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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.07421 |
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| _version_ | 1866915165076717568 |
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| author | Tancer, Martin |
| author_facet | Tancer, Martin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07421 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Simpler algorithmically unrecognizable 4-manifolds Tancer, Martin Geometric Topology Computational Geometry Group Theory 57-08, 57K40, 20F06, 20F10, 68Q17, 05C62 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. |
| title | Simpler algorithmically unrecognizable 4-manifolds |
| topic | Geometric Topology Computational Geometry Group Theory 57-08, 57K40, 20F06, 20F10, 68Q17, 05C62 |
| url | https://arxiv.org/abs/2310.07421 |