Floer homology groups for homology three-spheres

Floer Homology of Brieskorn Homology Spheres

Every Brieskorn homology sphere (p; q; r) is a double cover of the 3{sphere ramiied over a Montesinos knot k(p; q; r). We relate Floer homology of (p; q; r) to certain invariants of the knot k(p; q; r), among which are the knot signature and the Jones polynomial. We also deene an integer valued invariant of integral homology 3{spheres which agrees with the {invariant of W. Neu-mann and L. Siebe...

Floer Homology of Brieskorn Homology Spheres : Solution to Atiyah’s Problem

In this paper we answer the question posed by M. Atiyah, see [12], and give an explicit formula for Floer homology of Brieskorn homology spheres in terms of their branching sets over the 3–sphere. We further show how Floer homology is related to other invariants of knots and 3–manifolds, among which are the μ̄–invariant of W. Neumann and L. Siebenmann and the Jones polynomial. Essential progress...

Nonseparating spheres and twistedHeegaard Floer homology

Heegaard Floer homology was introduced by Ozsváth and Szabó [16]. For nullhomologous knots, there is a filtered version of Heegaard Floer homology, called knot Floer homology; see Ozsváth and Szabó [14] and Rasmussen [18]. Basically, if one knows the information about the knot Floer homology of a knot, then one can compute the Heegaard Floer homology of any manifold obtained by Dehn surgery on ...

Seiberg-Witten-Floer Theory for Homology 3-Spheres

We give the definition of the Seiberg-Witten-Floer homology group for a homology 3-sphere. Its Euler characteristic number is a Casson-type invariant. For a four-manifold with boundary a homology sphere, a relative Seiberg-Witten invariant is defined taking values in the Seiberg-Witten-Floer homology group, these relative Seiberg-Witten invariants are applied to certain homology spheres boundin...

Floer homology groups in Yang–Mills theory