An Elementary Differential Extension of Odd K-theory

TL;DR

This paper constructs a differential extension of odd K-theory using a Chern-Simons form-based equivalence relation on smooth maps into the stable unitary group, establishing its functoriality and exact sequences.

Contribution

It introduces a new differential extension of odd K-theory based on an equivalence relation involving Chern-Simons forms, with proofs of its functoriality and compatibility with existing theories.

Findings

Defines an equivalence relation using Chern-Simons forms

Establishes the abelian group structure under block sum

Proves the functoriality and exact sequences involving K-theory

Abstract

There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory and differential forms. To prove this we obtain along the way several results concerning even and odd Chern and Chern-Simons forms.