Cohomology of the classifying spaces of \(U(n)\)-gauge groups over the 2-sphere (Q2214736)

Let \(P_{n,k}\) be a principal \(U(n)\)-bundle over \(S^2\) such that \(c_1(P_{n,k})=k\in\mathbb{Z}\). Then the gauge group \(\mathcal{G}(P_{n,k})\) is the topological group of \(U(n)\)-equivariant automorphisms of \(P_{n,k}\) fixing \(S^2\). According to [\textit{M. F. Atiyah} and \textit{R. Bott}, Philos. Trans. R. Soc. Lond., Ser. A 308, 523--615 (1983; Zbl 0509.14014)] and [\textit{D. H. Gottlieb}, Trans. Am. Math. Soc. 171, 23--50 (1972; Zbl 0251.55018)], the classifying space of \(\mathcal{G}(P_{n,k})\) is homotopy equivalent to \(\text{Map}(S^2,BU(n);k)\), the connected component of \(\text{Map}(S^2,BU(n))\) that contains the the degree \(k\) map \(S^2\to BU(n)\). In this paper the author calculates the cohomology ring \(H^*(B\mathcal{G}(P_{n,k}))\cong H^*(\text{Map}(S^2,BU(n);k))\). In Section 2, the author introduces the free double suspension \(\hat{\sigma}^2_k:H^n(X)\to H^{n-2}(\text{Map}(S^2,X;k))\), which generalizes the suspension operation \(\sigma:H^n(X)\to H^{n-1}(\Omega X)\). Let \(\phi\) be the composition \[ \phi:\text{Map}(S^2,BU(n);k)\to\text{Map}(S^2,BU(\infty);k)\to\Omega^2_kBU(\infty)\to\Omega^2_0BU(\infty)\to BU(\infty) \] where the first map is induced by inclusion \(BU(n)\to BU(\infty)\), the second map is a retraction, the third map is induced by the concatenation with the degree \(-k\) map \(S^2\to BU(\infty)\), and the last map is the Bott periodicity homotopy equivalence. Let \(x_i=\phi^*(c_i)\) be the image of the \(i^{\text{th}}\) Chern class of the universal bundle. In Section 3, the author applies the Leray-Hirsch spectral sequence to the evaluation fibration \(\Omega^2_k(BU(n))\to\text{Map}(S^2,BU(n);k)\overset{ev}{\to}BU(n)\) and uses properties of \(\hat{\sigma}^2_f\) to show that there is a ring isomorphism \[ \mathbb{Z}[c_1,\dots,c_n,x_1,x_2,\dots]/(h_n,h_{n+1},\dots)\to H^*(\text{Map}(S^2,BU(n);k)), \] where \(h_i=kc_i+\sum_{1\leq j\leq i}(-1)^js_j(x_1,\dots,x_j)c_{i-j}\) and \(s_j(\sigma_1,\dots,\sigma_j)=\sum^n_{i=1}t^j_i\) is the \(j^{\text{th}}\) Newton polynomial. Moreover, he constructs a virtual bundle \(\zeta\in K(B\mathcal{G}(P_{n,k}))\) such that \(c_i(\zeta)=x_i\).