Homotopy decompositions of gauge groups over real surfaces

Abstract

We analyse the homotopy types of gauge groups of principal $U\left(n\right)$–bundles associated to pseudoreal vector bundles in the sense of Atiyah. We provide satisfactory homotopy decompositions of these gauge groups into factors in which the homotopy groups are well known. Therefore, we substantially build upon the low-dimensional homotopy groups as provided by Biswas, Huisman and Hurtubise.