The homotopy types of \(Sp(n)\)-gauge groups over \(\mathbb{C}P^2\) (Q6039229)

For a simply connected closed four dimensional manifold \(M\) and a simply connected simple compact Lie group \(G\), let \(P_k\to M\) denote the equivalence class of principal \(G\)-bundles over \(M\) whose second Chern class is \(k\), and we denote by \(\mathcal{G}_k(M)\) the gauge group of \(P_k\). Recall (from results due to Theriault and So) that there is a homotopy equivalence \[ \mathcal{G}_k(M)\simeq \begin{cases} \mathcal{G}_k(S^4)\times (\Omega^2G)^t &\text{ if }M\text{ is a spin 4-manifold}, \\ \mathcal{G}_k(\mathbb{C}P^2)\times (\Omega^2G)^{t-1} & \text{ if }M\text{ is a non-spin 4-manifold}, \end{cases} \] where \(t\) is the second Betti number of \(M\). So it suffices to classify the homotopy type of \(\mathcal{G}_k(X)\) \((X=S^4\text{ or }\mathbb{C}P^2\)) for studying the homotopy classification problem of \(\mathcal{G}_k(M)\). For two integers \(a\) and \(b\), let \((a,b)\) denote the greatest common divisor of \(a\) and \(b\), and let \(Q_2=S^3\cup e^7\) denote the symplectic quasi-projective space of \(Sp(2)\). In this article the author investigates the homotopy type of the space \(\mathcal{G}_k(\mathbb{C}P^2)\) for the case \(G=Sp(n)\) with \(n>2\). In particular, he shows that the equality \((k,4n(2n+1))=(k_1,4n(2n+1))\) holds if there is a homotopy equivalence \(\mathcal{G}_k(\mathbb{C}P^2)\simeq \mathcal{G}_{k_1}(\mathbb{C}P^2)\). To prove this, he shows that there is an isomorphism \([\Sigma^{4n-8}Q_2,B\mathcal{G}_k(\mathbb{C}P^2)]\cong \mathbb{Z}/(k,4n(2n+1))\) by using unstable \(K\)-theory.