Borel structurability by locally finite simplicial complexes

Abstract

We show that every countable Borel equivalence relation structurable by n-dimensional contractible simplicial complexes embeds into one which is structurable by such complexes with the further property that each vertex belongs to at most M-n := 2(n-1)(n(2) + 3n + 2)-2 edges; this generalizes a result of Jackson-Kechris-Louveau in the case n = 1. The proof is based on that of a classical result of Whitehead on countable CW-complexes.