Every finite complex has the homology of some $${\mathrm{CAT}(0)}$$ CAT ( 0 ) cubical duality group

Abstract

We prove that every finite connected simplicial complex has the homology of the classifying space for some $${\mathrm{CAT}(0)}$$ cubical duality group. More specifically, for any finite simplicial complex $$X$$ , we construct a locally $${\mathrm{CAT}(0)}$$ cubical complex $$T_{X}$$ and an acyclic map $$t_{X} : T_{X} \rightarrow X$$ such that $$\pi _{1}(T_{X})$$ is a duality group.