Cohomology of the moduli space of \(SO(n)\)-instantons with instanton number 1
type: Item
instance of: scholarly article
title: Cohomology of the moduli space of \(SO(n)\)-instantons with instanton number 1
zbMATH Open document ID: 1066.57036
author: Yasuhiko Kamiyama
author: Akira Kono
author: Michishige Tezuka
published in: Topology and its Applications
publication date: 2005-02-22T00:00:00Z
review text: 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)].
reviewed by: Daniel Beltiţă
Mathematics Subject Classification ID: 57T15
Mathematics Subject Classification ID: 58D27
zbMATH DE Number: 2136484
zbMATH Keywords: moduli space
zbMATH Keywords: squaring operations
zbMATH Keywords: homogeneous space
MaRDI profile type: Publication
full work available at URL: j.topol.2003.08.031
OpenAlex ID: W2069119140
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DOI: 10.1016/J.TOPOL.2003.08.031
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description: scientific article; zbMATH DE number 2136484