Homotopy types of gauge groups over non-simply-connected closed 4-manifolds

TL;DR

This paper determines the homotopy types of gauge groups over certain non-simply-connected 4-manifolds, extending understanding of gauge theory in topology for manifolds with specific fundamental groups.

Contribution

It computes the homotopy types of gauge groups over non-simply-connected 4-manifolds with particular fundamental groups, a novel extension in the topology of gauge groups.

Findings

Homotopy type of the suspension of M calculated.

Homotopy types of gauge groups over specified fundamental groups determined.

Results applicable to gauge groups over manifolds with fundamental groups involving free and cyclic components.

Abstract

Let $G$ be a simply-connected simple compact Lie group and let $M$ be an orientable smooth closed 4-manifold. In this paper we calculate the homotopy type of the suspension of $M$ and the homotopy types of the gauge groups of principal $G$-bundles over $M$ when $\pi_1(M)$ is: (1)~$\mathbb{Z}^{*m}$, (2)~$\mathbb{Z}/p^r\mathbb{Z}$, or (3)~$\mathbb{Z}^{*m}*(*^n_{j=1}\mathbb{Z}/p_j^{r_j}\mathbb{Z})$, where $p$ and the $p_j$'s are odd primes.