Dimensionally reduced sutured Floer homology as a string homology

We show that the sutured Floer homology of a sutured $3$ –manifold of the form $(D^{2} \times S^{1}, F \times S^{1})$ can be expressed as the homology of a string-type complex, generated by certain sets of curves on $(D^{2}, F)$ 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.

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Keywords

string homology, sutures, Floer homology

Mathematical Subject Classification 2010

Primary: 57M50

Secondary: 57R58, 57M27

References

Publication

Received: 9 March 2013

Revised: 13 November 2014

Accepted: 18 November 2014

Published: 22 April 2015

Authors

Daniel V Mathews
School of Mathematical Sciences Monash University Building 28, room 401 Clayton VIC 3800 Australia

Eric Schoenfeld
Department of Mathematics Michigan State University 619 Red Cedar Road East Lansing, MI 48824 USA

Volume 15, issue 2 (2015)

Daniel V Mathews and Eric Schoenfeld

Abstract