Nilpotent subgroups of the group of self-homotopy equivalences (Q1178329)

For a space \(X\) let \({\mathcal E}(X)\) be the group of homotopy classes of self homotopy equivalences. If \(E_ *\) is a homology theory, then \({\mathcal E}(X)\) acts naturally on \(E_ *(X)\). The main result of the paper is as follows: Let \(E\) be a connective, reduced, multiplicative homology theory such that \(E_ *(S^ 0)\approx {\mathbb{Z}}_ P\) (integers localized at some set of primes \(P\)). Let \(X\) be a connected, nilpotent, finite dimensional \(CW\)-complex and let \(X_ P\) be its \(P\)-localization. Then any subgroup \(G\subset{\mathcal E}(X)\) acting nilpotently on \(E_ *(X_ P)\) is nilpotent. Let \({\mathcal E}_ E(X)\subset{\mathcal E}(X)\) be the subgroup of all elements acting trivially on \(E_ *(X)\). Then for \(X\) as above it follows that \({\mathcal E}_{MU}(X)\), \({\mathcal E}_{MSp}(X)\) and \({\mathcal E}_{BP}(X_{(p)})\) are nilpotent groups. Here \(p\) is a prime and \(BP\) is the corresponding Brown-Peterson theory. It is interesting to note that the connectivity of \(E\) is crucial. The authors give an example of a space \(X\) where \(\tilde K_ *(X)=0\) whence \({\mathcal E}_ K(X)={\mathcal E}(X)\), but where this group is not nilpotent. The method of proof is to use the Adams spectral sequence and the Atiyah- Hirzebruch spectral sequence in order to reduce the problem from general \(E\) to ordinary homology. Then the results of \textit{E. Dror} and \textit{A. Zabrodsky} [Topology 18, 187-197 (1979; Zbl 0417.55008)] are applied.