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

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 ${\mathcal{G}}_k({\mathbb{C}\mathbb{P}}^2)$. In this paper we investigate properties of ${\mathcal{G}}_k({\mathbb{C}\mathbb{P}}^2)$ when $G=SU(n)$ that partly classify the homotopy types of the gauge groups.