A refinement of sutured Floer homology

TL;DR

This paper introduces a refined sutured Floer homology complex that generalizes existing knot and link invariants, providing a more detailed invariant for balanced sutured manifolds with applications to three-manifold topology.

Contribution

It develops a new chain complex invariant for sutured manifolds that extends Ozsvath-Szabo and Rasmussen invariants, incorporating algebraic and filtration structures.

Findings

The invariant generalizes knot and link Floer homologies.

Basic properties like exact triangles are established.

The construction is stable under certain topological operations.

Abstract

We introduce a refinement of the Ozsvath-Szabo complex associated to a balanced sutured manifold $(X,\tau)$ by Juhasz. An algebra $A_\tau$ is associated to the boundary of a sutured manifold and a filtration of its generators by $H^2(X,\partial X;\Z)$ is defined. For a fixed Spin^c structure $s$ over the manifold $X'$, which is obtained from $X$ by filling out the sutures, the Ozsvath-Szabo chain complex $CF(X,\tau,s)$ is then defined as a chain complex with coefficients in $A_\tau$ and filtered by $\SpinC(X,\tau)$. The filtered chain homotopy type of this chain complex is an invariant of $(X,\tau)$ and the Spin^c class $s\in\SpinC(X')$. The construction generalizes the construction of Juhasz. It plays the role of $CF^-(X,s)$ when $X$ is a closed three-manifold, and the role of $CFK^-(Y,K;s)$ when the sutured manifold is obtained from a knot $K$ inside a three-manifold $Y$. Our invariants generalize both the knot invariants of Ozsvath-Szabo and Rasmussen and the link invariants of Ozsvath and Szabo. We study some of the basic properties of the corresponding Ozsvath-Szabo complex, including the exact triangles, and some form of stabilization.