Seiberg-Witten-Floer Theory for Homology 3-Spheres

TL;DR

This paper defines the Seiberg-Witten-Floer homology for homology 3-spheres, linking it to Casson invariants and applying it to four-manifolds with boundary, advancing the understanding of 3- and 4-dimensional topology.

Contribution

It introduces the Seiberg-Witten-Floer homology for homology 3-spheres and relates it to existing invariants, providing new tools for studying 4-manifolds with boundary.

Findings

Seiberg-Witten-Floer homology's Euler characteristic equals a Casson-type invariant.

Defined a relative Seiberg-Witten invariant for 4-manifolds with homology sphere boundary.

Applied invariants to homology spheres bounding Stein surfaces.

Abstract

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 bounding Stein surfaces.