# A homotopy decomposition and the classification of gauge groups of principal bundles over 4-manifolds

Given a principal $G$-bundle $P$ over a manifold $M$, the associated gauge group $\mathcal{G}(P)$ is the topological group of $G$-equivariant automorphisms of $P$ which fix $M$. The topology of gauge groups has a deep connection to many topics in geometry and topology, and topologists have been extensively investigating the cases where $M$ is a spin 4-manifold. This talk will be divided into three parts. First I will give a brief review on the classification of gauge groups of principal bundles over simply-connected 4-manifold. In the second part I will introduce a homotopy decomposition method which can extend the results to certain non-simply-connected 4-manifolds. Lastly, I will talk about the classification of gauge groups where $M$ is a spin 4-manifold and my recent work on the non-spin 4-manifold case.