Commutative cocycles and stable bundles over surfaces

TL;DR

This paper constructs explicit representatives for real commutative K-theory classes on surfaces using unstable methods and analyzes their properties through commutative cocycles and point-wise inversion.

Contribution

It introduces a new unstable approach to explicitly represent commutative K-theory classes on surfaces with commutative cocycles.

Findings

Explicit representatives for real commutative K-theory classes on surfaces.

Analysis of classes via point-wise inversion on cocycles.

Connection between cocycles and K-theory classes.

Abstract

Commutative K-theory, a cohomology theory built from spaces of commuting matrices, has been explored in recent work of Adem, G\'{o}mez, Gritschacher, Lind, and Tillman. In this article, we use unstable methods to construct explicit representatives for the real commutative K-theory classes on surfaces. These classes arise from commutative O(2)-valued cocycles, and are analyzed via the point-wise inversion operation on commutative cocycles.