The homotopy types of \(\mathrm{SO}(4)\)-gauge groups
type: Item
instance of: scholarly article
title: The homotopy types of \(\mathrm{SO}(4)\)-gauge groups
zbMATH Open document ID: 1471.55011
author: Daisuke Kishimoto
author: Ingrid Membrillo-Solis
author: Stephen Theriault
published in: European Journal of Mathematics
publication date: 2021-08-24T00:00:00Z
review text: Let \(Z\) denote the subgroup of \((S^3)^n\) generated by an element \((-1,\dots ,-1)\) and define the topological group \(K_n\) by the quotient \(K_n=(S^3)^n/Z\). Note that \(K_1=\mathrm{SO}(3)\) and \(K_2=\mathrm{SO}(4)\). For each \(1\leq i\leq n\), let \(\epsilon_i\in \pi_3(K_n)\) denote the homotopy class of the map \(p\circ\tilde{\iota}_i\), where \(\tilde{\iota}_i:S^3\to (S^3)^n\) is the \(i\)-th inclusion and \(p:(S^3)^n\to K_n\) denotes the projection map. For each \(n\)-tuple \((k_1,\dots ,k_n)\in \mathbb{Z}^n\), let \(\mathcal{G}_{k_1,\dots ,k_n}\) denote the gauge group of a principal \(K_n\)-bundle over \(S^4\) classified by the characteristic element \(\sum_{i=1}^nk_i\epsilon_i\in \pi_2(K_n)\). Let \(\langle a_1,\dots ,a_n\rangle\) denote the multiset consisting of elements \(a_1,\dots,a_n\), and \((a,b)\) denote the greatest common divisor of integers \(a\) and \(b\). In this paper, the authors consider the classification problem of homotopy types of gauge groups \(\{\mathcal{G}_{k_1,\dots ,k_n}:k_1,\dots ,k_n\in \mathbb{Z}\}\) and they prove the following three results: \begin{itemize} \item[(i)] If \(\langle (k_1,12),\dots ,(k_n,12)\rangle = \langle (l_1,12),\dots ,(l_n,12)\rangle\), there is a homotopy equivalence \(\mathcal{G}_{k_1,\dots ,k_n}\simeq \mathcal{G}_{l_1,\dots ,l_n}\). \item[(ii)] If there is a homotopy equivalence \(\mathcal{G}_{k_1,\dots ,k_n}\simeq \mathcal{G}_{l_1,\dots ,l_n}\), then the equality \(\langle (k_1,m),\dots ,(k_n,m)\rangle = \langle (l_1,m),\dots ,(l_n,m)\rangle\) holds for \(m\in \{3,4\}\). \item[(iii)] Let \(p\geq 2\) be a prime. Then there is a \(p\)-local homotopy equivalence \(\mathcal{G}_{k_1,\dots ,k_n}\simeq_p \mathcal{G}_{l_1,\dots ,l_n}\) if and only if the equality \[ \langle \nu_p((k_1,12)),\dots,\nu_p((k_n,12))\rangle = \langle \nu_p((l_1,12)),\dots ,\nu_p((l_n,12))\rangle \] holds. \end{itemize} Recall that there is a homotopy equivalence \[ B\mathcal{G}_{k_1,\dots ,k_n}\simeq \mathrm{map}(S^4,BK_n;f_{k_1…
reviewed by: Kohhei Yamaguchi
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 57Q35
zbMATH DE Number: 7384707
zbMATH Keywords: gauge group
zbMATH Keywords: homotopy type
zbMATH Keywords: SO(4)
MaRDI profile type: Publication
full work available at URL: s40879-021-00453-0
OpenAlex ID: W3158716522
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DOI: 10.1007/S40879-021-00453-0
description: scientific article; zbMATH DE number 7384707