The group of homotopy equivalences of products of spheres and of Lie groups

Abstract.

We investigate the group ${\cal E}_\#(X)$ of self homotopy equivalences of a space X which induce the identity homomorphism on all homotopy groups. We obtain results on the structure of ${\cal E}_\#(X)$ provided the p-localization $X_{(p)}$ of X has the homotopy type of a p-local product of odd-dimensional spheres. In particular, we show that ${\cal E}_\#(X)_{(p)}$ is a semidirect product of certain homotopy groups $\pi_n(X_{(p)})$. We also show that ${\cal E}_\#(X)_{(p)}$ has a central series whose successive quotients are $\pi_n(X_{(p)})$, which are direct sums of homotopy groups of p-local spheres. This leads to a determination of the order of the p-torsion subgroup of ${\cal E}_\#(X)$ and an upper bound for its p-exponent. These results apply to any Lie group G at a regular prime p. We derive some general properties of ${\cal E}_\#(G)$ and give numerous explicit calculations.