Tautological characteristic classes I

TL;DR

This paper explores the formalism of tautological characteristic classes for flat bundles, connecting them to known classes like the Witt class and Euler class across different groups and fields.

Contribution

It introduces a formalism for tautological characteristic classes of flat bundles and applies it to various groups, generalizing existing classes such as the Euler class.

Findings

Connected tautological classes to the Witt class of Nekovar.

Generalized the Euler class for positive determinant groups over ordered fields.

Provided a unified framework for characteristic classes of flat bundles.

Abstract

We discuss the formalism of tautological characteristic classes of flat bundles. Applied to $PSL(2,K)$ it yields the Witt class of Nekovar. Applied to $PGL_+(2n,K)$, the general linear groups with positive determinant over an arbitrary ordered field, it yields (a generalization of) the Euler class.