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?