Secondary theories for simplicial manifolds and classifying spaces

TL;DR

This paper introduces secondary theories and characteristic classes for simplicial smooth manifolds, extending existing concepts like multiplicative K-theory and differential characters to a broader class of geometric objects.

Contribution

It generalizes Karoubi's multiplicative K-theory and differential characters to simplicial smooth manifolds, including classifying spaces of Lie groups and groupoids.

Findings

Defined secondary theories for simplicial manifolds

Extended differential characters to new geometric contexts

Provided examples involving classifying spaces

Abstract

We define secondary theories and characteristic classes for simplicial smooth manifolds generalizing Karoubi's multiplicative K-theory and multiplicative cohomology groups for smooth manifolds. As a special case we get versions of the groups of differential characters of Cheeger and Simons for simplicial smooth manifolds. Special examples include classifying spaces of Lie groups and Lie groupoids.