The Homotopy Type of Lie Semigroups in Semi-Simple Lie Groups

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