  A topological version of a longstanding conjecture of H. Hopf, originally
proposed by W. Thurston, states that the sign of the Euler characteristic of a
closed aspherical manifold of dimension $d=2m$ depends only on the parity of
$m$. Gromov defined several hyperbolization functors which produce an
aspherical manifold from a given simplicial or cubical manifold. We investigate
the combinatorics of several of these hyperbolizations and verify the Euler
Characteristic Sign Conjecture for each of them. In addition, we explore
further combinatorial properties of these hyperbolizations as they relate to
several well-studied generating functions.

Combinatorics;Geometric Topology