The homotopy types of \(\mathrm{SO}(4)\)-gauge groups
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 \[ B\mathcal{G}_{k_1,\dots ,k_n}\simeq \mathrm{map}(S^4,BK_n;f_{k_1,\dots ,k_n})\quad (f_{k_1,\dots ,k_n}=\sum_{i=1}^nk_i\epsilon_i), \] where \(\mathrm{map}(S^4,BK_n)\) is the space of maps \(S^4\to BK_n\) and \(\mathrm{map}(S^4,BK_n;f)\) denotes its path-component containing the map \(f\). Then by using a careful analysis of the homotopy fiber sequence \[ K_n\stackrel{\partial_{k_1,\dots ,k_n}}{\longrightarrow} \Omega^3_0K_n \longrightarrow\mathrm{map}(S^4,BK_n;f_{k_1,\dots ,k_n}) \longrightarrow BK_n \] they prove the assertions.
scientific article; zbMATH DE number 7384707