Holomorphic discs and sutured manifolds

Abstract

In this paper we construct a Floer-homology invariant for a natural and wide class of sutured manifolds that we call balanced. This generalizes the Heegaard Floer hat theory of closed three-manifolds and links. Our invariant is unchanged under product decompositions and is zero for nontaut sutured manifolds. As an application, an invariant of Seifert surfaces is given and is computed in a few interesting cases. 57M27, 57R58

The sutured Floer homology polytope

“…In [13], I introduced sutured Floer homology, in short SFH, which is an invariant of balanced sutured manifolds. SFH is an invariant of three-manifolds with boundary and generalizes b HF ; b HFK and b…”

Sutured Floer homology distinguishes between Seifert surfaces

“…The notion of a sutured manifold (M, γ) was first defined by Gabai [Ga83]. Here we give a less general definition that is suited to thinking about a particular class of so-called balanced sutured manifolds defined by Juhász [Ju06]. Definition 1.…”

“…Every balanced sutured manifold (M, γ) has an associated space of relative Spin c structures Spin c (M, γ); we define relative Spin c structures in the following paragraph. For each s ∈ Spin c (M, γ) there is a well-defined abelian group SF H(M, γ, s) [Ju06], and the direct sum of these groups forms the sutured Floer homology of (M, γ). That is,…”

“…The following definition of relative Spin c structures originates from Turaev's work [Tu90], but in the current phrasing comes from [Ju06].…”

“…Given a Seifert surface R, the complement S 3 (R) := S 3 \ Int(R × I) together with the curve ∂R × {1/2} on the boundary is a type of 3-manifold called a balanced sutured manifold. Therefore, it is reasonable to investigate the possibility of using sutured Floer homology, an invariant of balanced sutured manifolds introduced by Juhász [Ju06], to distinguish between equivalence classes of Seifert surfaces.…”