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

Abstract

Let $M_{l,m}$ be the total space of the $S^3$-bundle over $S^4$ classified by the element $l\sigma +m\rho \in {\pi }_4\left(SO\right(4\left)\right)$, $l,m\in Z$. In this paper we study the homotopy theory of gauge groups of principal G-bundles over manifolds $M_{l,m}$ when G is a simply connected simple compact Lie group such that ${\pi }_6\left(G\right)=0$. That is, G is one of the following groups: $SU\left(n\right)$ $(n\ge 4)$, $Sp\left(n\right)$ $(n\ge 2)$, $Spin\left(n\right)$ $(n\ge 5)$, $F_4$, $E_6$, $E_7$, $E_8$. If the integral homology of $M_{l,m}$ is torsion-free, we describe the homotopy type of the gauge groups over $M_{l,m}$ as products of recognisable spaces. For any manifold $M_{l,m}$ with non-torsion-free homology, we give a p-local homotopy decomposition, for a prime $p\ge 5$, of the loop space of the gauge groups.