Hsiang, W.-c.; Shaneson, J. L. Fake tori, the annulus conjecture, and the conjectures of Kirby. Proc. Nat. Acad. Sci. U.S.A. 62 1969 687–691.


The paper above classified all fake tori for dimension $\ge 5$. How about low dimension?


To be precise: Let $M^n$ be a topological manifold of dimension $n=3, 4$, which has the same homotopy type of the standard torus $T^n$. My question is whether $M^n$ is homeomorphic to the standard torus?



Are there any fake 3-tori?
@Ryan answers the question in three dimensions (and such a result cannot hold without the Poincare conjecture, since otherwise you could take a connected sum with a fake $S^3.)$ In general, this is a special case of the Borel Conjecture, which is known to hold in dimension four for groups of subexponential growth, such as $\mathbb{Z}^4$ (in the topological category). For more, see @Igor Belegradek's answer to [this question](https://mathoverflow.net/questions/69431/more-four-dimensional-counterexamples), and references therein.


