Homotopy types of $SU(n)$-gauge groups over non-spin 4-manifolds

TL;DR

This paper studies the homotopy types of $SU(n)$ gauge groups over certain non-spin 4-manifolds, providing partial classifications based on properties of gauge groups over $CP^2$.

Contribution

It investigates the homotopy types of $SU(n)$ gauge groups over non-spin 4-manifolds, offering new insights into their classification.

Findings

Homotopy type of gauge groups over non-spin 4-manifolds linked to those over $CP^2$.

Partial classification results for $SU(n)$ gauge groups.

Properties of gauge groups over $CP^2$ inform the homotopy classification.

Abstract

Let $M$ be an orientable, simply-connected, closed, non-spin 4-manifold and let $\mathcal{G}_k(M)$ be the gauge group of the principal $G$-bundle over $M$ with second Chern class $k\in\mathbb{Z}$. It is known that the homotopy type of $\mathcal{G}_k(M)$ is determined by the homotopy type of $\mathcal{G}_k(\mathbb{CP}^2)$. In this paper we investigate properties of $\mathcal{G}_k(\mathbb{CP}^2)$ when $G = SU(n)$ that partly classify the homotopy types of the gauge groups.