Homotopy types of gauge groups over high dimensional manifolds

TL;DR

This paper investigates the homotopy types of gauge groups over high-dimensional manifolds, providing decompositions and classifications using manifold topology and homotopy theory techniques.

Contribution

It offers new homotopy decompositions and classifications of gauge groups over specific high-dimensional manifolds, extending previous work in the field.

Findings

Homotopy decompositions of gauge groups over certain manifolds

Classification results for gauge groups over $(n-1)$-connected $2n$-manifolds

Analysis of gauge groups over sphere bundle total spaces

Abstract

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.