The equivalence of two Seiberg-Witten Floer homologies

TL;DR

This paper proves that monopole Floer homology is mathematically equivalent to the S^1-equivariant homology of the Seiberg-Witten Floer spectrum, establishing a fundamental connection between two Floer homology theories.

Contribution

It demonstrates the isomorphism between monopole Floer homology and the S^1-equivariant homology of the Seiberg-Witten Floer spectrum, clarifying their relationship.

Findings

Monopole Floer homology is isomorphic to the S^1-equivariant homology of the Seiberg-Witten Floer spectrum.

The result unifies two different approaches to Floer homology in gauge theory.

Provides a new perspective on the structure of Floer homologies in 3-manifold topology.

Abstract

We show that monopole Floer homology (as defined by Kronheimer and Mrowka) is isomorphic to the S^1-equivariant homology of the Seiberg-Witten Floer spectrum constructed by the second author.