Discrete homogeneity and ends of manifolds

TL;DR

This paper establishes a precise topological characterization of connected non-compact manifolds of dimension at least 2, linking strong discrete homogeneity to having exactly one end.

Contribution

It proves that such manifolds are strongly discrete homogeneous if and only if they possess a single end, clarifying the relationship between topology and homogeneity.

Findings

Manifolds with one end are strongly discrete homogeneous.

Manifolds with multiple ends are not strongly discrete homogeneous.

The result applies to all connected non-compact metrizable manifolds of dimension ≥ 2.

Abstract

It is shown that a connected non-compact metrizable manifold of dimension $\ge 2$ is strongly discrete homogeneous if and only if it has one end (in the sense of Freudenthal compactification).