Counting homotopy types of gauge groups (Q2766381)

For \(K\) a finite complex and \(G\) a compact connected Lie group, a finiteness result is proved for gauge groups \(G(P)\) of principal \(G\)-bundles \(P\) over \(K\): as \(P\) ranges over all principal \(G\)-bundles with base \(K\), the number of homotopy types of \(G(P)\) is finite; indeed this remains true when these gauge groups are classified by \(H\)-equivalence, i.e. homotopy equivalences which respect multiplication up to homotopy. A case study is given for \(K= S^4\), \(G= SU(2)\): there are eighteen \(H\)-equivalence classes of gauge groups in this case. These questions are studied via fibre homotopy theory of bundles of groups; the calculations in the case study involve \(K\)-theories and \(e\)-invariants.