On homogeneous locally conical spaces. (English) Zbl 1403.54003
Before providing the results of this paper, we shall recall a couple of definitions.
Definition. A space is locally conical if each of its points has an open neighborhood that is homeomorphic to the open cone over a compact space (we presume with the cone point matching up with the given point of \(X\)).
Definition. A space \(X\) is strongly \(n\)-homogeneous if every bijection between \(n\)-element subsets of \(X\) can be extended to a homeomorphism of \(X\).
Here are the main results as stated in the Abstract.
Theorem. Every homogeneous locally connected separable metric space that is not a \(1\)-manifold is strongly \(n\)-homogeneous for each \(n\geq2\). Furthermore, every homogeneous locally conical separable metric space is countable dense homogeneous.
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 \(p\) to \(q\)).
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 \(n\geq2\) and \(X\) is countable dense homogeneous.
Definition. A space is locally conical if each of its points has an open neighborhood that is homeomorphic to the open cone over a compact space (we presume with the cone point matching up with the given point of \(X\)).
Definition. A space \(X\) is strongly \(n\)-homogeneous if every bijection between \(n\)-element subsets of \(X\) can be extended to a homeomorphism of \(X\).
Here are the main results as stated in the Abstract.
Theorem. Every homogeneous locally connected separable metric space that is not a \(1\)-manifold is strongly \(n\)-homogeneous for each \(n\geq2\). Furthermore, every homogeneous locally conical separable metric space is countable dense homogeneous.
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 \(p\) to \(q\)).
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 \(n\geq2\) and \(X\) is countable dense homogeneous.
Reviewer: Leonard R. Rubin (Norman)
MSC:
| 54B15 | Quotient spaces, decompositions in general topology |
| 54F15 | Continua and generalizations |
| 54H99 | Connections of general topology with other structures, applications |
| 54H15 | Transformation groups and semigroups (topological aspects) |
| 57N15 | Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010) |
| 57S05 | Topological properties of groups of homeomorphisms or diffeomorphisms |
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