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 $\mathbb Q$.



Example of 4-manifold with $\pi_1=\mathbb Q$
Since any compact manifold has the homotopy type of a finite CW-complex (see this MathOverflow question: [Are non-PL manifolds CW-complexes?](https://mathoverflow.net/questions/36838/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*](http://dx.doi.org/10.1090/S0002-9947-1972-0301742-6), Transactions of the American Mathematical Society **168** (1972), see in particular page 209.


Now it suffices to take $X=M \times \mathbb{R}$.


