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.

arXiv comment12 pages; minor revisions