Gauge groups and classification of bundles with simple structural group

Abstract

Suppose 7, = l,2 are principal K-bundles which are C r -isomorphic in the sense that there exists a K-equvariant C r -diffeomorphism /:0\-^ If h belongs to the gauge group H 2 of ^2 then h f lies in Hi and we have a group isomorphism H 2 -+ H which is C. It is the purpose of this paper to investigate the converse in the case where K is a simple Lie group. (If K is abelian the gauge group of every K bundle over X is C r (X, K) so there is no hope of a converse. However for simple groups the situation is much better). 0. Introduction. Let K be a compact connected Lie group with Lie algebra JC. Let : ^-^Xbea principal IC-bundle of class C 00 where X is a compact, connected C-manifold.