Counting Homotopy Types of Gauge Groups

Abstract

For K a connected 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, that is, homotopy equivalences which respect multiplication up to homotopy. A case study is given for K = S4, G = SU(2): there are eighteen H-equivalence classes of gauge group 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. 1991 Mathematics Subject Classification: 54C35, 55P15, 55R10.