Topological median algebra structures on ER homology manifolds I: local cubulation

TL;DR

This paper investigates topological median algebra structures on Euclidean spaces and ER homology manifolds, demonstrating their local cubulation properties and conditions for metrizability, with examples illustrating diverse structures.

Contribution

It establishes that all such median structures admit local CAT(0) cubulation and characterizes when they are metrizable, providing new examples of median algebra structures.

Findings

Median structures have local CAT(0) cubulation.

Metrizability of median structures depends on compactness of intervals.

Constructs numerous non-locally cubulated median algebra examples.

Abstract

We study topological median algebra structures on Euclidean spaces and, more generally, ER homology manifolds. We show that all such median structures have a local CAT(0) cubulation structure. We also show that topological median algebra structures are completely metrizable as median metric spaces if and only if intervals are compact. We give examples of both metrizable and non-metrizable such structures, as well as provide a construction for producing many non-locally cubulated topological median algebra structures on the unit ball in Euclidean space.