TITLE: Example of 4-manifold with $\pi_1=\mathbb Q$
QUESTION [28 upvotes]: This might be well known for algebraic topologist. So I am looking for an explicit example of a 4 dimensional manifold with fundamental group isomorphic to the rationals $\mathbb Q$.

REPLY [58 votes]: Since any compact manifold has the homotopy type of a finite CW-complex (see this MathOverflow question: Are non-PL manifolds CW-complexes?)  and $\mathbb{Q}$ is not finitely presented, the manifold $X$ you are looking for is necessarily non-compact.
An explicit construction of a non-compact three-manifold $M$ with $\pi_1(M)=\mathbb{Q}$   can be found in the paper
B. Evans and L. Moser: Solvable Fundamental Groups of Compact 3-Manifolds, Transactions of the American Mathematical Society 168 (1972), see in particular page 209.
Now it suffices to take $X=M \times \mathbb{R}$.