A quadratic estimation for the Kühnel conjecture on embeddings
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866911024848830464 |
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| author | Dzhenzher, S. Skopenkov, A. |
| author_facet | Dzhenzher, S. Skopenkov, A. |
| contents | The classical Heawood inequality states that if the complete graph $K_n$ on $n$ vertices is embeddable in the sphere with $g$ handles, then $g \ge\dfrac{(n-3)(n-4)}{12}$. A higher-dimensional analogue of the Heawood inequality is the Kühnel conjecture. In a simplified form it states that for every integer $k>0$ there is $c_k>0$ such that if the union of $k$-faces of $n$-simplex embeds into the connected sum of $g$ copies of the Cartesian product $S^k\times S^k$ of two $k$-dimensional spheres, then $g\ge c_k n^{k+1}$. For $k>1$ only linear estimates were known. We present a quadratic estimate $g\ge c_k n^2$. The proof is based on beautiful and fruitful interplay between geometric topology, combinatorics and linear algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_04188 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A quadratic estimation for the Kühnel conjecture on embeddings Dzhenzher, S. Skopenkov, A. Combinatorics Discrete Mathematics Algebraic Topology Geometric Topology 57Q35, 15B99, 68R99, 55N45 The classical Heawood inequality states that if the complete graph $K_n$ on $n$ vertices is embeddable in the sphere with $g$ handles, then $g \ge\dfrac{(n-3)(n-4)}{12}$. A higher-dimensional analogue of the Heawood inequality is the Kühnel conjecture. In a simplified form it states that for every integer $k>0$ there is $c_k>0$ such that if the union of $k$-faces of $n$-simplex embeds into the connected sum of $g$ copies of the Cartesian product $S^k\times S^k$ of two $k$-dimensional spheres, then $g\ge c_k n^{k+1}$. For $k>1$ only linear estimates were known. We present a quadratic estimate $g\ge c_k n^2$. The proof is based on beautiful and fruitful interplay between geometric topology, combinatorics and linear algebra. |
| title | A quadratic estimation for the Kühnel conjecture on embeddings |
| topic | Combinatorics Discrete Mathematics Algebraic Topology Geometric Topology 57Q35, 15B99, 68R99, 55N45 |
| url | https://arxiv.org/abs/2208.04188 |