On the bounded cohomology of Lie groups

TL;DR

This paper investigates when integral Borel cohomology classes of connected Lie groups can be represented by bounded cocycles, linking this property to the linearity of the group's radical, and generalizing Gromov's boundedness theorem.

Contribution

It establishes a criterion connecting bounded cocycle representations to the linearity of the radical of Lie groups, extending Gromov's boundedness results.

Findings

Bounded cocycle representation is equivalent to the radical being linear.

Generalization of Gromov's boundedness theorem for characteristic classes.

Provides a criterion for boundedness in the cohomology of Lie groups.

Abstract

We show that each integral Borel cohomology class of a connected Lie group G can be represented by a Borel bounded cocycle if and only if the radical of G is linear. This leads to a generalization of Gromov's boundedness theorem on characteristic classes of flat bundles.