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 $kinmathbb{Z}$. 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.