Framed Floer Homology

TL;DR

This paper introduces Framed Floer homology, a new invariant derived from hypercube chain complexes associated with framed links, which helps understand surgeries on three-manifolds and their invariance properties.

Contribution

It establishes Framed Floer homology as a link invariant, explores its relation to Heegaard Floer homology, and develops a combinatorial approach to surgery exact triangles.

Findings

Framed Floer homology is an invariant of oriented framed links.

It satisfies a surgery exact triangle similar to Heegaard Floer homology.

Provides a combinatorial construction for surgery exact triangles.

Abstract

For any three-manifold presented as surgery on a framed link (L,\Lambda) in an integral homology sphere, Manolescu and Ozsv\'ath construct a hypercube of chain complexes whose homology calculates the Heegaard Floer homology of \Lambda-framed surgery on Y. This carries a natural filtration that exists on any hypercube of chain complexes; we study the E_2 page of the associated spectral sequence, called Framed Floer homology. One purpose of this paper is to show that Framed Floer homology is an invariant of oriented framed links, but not an invariant of the surgered manifold. We discuss how this relates to an attempt at a combinatorial proof of the invariance of Heegaard Floer homology. We also show that Framed Floer homology satisfies a surgery exact triangle analogous to that of Heegaard Floer homology. This setup leads to a completely combinatorial construction of a surgery exact triangle in Heegaard Floer homology. Finally, we study the Framed Floer homology of two-component links which surger to S^2 x S^1 # S^2 x S^1 and have Property 2R.