Classifying spaces and a subgroup of the exceptional Lie group \(G_2\) (Q2734865)

The author defines a space as a \(p\)-compact classifying space if its loop space is a \(p\)-compact group. The \(p\)-compact groups \(X\) were introduced in [\textit{W. G. Dwyer} and \textit{C. W. Wilkerson}, Ann. Math. (2) 139, No. 2, 395-442 (1994; Zbl 0801.55007)]. They behave like compact Lie groups, and the classifying spaces \(BX\) and \((BG)_p^\wedge\) are similar. In [\textit{K. Ishiguro}, Toral groups and classifying spaces of \(p\)-compact groups, in `Homotopy methods in algebraic topology', Proc. AMS-IMS-SIAM joint Summer research conf., Univ. of Colorado, Boulder, CO, USA, June 20-24, 1999, Contemp. Math. 271, 155-167 (2001; Zbl 1005.55005)], the author gave (in a special case) necessary and sufficient conditions on \(G\) such that \((BG)_p^\wedge\) is a \(p\)-compact classifying space, and he stated a necessary condition in the general case. In this paper, the author shows that this condition is not sufficient. For that he constructs a counterexample by using a subgroup of the exceptional Lie group \(G_2\) at \(p=3\).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00050].