Seiberg-Witten Floer spectra for $b_1>0$

TL;DR

This paper generalizes the Seiberg-Witten Floer spectrum to three-manifolds with positive first Betti number, introduces Bauer-Furuta maps for cobordisms, and provides example computations.

Contribution

It extends Floer spectrum constructions to a broader class of three-manifolds and defines new maps for cobordisms, with explicit example calculations.

Findings

Successfully constructed generalized Floer spectra for $b_1>0$

Defined Bauer-Furuta maps on these spectra for cobordisms

Provided example computations demonstrating the theory

Abstract

We construct a generalization of the Seiberg-Witten Floer spectrum for suitable three-manifolds $Y$ with $b_1(Y)>0$. For a cobordism between three-manifolds we define Bauer-Furuta maps on these new spectra, and additionally compute some examples.