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On homogeneous locally conical spaces

Volume 241 / 2018

Fredric D. Ancel, David P. Bellamy Fundamenta Mathematicae 241 (2018), 1-15 MSC: Primary 54B15, 54F15, 54H99; Secondary 54H15, 57N15, 57S05. DOI: 10.4064/fm282-4-2017 Published online: 30 August 2017

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 $n \geq 2$. 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 $n \geq 2$ and $X$ is countable dense homogeneous.

Authors

  • Fredric D. AncelDepartment of Mathematical Sciences
    University of Wisconsin-Milwaukee
    Box 413
    Milwaukee, WI 53201-0413, U.S.A.
    e-mail
  • David P. BellamyDepartment of Mathematics
    University of Delaware
    Newark, DE 19716, U.S.A.
    e-mail

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