Every finite complex is the classifying space for proper bundles of a virtual Poincar\'e duality group

TL;DR

This paper demonstrates that any finite connected simplicial complex can be represented as the classifying space for proper bundles of a virtual Poincaré duality group, linking topology and group actions.

Contribution

It proves that every finite connected simplicial complex is homotopy equivalent to such a classifying space, establishing a new connection between simplicial complexes and virtual Poincaré duality groups.

Findings

Every finite connected simplicial complex is homotopy equivalent to a classifying space for a virtual Poincaré duality group.

Constructs a proper action of a virtually torsion-free group on a contractible manifold.

Shows the homotopy equivalence between simplicial complexes and quotients of manifolds by group actions.

Abstract

We prove that every finite connected simplicial complex is homotopy equivalent to the quotient of a contractible manifold by proper actions of a virtually torsion-free group. As a corollary, we obtain that every finite connected simplicial complex is homotopy equivalent to the classifying space for proper bundles of some virtual Poincar\'e duality group.