings. We also give some generalisations of this isomorphism, computing “hat” and “infinity” versions of this string homology. In addition to giving interesting elementary facts about the algebra of curves on surfaces, these isomorphisms are inspired by, and establish further, connections between invariants from Floer homology and string topology.

Article information

Source
Algebr. Geom. Topol., Volume 15, Number 2 (2015), 691-731.

Dates
Received: 9 March 2013
Revised: 13 November 2014
Accepted: 18 November 2014
First available in Project Euclid: 28 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1511895786

Digital Object Identifier
doi:10.2140/agt.2015.15.691

Mathematical Reviews number (MathSciNet)
MR3342673

Zentralblatt MATH identifier
1330.57026

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57R58: Floer homology 57M27: Invariants of knots and 3-manifolds

Keywords
string homology sutures Floer homology

Citation

Mathews, Daniel V; Schoenfeld, Eric. Dimensionally reduced sutured Floer homology as a string homology. Algebr. Geom. Topol. 15 (2015), no. 2, 691--731. doi:10.2140/agt.2015.15.691. https://projecteuclid.org/euclid.agt/1511895786


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References

  • M Chas, D Sullivan, String topology (2008)
  • K Cieliebak, J Latschev, The role of string topology in symplectic field theory, from: “New perspectives and challenges in symplectic field theory”, CRM Proc. Lecture Notes 49, Amer. Math. Soc. (2009) 113–146
  • V Colin, P Ghiggini, K Honda, Embedded contact homology and open book decompositions (2010)
  • V Colin, P Ghiggini, K Honda, Equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions, Proc. Natl. Acad. Sci. USA 108 (2011) 8100–8105
  • V Colin, P Ghiggini, K Honda, The equivalence of Heegaard–Floer homology and embedded contact homology, III: From hat to plus (2012)
  • V Colin, P Ghiggini, K Honda, The equivalence of Heegaard–Floer homology and embedded contact homology via open book decompositions, I (2012)
  • V Colin, P Ghiggini, K Honda, The equivalence of Heegaard–Floer homology and embedded contact homology via open book decompositions, II (2012)
  • V Colin, P Ghiggini, K Honda, M Hutchings, Sutures and contact homology I, Geom. Topol. 15 (2011) 1749–1842
  • Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560–673 GAFA 2000 (Tel Aviv, 1999)
  • W,M Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85 (1986) 263–302
  • R Golovko, The embedded contact homology of sutured solid tori, Algebr. Geom. Topol. 11 (2011) 1001–1031
  • K Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000) 309–368
  • K Honda, W,H Kazez, G Matić, Right-veering diffeomorphisms of compact surfaces with boundary II, Geom. Topol. 12 (2008) 2057–2094
  • M Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc. $($JEMS$)$ 4 (2002) 313–361
  • A Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006) 1429–1457
  • A Juhász, Floer homology and surface decompositions, Geom. Topol. 12 (2008) 299–350
  • P Kronheimer, T Mrowka, Monopoles and three-manifolds, New Mathematical Monographs 10, Cambridge Univ. Press, Cambridge (2007)
  • C Kutluhan, Y-J Lee, C,H Taubes, ${HF}={HM}$ I: Heegaard–Floer homology and Seiberg–Witten–Floer homology (2010)
  • C Kutluhan, Y-J Lee, C,H Taubes, ${HF}={HM}$ II: Reeb orbits and holomorpic curves for the ech/Heegaard–Floer correspondence (2010)
  • C Kutluhan, Y-J Lee, C,H Taubes, ${HF}={HM}$ III: Holomorphic curves and the differential for the ech/Heegaard–Floer correspondence (2010)
  • C Kutluhan, Y-J Lee, C,H Taubes, ${HF}={HM}$ IV: The Seiberg–Witten–Floer homology and ech correspondence (2011)
  • C Kutluhan, Y-J Lee, C,H Taubes, ${HF}={HM}$ V: Seiberg–Witten–Floer homology and handle additions (2012)
  • R Lipshitz, P Ozsvath, D Thurston, Bordered Heegaard–Floer homology: invariance and pairing (2008)
  • D,V Mathews, Chord diagrams, contact-topological quantum field theory and contact categories, Algebr. Geom. Topol. 10 (2010) 2091–2189
  • D,V Mathews, Sutured Floer homology, sutured TQFT and noncommutative QFT, Algebr. Geom. Topol. 11 (2011) 2681–2739
  • D,V Mathews, Sutured TQFT, torsion and tori, Internat. J. Math. 24 (2013) 1350039, 35
  • D,V Mathews, Itsy bitsy topological field theory, Ann. Henri Poincaré 15 (2014) 1801–1865
  • P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159–1245
  • P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027–1158
  • E Schoenfeld, Higher symplectic field theory invariants for cotangent bundles of surfaces, PhD thesis, Stanford University, Ann Arbor, MI (2009) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/305000149 {\unhbox0
  • V,G Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. École Norm. Sup. 24 (1991) 635–704
  • H Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937) 276–284
  • R Zarev, Bordered Floer homology for sutured manifolds (2009)