On homogeneous locally conical spaces

The main result of this article is: THEOREM. Every homogeneous locally conical connected separable metric space that is not a 1-manifold is strongly n-homogeneous for each n >= 2. Furthermore, every homogeneous locally conical separable metric space is countable dense homogeneous. This theorem has the following two consequences. COROLLARY 1. If X is a homogeneous compact suspension, then X is an absolute suspension (i.e., for any two distinct points p and q of X there is a homeomorphism from X to a suspension that maps p and q to the suspension points). COROLLARY 2. If there exists a locally conical counterexample X to the Bing-Borsuk Conjecture (i.e., X is a locally conical homogeneous Euclidean neighborhood retract that is not a manifold), then each component of X is strongly n-homogeneous for all n >= 2 and X is countable dense homogeneous.