Lescop's Invariant and Gauge Theory

TL;DR

This paper extends the relationship between the Lescop invariant and instanton Floer homology from homology 3-spheres to all closed oriented 3-manifolds with positive first Betti number, using surgery techniques.

Contribution

It generalizes Taubes' result linking the Casson invariant and Floer homology to a broader class of 3-manifolds via the Lescop invariant.

Findings

Established a formula relating Lescop invariant to instanton Floer homology for all such 3-manifolds.

Used surgery techniques to prove the extended relationship.

Demonstrated the invariance and applicability of the formula across different manifold classes.

Abstract

Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds with positive first Betti number by establishing a similar relationship between the Lescop invariant of the manifold and its instanton Floer homology. The proof uses surgery techniques.