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 $\mathbb Z$.