Hopf–Thurston Euler characteristic sign conjecture

Let $M$ be a closed, aspherical manifold of dimension

$$d=2m.$$

Hopf–Thurston sign conjecture. The Euler characteristic of $M$ satisfies

$$(-1)^m\chi(M)\geq 0.$$

This is a topological version of Hopf's longstanding conjecture, originally proposed in this form by Thurston, concerning the sign of the Euler characteristic of closed aspherical manifolds. It is a fundamental unsolved problem in geometric topology; the paper verifies it for the hyperbolization functors under study.