Periodic Floer pro-spectra from the Seiberg-Witten equations

TL;DR

This paper constructs periodic pro-spectra invariants from Seiberg-Witten equations on certain three-manifolds, leading to new Floer homology theories and stable homotopy invariants for four-manifolds with boundary.

Contribution

It introduces a novel approach using finite dimensional approximation to produce periodic pro-spectra invariants, expanding the toolkit for studying three- and four-manifolds.

Findings

Constructed periodic pro-spectra from Seiberg-Witten equations.

Derived various Floer homology theories via functors applied to these invariants.

Developed stable homotopy versions of relative Seiberg-Witten invariants.

Abstract

Given a three-manifold with b_1=1 and a nontorsion spin^c structure, we use finite dimensional approximation to construct from the Seiberg-Witten equations two invariants in the form of a periodic pro-spectra. Various functors applied to these invariants give different flavors of Seiberg-Witten Floer homology. We also construct stable homotopy versions of the relative Seiberg-Witten invariants for certain four-manifolds with boundary.