Homotopy types of gauge groups related to $S^3$-bundles over $S^4$

TL;DR

This paper investigates the homotopy types of gauge groups over certain $S^3$-bundles on $S^4$, providing explicit descriptions for torsion-free cases and $p$-local decompositions for more complex cases.

Contribution

It offers a comprehensive analysis of gauge groups over $S^3$-bundles on $S^4$, including explicit homotopy decompositions for various cases and Lie groups.

Findings

Homotopy types of gauge groups are described as products of recognizable spaces for torsion-free homology.

A $p$-local homotopy decomposition of the loop space of gauge groups is provided for non-torsion-free cases.

Results apply to a wide class of simply connected simple compact Lie groups.

Abstract

Let $M_{l,m}$ be the total space of the $S^3$-bundle over $S^4$ classified by the element $l\sigma+m\rho\in{\pi_4(SO(4))}$, $l,m\in\mathbb Z$. 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 $\pi_6(G)=0$. That is, $G$ is one of the following groups: $SU(n)$ $(n\geq4)$, $Sp(n)$ $(n\geq2)$, $Spin(n)$ $(n\geq5)$, $F_4$, $E_6$, $E_7$, $E_8$. 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 $p\geq 5$, of the loop space of the gauge groups.