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I saw the following question from the "Problem Section" in Schoen and Yau, page 281, problem 12:

Let $M_1, M_2$ each have negative curvature. If $\pi_1 (M_1)=\pi_1 (M_2)$, prove that $M_1$ is differeomorphic to $M_2$.

The authors then commented: " There is some progress due to Cheeger, Gromov, Farrell and Hsiang. Cheeger proved that $\pi_1(M)$ determines the second Stiefel bundle of $M$. Gromov proved $\pi_{1}(M)$ determines the unit tangent bundle of $M$. Farrell-Hsiang proved $\pi_{1}(M)$ determines $M\times \mathbb{R}^{3}$. Farrel-Hsiang have only to assume that one of the manifolds has negative curvature."

Here the paper due to Cheeger&Gromov is this one. I am unable to find any reference on Farrell&Hsiang's work as the book says it was to be published. May I ask if there is any significant progress on this problem? This is obviously some strong rigidity theorem (if turned out to be true). A quick "scan" of the papers involved in the end of Gromov's paper did not reveal anything particularly relevant.

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Disproven by Farrel and Jones about 20 years ago. –  Misha Jul 20 at 23:32
    
:( I did not know.... –  Bombyx mori Jul 20 at 23:33
    
@Misha: May I ask where can I find a counter-example? –  Bombyx mori Jul 20 at 23:40

1 Answer 1

up vote 14 down vote accepted

Counter-example was constructed by T.Farrell and L.Jones, in "Negatively curved manifolds with exotic smooth structures", JAMS 1989, volume 2, number 4.

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Thanks a lot! This closed the question. –  Bombyx mori Jul 21 at 13:19

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