Flat bundles with complex analytic holonomy

TL;DR

This paper characterizes when flat principal G-bundles over finite CW-complexes become trivial after finite covers, linking this to characteristic classes and properties of the Lie group G, especially for complex and compact groups.

Contribution

It establishes new equivalences between trivializability of flat G-bundles over finite covers and conditions on characteristic classes and the structure of G, extending to amenable and compact Lie groups.

Findings

Flat principal G-bundles become trivial over finite covers if characteristic classes vanish.

For connected amenable G, the same trivialization condition holds.

If G is compact, all flat G-bundles over finite CW-complexes trivialize over finite covers.

Abstract

Let G be a connected complex Lie group. We show that any flat principal G-bundle over any finite CW-complex pulls back to a trivial bundle over some finite covering space of the base space if and only if each real characteristic class of positive degree of G vanishes. A third equivalent condition is that the derived group of the radical of G is simply connected. As a corollary, the same conditions are equivalent if G is a connected amenable Lie group. In particular, if G is a connected compact Lie group then any flat principal G-bundle over any finite CW-complex pulls back to a trivial bundle over some finite covering space of the base space.