Cohomology of the moduli space of \(SO(n)\)-instantons with instanton number 1
For simply connected compact simple Lie groups \(G\), the equivalence classes of principal \(G\)-bundles over \(S^4\) are naturally parameterized by \({\mathbb Z}\), because of the fact that \(\pi_3(G)\simeq{\mathbb Z}\). Now pick a principal \(G\)-bundle \(P_k\) in the equivalence class corresponding to some \(k\in{\mathbb Z}\) and let \(M(k,G)\) be the moduli space of based gauge equivalence classes of \(G\)-instantons on \(P_k\). The purpose of the paper under review is to derive cohomological information on the space \(M(k,G)\) in the case when \(k=1\) and \(G=\text{SO}(n)\). More precisely, the cohomology ring with coefficients in \({\mathbb Z}/2\) with squaring operations, as well as the integral cohomology group, are determined for the space \(M(1,\text{SO}(n))\). To this end, the later space is shown to be diffeomorphic to \({\mathbb R}^5\times X_n\), where \(X_n=\text{SO}(n)/(\text{SO}(n-4)\times\text{SU}(2))\) for \(n\geq5\). The proof of this fact relies on a result due to \textit{C. P.~Boyer, B. M.~Mann} and \textit{D.~Waggoner} [Trans. Am. Math. Soc. 323, No. 2, 529--561 (1991; Zbl 0725.53065)].
scientific article; zbMATH DE number 2136484
scientific article