Characteristic classes of fiber bundles

TL;DR

This paper introduces new characteristic classes for fiber bundles using flat connections in infinite-dimensional Lie algebras, linking to known classes like Morita-Mumford-Miller classes.

Contribution

It develops a novel method to construct characteristic classes via flat connections in infinite-dimensional Lie algebras, extending the understanding of fiber bundle invariants.

Findings

Constructed characteristic classes via flat connections in infinite-dimensional Lie algebras.

Proved independence of the cohomology map from choices made in the construction.

Connected the new classes to Morita-Mumford-Miller classes for surface bundles.

Abstract

In this paper, we construct new characteristic classes of fiber bundles via flat connections with values in infinite-dimensional Lie algberas of derivations. In fact, choosing a fiberwise metric, we construct a chain map to the de Rham complex on the base space, and show that the induced map on cohomology groups is independent of the choices. Moreover, we show that applying the construction to a surface bundle, our construction gives Morita-Mumford-Miller classes.