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It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric. Is it known if there are closed Riemannian manifolds which are isospectral but not homeomorphic? (By isospectral, I mean that the Laplace-Beltrami operator on functions has the same spectrum on both manifolds.)

Thanks,

Dmitri

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There's an example due to Doyle and Rossetti ("Tetra and Didi, the cosmic spectral twins"; http://arxiv.org/abs/math.DG/0407422) of 3-manifolds that are isospectral but not even homeomorphic. I don't know if this was the first such example.

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    $\begingroup$ There's older examples, e.g. of Vigneras: ams.org/mathscinet-getitem?mr=584073 $\endgroup$
    – Ian Agol
    May 16, 2012 at 4:40
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    $\begingroup$ Sunada (ams.org/mathscinet-getitem?mr=782558) gave a general framework in which the results of Vigneras fit. $\endgroup$ May 16, 2012 at 9:43
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    $\begingroup$ Today I learned that you can't hear the topology of a drum. $\endgroup$ May 16, 2012 at 15:27
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    $\begingroup$ Agol, are you sure, that Vigneras constructs isospectral varieties that are not homeomorphic, as opposed as only not isometric? I haven't read the paper in details, but the mathscinet reviews, the introduction of the paper, and the last theorem (theorem 8) says "isometric". $\endgroup$
    – Joël
    May 16, 2012 at 16:14
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    $\begingroup$ @Jan: It follows from an argument of Chen ( ams.org.proxy.lib.umich.edu/mathscinet/search/… ) that Vigneras' examples cannot be constructed from Sunada's method. $\endgroup$
    – user1073
    Oct 29, 2014 at 5:04

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