# zbMATH — the first resource for mathematics

Counting homotopy types of gauge groups. (English) Zbl 1024.55005
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.
Reviewer: W.A.Sutherland

##### MSC:
 55P15 Classification of homotopy type 55R10 Fiber bundles in algebraic topology
##### Keywords:
gauge groups; homotopy
Full Text: