The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high-dimensional manifolds. To be more specific, we study gauge groups of bundles over (n − 1)-connected closed 2n-manifolds, the classification of which was determined by Wall and Freedman in the combinatorial category. We also investigate the gauge groups of the total manifolds of sphere bundles based on the classical work of James and Whitehead. Furthermore, other types of 2n-manifolds are also considered. In all the cases, we show various homotopy decompositions of gauge groups. The methods are combinations of manifold topology and various techniques in homotopy theory.

Let P be a principal bundle with semisimple compact simply connected structure group G over a compact simply connected four-manifold M. In this paper we give explicit formulas for the rational homotopy groups and cohomology algebra of the gauge group and of the space of (irreducible) connections modulo gauge transformations for any such bundle.

For $K$ a connected finite complex and $G$ a compact connected Lie group, a finiteness result is proved for gauge groups ${\mathcal G}(P)$ of principal $G$-bundles $P$ over $K$: as $P$ ranges over all principal $G$-bundles with base $K$, the number of homotopy types of ${\mathcal G}(P)$ 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 $K = S^4$, $G = \text{SU}(2)$: 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.