$Pin^{-}(2)$ Bauer-Furuta invariants

TL;DR

This paper extends Bauer-Furuta's stable cohomotopy refinement to $Pin^{-}(2)$ monopole invariants, providing a new connected sum formula based on Nakamura's proposal.

Contribution

It introduces a stable cohomotopy refinement for $Pin^{-}(2)$ monopole invariants and establishes a connected sum formula, advancing the understanding of these invariants.

Findings

Established the $Pin^{-}(2)$ Bauer-Furuta invariants as a stable cohomotopy refinement.

Derived a connected sum formula for the $Pin^{-}(2)$ invariants.

Extended Bauer-Furuta's constructions to a new class of monopole invariants.

Abstract

Adapting Bauer and Furuta's constructions of the refinement of the Seiberg-Witten invariants, we establish the analogous stable cohomotopy refinement of the $Pin^{-}(2)$ monopole invariants proposed by Nakamura \cite{nakamura2015pin}, and give the corresponding connected sum formula.