TITLE: Smooth four-manifolds with contractible universal cover
QUESTION [11 upvotes]: Let $X$ be a smooth compact four-manifold with definite non-trivial intersection form. Can the universal cover of $X$ be contractible?
It semms to me that the answer is negative when $X$ is simply connected using results of Freedman and Donaldson. Is anything known when $X$ is not simply connected? Donaldson proved that also in this case the intersection form is diagonalizable over $\mathbb Z$.

REPLY [14 votes]: In algebraic geometry, there are examples of "fake projective planes", which in this context means smooth complex surfaces of general type with the same cohomology ring as the complex projective plane. It is known that the universal cover of such spaces is the complex hyperbolic ball. So the answer to your question is yes. (The first such fake projetive plane was shown to exist by Mumford.)