Simpler algorithmically unrecognizable 4-manifolds

TL;DR

The paper advances the study of unrecognizable $4$-manifolds by removing Markov's trick from the construction and achieving unrecognizability with deficiency-based Adian-Rabin sets. By combining Rohlin/Freedman theory with a modified Borisov encoding grounded in small cancellation, the authors prove that $ obreak{\\#}_{9}(S^2 imes S^2)$ is unrecognizable, improving previous bounds and offering a concrete path toward near-S$^4$ examples. The core method links undecidable word problems (via Matiyasevich's Thue system and Borisov-type presentations) to topological realizations through carefully controlled encodings, ensuring undecidability without stabilization by extra handles. The work has potential implications for broader classes of simply connected 4-manifolds and underscores the role of deficiency and encoding schemes in algorithmic topology.

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.

Figures (12)

• Figure 1: An example of a part of a diagram showing the graph $G$ and twins. The vertices of $G$ are marked by small boxes, the edges of $G$ are dashed. For example, the pairs $(\sigma_1, e_1)$ and $(\sigma_2,e_2)$ are twins where $e_1$ is the edge labelled $a$ in $\sigma_1$ and $e_2$ is the edge labelled $a$ in $\sigma_2$. Also the pairs $(\sigma_2, e_3)$ and $(\sigma_3, e_3)$ are twins where $e_3$ is the joint edge of $\sigma_2$ and $\sigma_3$.
• Figure 2: A contraction of the disk bounded by $\gamma$.
• Figure 3: A grid diagram $\mathcal{G}(aba,2)$.
• Figure 4: Labels of edges of $\sigma_1$ and $\sigma_2$.
• Figure 5: A contraction of a reducing pair.

Theorems & Definitions (43)

• Theorem 1
• Theorem 2
• Corollary 3
• Corollary 4
• Proposition 5
• proof
• Theorem 6: Rohlin; see, e.g. scorpan05
• Theorem 7: Freedman
• Theorem 8
• proof