Homotopy types of gauge groups related to S3-bundles over S4

Abstract

Let Ml,m be the total space of the S3-bundle over S4 classified by the element lσ+mρ∈π4(SO(4)), l,m∈Z. In this paper we study the homotopy theory of gauge groups of principal G-bundles over manifolds Ml,m when G is a simply connected simple compact Lie group such that π6(G)=0. That is, G is one of the following groups: SU(n)(n≥4), Sp(n)(n≥2), Spin(n)(n≥5), F4, E6, E7, E8. If the integral homology of Ml,m is torsion-free, we describe the homotopy type of the gauge groups over Ml,m as products of recognisable spaces. For any manifold Ml,m with non-torsion-free homology, we give a p-local homotopy decomposition, for a prime p≥5, of the loop space of the gauge groups.