On homogeneous locally conical spaces

TL;DR

This paper proves that homogeneous locally conical spaces are highly symmetric and explores implications for suspensions and the Bing-Borsuk Conjecture, advancing understanding of their topological structure.

Contribution

It establishes that such spaces are strongly n-homogeneous and countable dense homogeneous, even without connectedness, and derives new consequences for suspensions and potential counterexamples.

Findings

Homogeneous locally conical spaces are strongly n-homogeneous for all n ≥ 2.

Such spaces are countable dense homogeneous.

Implications for suspensions and the Bing-Borsuk Conjecture.

Abstract

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 $n \geq 2$ and countable dense homogeneous. Furthermore, countable dense homogeneity can be proven without assuming the space is connected. 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 $X$ is strongly $n$-homogeneous for all $n \geq 2$ and countable dense homogeneous.