Homotopy types of gauge groups over high dimensional manifolds

The homotopy theory of gauge groups received considerable attentions in the recent decades, in the theme of which, the works mainly focus on bundles over $4$-dimensional manifolds and vary the structure groups case by case. In this work, we study the homotopy theory of gauge groups over higher dimensional manifolds with mild restrictions on the structure groups of principal bundles. In particular, we study gauge groups of bundles over $(n-1)$-connected closed $2n$-manifolds, the classification of which was determined by Wall and Freedman. We also investigate the gauge groups of the total spaces 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 of combinations of manifold topology and various techniques in homotopy theory.