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Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum?

I would be happy with any $X,Y$ that are not diffeomorphic but are homeomorphic. This is a copy of the (unanswered) question here.

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    $\begingroup$ I don’t know the answer, but an immediate observation is that the dimensions have to be the same by Weyl’s law, so any such examples will be homeomorphic. $\endgroup$
    – Gabe K
    Commented May 13 at 1:29

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