The homotopy types of SU(n)-gauge groups over \(\mathbb {C}P^3\)

Let m and n be two positive integers such that \(m \le n\) $m\le n$ and \(n \ge 3\) $n\ge 3$ . In this article, by the unstable K-theory method, we will study the homotopy types of gauge groups of the principal SU(n)-bundles over \(\mathbb {C}P^3\) $\mathbb{C}{P}^3$ . Let \(\mathcal {G}_{l,k}(\mathbb {C}P^3)\) ${\mathcal{G}}_{l,k}(\mathbb{C}{P}^3)$ be the gauge groups of the principal SU(n)-bundles over \(\mathbb {C}P^3\) $\mathbb{C}{P}^3$ , we will partially classify the homotopy types of \(\mathcal {G}_{0,k}(\mathbb {C}P^3)\) ${\mathcal{G}}_{0,k}(\mathbb{C}{P}^3)$ by showing that if there is a homotopy equivalence \(\mathcal {G}_{0,k}(\mathbb {C}P^3)\simeq \mathcal {G}_{0,k'}(\mathbb {C}P^3)\) ${\mathcal{G}}_{0,k}(\mathbb{C}{P}^3)\simeq {\mathcal{G}}_{0,k^{\prime }}(\mathbb{C}{P}^3)$ then we have \((\frac{1}{2}(n-1)n(n+1)(n + 2), k)=(\frac{1}{2}(n-1)n(n+1)(n+2), k')\) $(\frac{1}{2}(n-1)n(n+1)(n+2),k)=(\frac{1}{2}(n-1)n(n+1)(n+2),k^{\prime })$ , when n is odd and \((\frac{1}{4}(n-1)n(n+1)(n + 2), k) = (\frac{1}{4}(n-1)n(n+1)(n+2), k')\) $(\frac{1}{4}(n-1)n(n+1)(n+2),k)=(\frac{1}{4}(n-1)n(n+1)(n+2),k^{\prime })$ , when n is even.