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Let me recall the conjecture: $M$ and $N$ two aspherical closed $n$-manifolds with isomorphic fundamental groups, then $M$ and $N$ are homeomorphic.

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    $\begingroup$ See the manifold atlas here for a summary of what I think are fairly up to date results. $\endgroup$
    – mme
    Commented Jan 16, 2016 at 20:44
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    $\begingroup$ This is a question better answered by 5 minutes in front of a Google prompt. $\endgroup$
    – Ryan Budney
    Commented Jan 17, 2016 at 1:22

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Surely the state of the art is in this paper from a few days ago at http://arxiv.org/abs/1601.00262 by some of the leading experts in the field.

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  • $\begingroup$ I may have misunderstood the abstract/introduction. But, it seems to me that they have proved the conjecture as they state it as questions of Borel. Is this correct?! $\endgroup$
    – user51223
    Commented Jan 18, 2016 at 8:30
  • $\begingroup$ They proved a brilliant result but it's not the same conjecture :-) $\endgroup$
    – Mikhail Katz
    Commented Jan 18, 2016 at 8:35

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