Mod \(p\) homology of \(F_{4}\)-gauge groups over \(S^4\) (Q2315295)

Let \(G\) be a compact connected Lie group and let \(\pi:P\to X\) be a principal \(G\)-bundle over a space \(X\). Then the gauge group of \(P\) is the topological group consisting of automorphisms \(\phi:P\to P\) which fix the base space \(X\) and preserve the \(G\)-action on \(P\), that is \(\pi(\phi(\xi))=\pi(\xi)\) and \(\phi(\xi\cdot g)=\phi(\xi)\cdot g\). If \(X=S^4\) and \(G\) is simple and 1-connected, then the isomorphism classes of principal \(G\)-bundles \(P\to S^4\) are classified by their second Chern classes \(k\in\mathbb{Z}\). Denote the gauge group of the principal \(G\)-bundle with second Chern class \(k\) by \(\mathcal{G}_k(G)\). The homotopy types of \(\mathcal{G}_k(G)\) are difficult to obtain in general. When \(G\) is an exceptional Lie group, only partial results are known. In this article the author aims at computing the mod-\(p\) homology of the \(F_4\)-gauge groups \(\mathcal{G}_k(F_4)\) for any prime \(p\). The algebraic structure of the mod-\(p\) cohomology of \(F_4\) is well understood, from which the mod-\(p\) homology of \(\Omega^4_0F_4\) can be obtained. Consequently, the author computes the mod-\(p\) homology of \(\mathcal{G}_k(F_4)\) with the Serre spectral sequence associated to the evaluation fibration \(\Omega^4_0F_4\to\mathcal{G}(F_4)\to F_4\). In Section 2 he recalls the material for latter sections, including the \(p\)-localized decomposition of \(F_4\), the mod-\(p\) homology of \(F_4\) and related spaces, and the criterion for which spectral sequences associated to path loop fibrations collapse at the \(E_2\) term. In Sections 3 and 4 he uses spectral sequences to compute \(H_*(\Omega^4_0F_4;\mathbb{F}_p)\) and \(H_*(\mathcal{G}_k(F_4);\mathbb{F}_p)\). In particular, the Serre spectral sequence collapses at the \(E_2\)-term for \(p\neq2, 13\), so \(H_*(\mathcal{G}_k(F_4);\mathbb{F}_p)\cong H_*(F_4;\mathbb{F}_p)\otimes H_*(\Omega^4_0F_4;\mathbb{F}_p)\) in this case.