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More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds?

Related: if such a manifold is not known, are there any restrictions on properties of such manifolds, such as curvature?

What I have so far found works only for dimension $n\geq 5$. There are of course the results by Hsiang and Shaneson in "Fake Tori, the Annulus Conjecture, and the Conjectures of Kirby", classifying "fake tori" in $n\geq 5$ (homotopy equivalent to standard tori), which were then shown to be homeomorphic, but not diffeomorphic to the standard tori, by Hsiang and Wall in "On Homotopy Tori II".

Also, if I interpret the Bieberbach theorem on flat manifolds correctly, these exotic tori cannot be flat, since this would mean they are isometric to standard tori.

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    $\begingroup$ My understanding is that this is open, and in fact no exotic closed aspherical 4-manifolds are known. $\endgroup$
    – Igor Belegradek
    Commented Jun 14, 2013 at 22:47
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    $\begingroup$ As for curvature restrictions no exotic 4-torus has a metric of nonnegative Ricci curvature (this is due to Cheeger-Gromoll and is a corollary of their splitting theorem). $\endgroup$
    – Igor Belegradek
    Commented Jun 14, 2013 at 22:50
  • $\begingroup$ @Igor Belegradek: Thank you. I suspected it was open, but not being an expert in the field made me hesitate. The reference to Cheeger-Gromoll is very much appreciated, I wouldn't have found it myself. $\endgroup$
    – Jan Jitse Venselaar
    Commented Jun 14, 2013 at 23:11
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    $\begingroup$ Building on Igor Belegradek's comment, exotic tori are spin and enlargeable (in the sense of Gromov and Lawson), so they do not admit metrics of non-negative scalar curvature. $\endgroup$
    – Michael Albanese
    Commented Dec 16, 2019 at 3:50

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