The group of homotopy equivalences of products of spheres and of Lie groups (Q1849484)

The authors obtain general and specific results on the \(p\)-localization of \({\mathcal E}(X)\) for spaces such as Lie groups and products of spheres. Let \({\mathcal E}_\sharp (X)\) denote the group of those homotopy self-equivalence classes which induce the identity automorphism on all homotopy groups. Given a \(1\)-\( \)connected finite complex \(X\) which has the rational homotopy type of a product of odd-dimensional spheres, the \(p\)-localization \(X_{(p)}\) has the homotopy type of \(P=S^{n_1}_{(p)}\times \dots \times S^{n_k}_{(p)}\) for all primes \(p\) which are regular for \(X\); all except finitely many primes are regular for \(X\). If \(p\) is regular for \(X\), then \({\mathcal E}_\sharp (X)_{(p)}={\mathcal E}_\sharp (X_{(p)})= {\mathcal E}_\sharp (P)\). The authors describe \({\mathcal E}_\sharp (P)\) as an iterated semi-direct product of certain homotopy groups of \(P\) and obtain a central series for \({\mathcal E}_\sharp (P)\) whose quotients are direct summands of such homotopy groups. Results are deduced on the order of the \(p\)-torsion subgroup of \({\mathcal E}_\sharp (P)\) in general, for a large number of specific Lie groups and for some products of spheres. The computer algebra system MAPLE was used for these calculations.