TITLE: Manifolds covered by an n-dimensional torus
QUESTION [12 upvotes]: It is well-known that classification of manifolds up to homemorphism is, in general, out of question. However, this task is sometimes tractable under some additional assumptions on manifolds one would like to classify. I want to ask about one specific example of this situation.
Let $T^n$ denote the $n$-dimensional torus $(S^1)^n$. Suppose that a closed manifold is finitely covered by $T^n$. I would like to know to what extent (and whether at all) it is possible to classify such manifolds $M$. If $n=1$, then $M=S^1$; if $n=2$, then $M=S^1 \times S^1$ or $M =K$, the Klein bottle. (But then again, this is not really interesting, because in these dimensions all manifolds are classified.) What happens if $n \geq 3$, or at least if $n=3$?

REPLY [12 votes]: People have already mentioned the Bieberbach theorems, which imply that your manifold is homotopy equivalent to a Euclidean manifold.  In fact, it is homeomorphic to the Euclidean manifold, at least if the dimension is at least $5$.  This is predicted by the Borel conjecture, which asserts that homotopy equivalent aspherical manifolds are homeomorphic in high dimensions.  The necessary case of the Borel conjecture is proven in the following paper of Farrell and Jones (building on lots of earlier work of theirs): 
F. T. Farrell and L. E. Jones. Topological rigidity for compact non-positively curved manifolds.
In Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), pages 229–
274. Amer. Math. Soc., Providence, RI, 1993.

EDIT : The question inspired me to do a little more reading about the history of the Borel conjecture, and I learned that for virtually abelian fundamental groups like the ones in this question, the Borel conjecture was proven in the earlier paper
MR0704219 (84k:57017)
Farrell, F. T.(1-MI); Hsiang, W. C.(1-PRIN)
Topological characterization of flat and almost flat Riemannian manifolds Mn (n≠3,4). 
Amer. J. Math. 105 (1983), no. 3, 641–672.