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 $\phi_*=p$?
EDIT. (Thanks for helpful comments.) There are some relevant results: "Algorithmic aspects of homeomorphism problems" by Nabutovsky and Weinberger. There is a preprint version https://arxiv.org/abs/math/9707232 (click ps, pdf is a mess!), and a published version http://www.ams.org/books/conm/231/. A sort of review of their methods was written by R.I.Soare, "Computability theory and differential geometry" (Bulletin of Symbolic Logic, 2004).
According to the paper, the problem is decidable for simply connected manifolds of dimension at least 5. They also construct a counterexample for a non-simply connected case (Proposition 0.1 in the published paper). However, I would like to point out that this counteraxample does not answer the question. Even if algorithm for solving a problem formulated above exists, it cannot be used unless you have an explicit isomorphism between fundamental groups. In the situation of Proposition 0.1, it leads to no contradiction.
Interestingly, the results of Nabutovsky and Weinberger may be taken as a hint that the answer to the question is positive. At least, this possibility is not excluded for all I know.
EDIT. Actually, the example in Proposition 0.1 does show that the problem is undecidable. (So, I got it wrong. Thanks to Achim Krause for clarifying the details.)