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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?

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Could you remind us what definition of "fake" you're using? –  Ryan Budney Oct 11 '12 at 20:27
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Presumably this answers your question: if a 3-manifold has fundamental group isomorphic to $\mathbb Z^3$ then it is homeo/diffeo/PL equivalent to $(S^1)^3$. The follows from geometrization. –  Ryan Budney Oct 11 '12 at 20:28
    
@Ryan Fake torus means the manifold which is homotopic to the torus. Could you please give more details on how to derive it from geometrization? Or does it follows from some easiler fact other than this big theorm? –  J. GE Oct 11 '12 at 20:36
    
This is the reference you want then: en.wikipedia.org/wiki/Moise%27s_theorem it's a theorem of Moise that topological, PL and smooth manifolds all have a unique compatible structure among these three categories. –  Ryan Budney Oct 11 '12 at 20:44
    
By "homotopic" do you mean homeomorphic or homotopy-equivalent? My previous comment answers the homeomorphic interpretation. If you really mean homotopy-type then you need geometrization to show the manifold is prime. Once you know the manifold is prime, it's an old theorem of Waldhausen's. –  Ryan Budney Oct 11 '12 at 20:46
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1 Answer

up vote 9 down vote accepted

@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, and references therein.

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Thanks Igor. Could you point to me the paper where the Borel Conjecture was proved for subexponential growth. Since in Belegradek's answer he mentioned it is true for "closed 4-manifolds homotopy equivalent to an infranil 4-manifold". –  J. GE Oct 12 '12 at 8:51
    
This is in Freedman and Quinn's book "Topology of 4-manifolds". The reference for the first paper it appears in should be there, as well. –  Ryan Budney Oct 12 '12 at 15:50
    
That's great, thanks Ryan. –  J. GE Oct 13 '12 at 6:21
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