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

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.