LOCALLY COMPACT SUBGROUP ACTIONS ON TOPOLOGICAL GROUPS

Abstract. Let X be a Hausdorff topological group and G a locally compact subgroup of X. We show that X admits a locally finite σ-discrete G-functionally open cover each member of which is G-homeomorphic to a twisted product G ×H Si, where H is a compact large subgroup of G (i.e., the quotient G/H is a manifold). If, in addition, the space of connected components of G is compact and X is normal, then X itself is G-homeomorphic to a twisted product G ×K S, where K is a maximal compact subgroup of G. This implies that X is K-homeomorphic to the product G/K × S, and in particular, X is homeomorphic to the product Rn × S, where n = dim G/K. Using these results we prove the inequality dim X ≤ dim X/G + dim G for every Hausdorff topological group X and a locally compact subgroup G of X. 1