Which free abelian groups can be realized as the fundamental group of a closed 3-manifold? The only one I can come up with is $\mathbb{Z}$, which is the fundamental group of $S^1 \times S^2$. For the application I have in mind, the key case is $\mathbb{Z}^2$. Here it is easy if you allow boundary (just take $T^2 \times [0,1]$), but I don't see how to do it without the boundary.



Closed 3-manifolds with free abelian fundamental groups
(I assume all occuring 3-manifolds to be orientable and closed)


A manifold with a free abelian fundamental group cannot be a connected sum of non-trivial 3-manifolds since its fundamental group is not a free product. A prime manifold is either $S^1\times S^2$ or irreducible (Hatcher's notes on 3-manifolds, 1.4). By 3.9 in the same source, an irreducible 3-manifold $M$ with infinite fundamental group is a $K(\pi,1)$. If $\pi = \mathbb{Z}^n$, then $M$ needs therefore to be homotopy equivalent to $(S^1)^n$. This can only be if $n=3$. 


