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YCor
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A homeomorphism with a prescribed action on the fundamental group - decidable or not?

I am curious if the following topological problem is decidable. Let $M,N$ be two closed manifolds. Given a group isomorphism $p: \pi_1(M)\to \pi_1(N)$, is there a homeomorphism $\phi: M\to N$ such that $\phi_*=p$?

Alex Gavrilov
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