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Unrecognizability of manifolds. (English) Zbl 1115.57014

A celebrated result of A. Markov [Dokl.Akad.Nauk SSSR 121, 218–220 (1958; Zbl 0092.00702)] says that for every \(n\geq 4\) the homeomorphism problem for \(n\)-dimensional manifolds (given, for instance, as finite simplicial complexes) is (Turing-)undecidable. The authors sketch a new proof, extending ideas of M. A. Shtan’ko [Izv.Math.68, No. 1, 205–221 (2004; Zbl 1069.57013)]. Additionally, it is shown that each \(n\)-manifold is unrecognizable within the class of all \(n\)-manifolds, provided that \(n\geq 5\). This strengthens a result of S. P. Novikov on the non-recognizability of spheres in dimension five and above. All results are based on the unsolvability of the isomorphism problem for finitely presented groups.

MSC:

57Q25 Comparison of PL-structures: classification, Hauptvermutung
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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