Unfolded Seiberg-Witten Floer spectra, II: Relative invariants and the gluing theorem

TL;DR

This paper extends the unfolded Seiberg-Witten Floer spectra to define relative invariants for 4-manifolds with boundary and provides a detailed proof of the gluing theorem for these invariants.

Contribution

It introduces a new framework for relative invariants in 4-manifold topology using unfolded Seiberg-Witten Floer spectra and proves the gluing theorem in this context.

Findings

Defined relative Bauer-Furuta invariants for 4-manifolds with boundary

Proved the gluing theorem for these invariants

Extended the spectral construction to general 3-manifolds

Abstract

We use the construction of unfolded Seiberg-Witten Floer spectra of general 3-manifolds defined in our previous paper to extend the notion of relative Bauer-Furuta invariants to general 4-manifolds with boundary. One of the main purposes of this paper is to give a detailed proof of the gluing theorem for the relative invariants.