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 $$S^4$$ S 4 with $${{\textrm{Spin}}}^{{\textrm{c}}}(n)$$ Spin c ( n ) , the complex spin group, as structure group and show how the study of their homotopy types reduces to that of $${{\textrm{Spin}}}(n)$$ Spin ( n ) -gauge groups over $$S^4$$ S 4 . We then advance on what is known by providing a partial classification for $${{\textrm{Spin}}}(7)$$ Spin ( 7 ) - and $${{\textrm{Spin}}}(8)$$ Spin ( 8 ) -gauge groups over $$S^4$$ S 4 .