2023/10/11 by Martin Tancer, Tancer, Martin
Mathematics · #05C62 #20F06 #20F10 #57-08 #57K40 #68Q17 #Advanced Topology and Set Theory #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2310.07421
openalex publication_date 2023/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 S4 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(S2 x S2) is unrecognizable while the previous best result is the unrecognizability of #12(S2 x S2) due to Gordon.