Spin structures and the divisibility of Euler classes

TL;DR

This paper explores the geometric interpretation of divisibility of Euler classes in KO-theory for spin modules, motivated by inequalities in 4-manifold topology and connections to Seiberg-Witten theory.

Contribution

It provides a new geometric perspective on Euler class divisibility in KO-theory for spin modules, linking it to monopole equations and 4-manifold inequalities.

Findings

Clarifies the role of reducibles in Seiberg-Witten monopole equations

Provides a geometric interpretation of Euler class divisibility

Connects KO-theory divisibility to 4-manifold inequalities

Abstract

In this short article we give a geometric meaning of the divisibility of $KO$-theoretical Euler classes for given two spin modules. We are motivated by Furuta's 10/8-inequality for a closed spin $4$-manifold. The role of the reducibles is clarified in the monopole equations of Seiberg-Witten theory, as done by Donaldson and Taubes in Yang-Mills theory.