Higher homotopy commutativity in localized Lie groups and gauge groups (Q1625450)

The Hopf theorem says that the rational cohomology of a compact connected Lie group \(G\) is the exterior algebra \[ H^*(G; \mathbb{Q}) = \Lambda_{\mathbb{Q}} (x_1, x_2, \ldots, x_l), \] where \(x_i \in H^{2n_i -1}(G; \mathbb{Q})\) and \(n_1 \leq n_2 \leq \cdots \leq n_l\). The sequence of numbers \(\{n_1, n_2,\ldots, n_l \}\) is said to be the \textit{type} of the Lie group \(G\). Recall that there are two major considerations of higher homotopy commutativity: one is \textit{F. D. Williams}' \(C_k\)-space [Trans. Am. Math. Soc. 139, 191--206 (1969; Zbl 0185.27103)] and the other is \textit{M. Sugawara}'s \(C_k\)-space [Mem. Coll. Sci., Univ. Kyoto, Ser. A 33, 257--269 (1960; Zbl 0113.16903)]. Let \(EG \rightarrow BG\) be the universal bundle of \(G\) and let \(E_n G \rightarrow B_n G\) be the restriction over the \(n\)-th projective space \(B_n G \subset BG\). Let \(\mathcal G (P)\) be the gauge group of \(P\), the total space of a principal \(G\)-bundle \(P \rightarrow B\). In this paper, the authors refine \textit{C. A. McGibbon}'s work [Math. Z. 201, No. 3, 363--374 (1989; Zbl 0682.55006)] by considering the higher homotopy commutativity in the sense of Sugawara. Let \(G\) be a compact connected simple Lie group of type \(\{n_1, n_2,\ldots, n_l \}\) and let \(p\) be a prime number. The authors show that if \(p> (n+k)n_l\), then the \(p\)-localized gauge group \(\mathcal G (E_n G)_{(p)}\) is a Sugawara \(C_k\)-space, and if \((n+1)n_l < p < (n+k) n_l\), then \(\mathcal G (E_n G)_{(p)}\) is not a Williams \(C_k\)-space, where \(n\) and \(k\) are positive integers. They also show that if \(p \geq kn_l +n_i\), then the \(p\)-localized gauge group \(\mathcal G (P)_{(p)}\) of any principal \(G\)-bundle \(P\) over \(S^{2n_i}\) is a Sugawara \(C_k\)-space. Finally, they prove that the localized Lie group \((G_2 )_{(5)}\) is not a Williams \(C_3\)-space.