Simpler algorithmically unrecognizable 4-manifolds

TL;DR

This paper advances the understanding of unrecognizable 4-manifolds by reducing the complexity of their construction, showing that a connected sum of nine S^2 x S^2 manifolds is unrecognizable, improving previous results.

Contribution

It introduces a new method to construct unrecognizable 4-manifolds without Markov's trick and reduces the complexity of the Adian-Rabin set used.

Findings

Connected sum #_9(S^2 x S^2) is unrecognizable.

Removed Markov's trick from the proof.

Reduced the number of relations in the Adian-Rabin set.

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.