Homotopy types of {{textrm{Spin}}}^{{textrm{c}}}(n)-gauge groups over S^4

Abstract

The gauge group of a principal G-bundle P over a space X is the group of G-equivariant homeomorphisms of P that cover the identity on X. We consider the gauge groups of bundles over S^4 with {{textrm{Spin}}}^{{textrm{c}}}(n), the complex spin group, as structure group and show how the study of their homotopy types reduces to that of {{textrm{Spin}}}(n)-gauge groups over S^4. We then advance on what is known by providing a partial classification for {{textrm{Spin}}}(7)- and {{textrm{Spin}}}(8)-gauge groups over S^4.