A Kuratowski-Type Classification of Critical Complexes for the 3-Sphere
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916001896988672 |
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| author | Eudave-Muñoz, Mario Ozawa, Makoto |
| author_facet | Eudave-Muñoz, Mario Ozawa, Makoto |
| contents | We give a Kuratowski-type classification of a graph-defined class of minimal piecewise-linear obstructions to embeddability in the 3-sphere. A finite simplicial complex \(X\) is called critical for \(S^3\) if \(|X|\) does not embed in \(S^3\), whereas deleting the open star of any simplex in the second barycentric subdivision of \(X\) yields a polyhedron embeddable in \(S^3\).
The main theorem completely classifies critical complexes of the form \((G\times S^1)\cup H\), where \(G\) and \(H\) are graphs and \(H\) is attached along vertices of the branch set of \(G\times S^1\). We prove that there are exactly seven such complexes up to homeomorphism: two \(K_4\)-type complexes, four \(Θ_4\)-type complexes, and one \(K_{2,3}\)-type complex.
The proof is combinatorial in nature. By collapsing the \(S^1\)-factor of \(G\times S^1\), we associate to \(X\) a reduction graph \(\widehat X=G\cup H\). Criticality implies that \(H\) is a forest, \(G\) is planar, and \(\widehat X\) is inclusion-minimal non-planar. Kuratowski's theorem therefore reduces the classification to the cases \(K_5\) and \(K_{3,3}\). A finite analysis of forest attachments, together with a face-incidence criterion for embeddability, leaves precisely the seven models listed above.
We also prove that every non-embeddable regular multibranched surface in \(S^3\) contains a critical subcomplex of the form \(M\cup H\), where \(M\) is a regular multibranched surface and \(H\) is a graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18279 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Kuratowski-Type Classification of Critical Complexes for the 3-Sphere Eudave-Muñoz, Mario Ozawa, Makoto Geometric Topology Combinatorics Primary 05C10, Secondary 57M15, 57Q35 We give a Kuratowski-type classification of a graph-defined class of minimal piecewise-linear obstructions to embeddability in the 3-sphere. A finite simplicial complex \(X\) is called critical for \(S^3\) if \(|X|\) does not embed in \(S^3\), whereas deleting the open star of any simplex in the second barycentric subdivision of \(X\) yields a polyhedron embeddable in \(S^3\). The main theorem completely classifies critical complexes of the form \((G\times S^1)\cup H\), where \(G\) and \(H\) are graphs and \(H\) is attached along vertices of the branch set of \(G\times S^1\). We prove that there are exactly seven such complexes up to homeomorphism: two \(K_4\)-type complexes, four \(Θ_4\)-type complexes, and one \(K_{2,3}\)-type complex. The proof is combinatorial in nature. By collapsing the \(S^1\)-factor of \(G\times S^1\), we associate to \(X\) a reduction graph \(\widehat X=G\cup H\). Criticality implies that \(H\) is a forest, \(G\) is planar, and \(\widehat X\) is inclusion-minimal non-planar. Kuratowski's theorem therefore reduces the classification to the cases \(K_5\) and \(K_{3,3}\). A finite analysis of forest attachments, together with a face-incidence criterion for embeddability, leaves precisely the seven models listed above. We also prove that every non-embeddable regular multibranched surface in \(S^3\) contains a critical subcomplex of the form \(M\cup H\), where \(M\) is a regular multibranched surface and \(H\) is a graph. |
| title | A Kuratowski-Type Classification of Critical Complexes for the 3-Sphere |
| topic | Geometric Topology Combinatorics Primary 05C10, Secondary 57M15, 57Q35 |
| url | https://arxiv.org/abs/2403.18279 |