The homotopy type of Lie semigroups in semi-simple Lie groups

Let G be a noncompact semi-simple Lie group and S subset of G a Lie semigroup with nonempty interior. We study the homotopy groups pi(n)(S), n greater than or equal to 1, of S, Generalizing a well known fact for G, it is proved that there exists a compact and connected subgroup K(S) subset of G such that pi(n)(S) is isomorphic to pi(n)(K(S)). Furthermore, there exists a coset K(S) z contained in int S which is a deformation retract of S.136215117