Euler characteristic of a closed manifold whose universal cover is Euclidean

Question

Let $M$ be $n$ dimensional compact connected smooth manifold without boundary whose universal cover is diffeomorphic to $\mathbb{R}^n$, must the Euler characteristic of $M$ vanish?

This is true in the simplest case, when $M= \mathbb R^n/\Lambda$ is a torus.

Answer 1

There is a difficult open problem along the lines of your question, known as the Hopf-Thurston Conjecture:

If $M$ is a closed aspherical manifold of dimension $2k$, then $(-1)^k\chi(M)\ge 0$.