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.