The \(K\)-theory of \(p\)-compact groups
A \(p\)-compact group is a loop space \(X\simeq \Omega (BX)\), with \(BX\) connected, \(H^*(X; \mathbb{Z}/p)\) finite, \(\pi_0(X)\) a finite \(p\)-group and \(\pi_n(X)\) a finitely generated \(\mathbb{Z}_p\)-module for each \(n\). By results of R. Kane and J. Lin the \(p\)-adic \(K\)-theory of a \(p\)-compact group is a primitively generated exterior algebra. Let \(T\to X\) be a maximal torus for \(X\) then the authors prove that \(K^* (BX;\mathbb{Z}_p) \cong K^* (BT: \mathbb{Z}_p)^W\) where \(W\) is the Weyl group of \(X\). This is an analogue of a well-known result of [\textit{M. F. Atiyah} and \textit{F. Hirzebruch}, Proc. Sympos. Pure Math. 3, 7-38 (1961; Zbl 0108.17705)] for compact Lie groups. As a corollary, the authors show that a connected finite loop space admits a maximal torus if and only if \(K^*(BL;\mathbb{Z})\) is isomorphic to \(K^*(BG; \mathbb{Z})\) for some compact Lie group, \(G\).
scientific article; zbMATH DE number 1129398