Simpler algorithmically unrecognizable 4-manifolds
Abstract
Markov proved that there exists an unrecognizable 4-manifold, that is, a 4-manifold for which the homeomorphism problem is undecidable. In this paper we consider the question how close we can get to S^4 with an unrecognizable manifold. One of our achievements is that we show a way to remove so-called Markov's trick from the proof of existence of such a manifold. This trick contributes to the complexity of the resulting manifold. We also show how to decrease the deficiency (or the number of relations) in so-called Adian-Rabin set which is another ingredient that contributes to the complexity of the resulting manifold. Altogether, our approach allows to show that the connected sum #_9(S^2 x S^2) is unrecognizable while the previous best result is the unrecognizability of #_12(S^2 x S^2) due to Gordon.
Keywords
Cite
@article{arxiv.2310.07421,
title = {Simpler algorithmically unrecognizable 4-manifolds},
author = {Martin Tancer},
journal= {arXiv preprint arXiv:2310.07421},
year = {2025}
}
Comments
27 pages, 12 figures. Version 2 has a number of corrections mostly in the proof of Theorem~15. (They are not really essential but some statements/proofs had to be slightly reworded in order to be correct.) HTML version does not compile pictures correctly