A refinement of sutured Floer homology

We introduce a refinement of the Ozsvath-Szabo complex associated to a balanced sutured manifold (X,τ) by Juhasz. An algebra A_τ is associated to the boundary of a sutured manifold and a filtration of its generators by H²(X,∂ X;) 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,τ,s) is then defined as a chain complex with coefficients in A_τ and filtered by (X,τ). The filtered chain homotopy type of this chain complex is an invariant of (X,τ) and the Spin^c class s∈(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.