On stable summands of the classifying space of a compact Lie group (Q1317057)

Some methods and results which succeeded in determining the stable splittings of the classifying space \(BG\) when \(G\) is a finite group are extended to the case where \(G\) is a compact Lie group. One of these involves extending Nishida's notion of dominant summands from the case of finite \(G\) to the case where \(G\) is a compact Lie group. It is proved that if the dominant summands of \(BG_{1+p}^ \wedge\) and \(BG_{2+p}^ \wedge\) are homotopy equivalent then the compact Lie groups \(G_ 1\) and \(G_ 2\) have isomorphic \(p\)-Sylow subgroups. Here the classifying spaces have been given a disjoint basepoint, and then \(p\)-completed. The other main result deals with a \(p\)-toral compact Lie group \(G\), and supposes that \(F_ n\) are finite \(p\)-groups approximating \(G\) homologically, with inclusions \(F_ m\to F_ n\). Then stable summands of \(BG_{+p}^ \wedge\) correspond to compatible stable summands of \(BF_{n+p}^ \wedge\) for arbitrarily large \(n\).