Dimensionally-reduced sutured Floer homology as a string homology

TL;DR

This paper demonstrates that sutured Floer homology for certain 3-manifolds can be represented as the homology of a string complex, linking Floer invariants with string topology through explicit algebraic and topological constructions.

Contribution

It introduces a novel string homology model for sutured Floer homology of specific 3-manifolds, extending the understanding of Floer invariants via surface curve algebra.

Findings

Sutured Floer homology expressed as string complex homology

Generalizations to 'hat' and 'infinity' versions of string homology

Connections established between Floer invariants and string topology

Abstract

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.