Holomorphic disks and topological invariants for closed three-manifolds

Abstract

The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y , equipped with a Spin c structure. Given a Heegaard splitting of Y = U 0 ∪ Σ U 1 , these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of Σ relative to certain totally real subspaces associated to U 0 and U 1 .

Floer homology and singular knots

“…Definition of HFS(K) and HFS(K). Suppose now that K ⊂ S 3 is a singular knot and (Σ, α, β, w, z) is a balanced Heegaard diagram compatible with K. As usual (compare [8], [10]), we consider the (g + ℓ)-fold symmetric power Sym g+ℓ (Σ) of the genus-g surface Σ, equipped with the tori…”

The sutured Floer homology polytope

“…Heegaard Floer homology is an invariant of closed oriented three-manifolds defined by Ozsváth and Szabó in [19]. It comes in four different flavors: b HF , HF C , HF and HF 1 .…”

Heegaard Diagrams and Holomorphic Disks

“…The material in Sections 2.1-2.4 is derived from [72]. The material from Subsection 2.5 is an account of the material starting in Section 8 if [72] and continued in [73]. The material from Subsection 2.6 can be found in [78], see also [88].…”

“…In Subsection 2.5, we sketch the construction of the maps induced by cobordisms, and in Subsection 2.6 we give preliminaries on the construction of the invariants for knots in S 3 . The material in Sections 2.1-2.4 is derived from [72]. The material from Subsection 2.5 is an account of the material starting in Section 8 if [72] and continued in [73].…”

“…In attempting to formulate an answer to this question, we came upon a construction which has, as its starting point a Heegaard diagram (Σ, α, β) for a three-manifold Y [72]. That is, Σ is an oriented surface of genus g, and α = {α 1 , ...α g } and β = {β 1 , ...β g } are a pair of g-tuples of embedded, homologically linearly independent, mutually disjoint, closed curves.…”