Embeddings of $k$-complexes in $2k$-manifolds and minimum rank of partial symmetric matrices
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866918519567810560 |
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| author | Skopenkov, A. |
| author_facet | Skopenkov, A. |
| contents | Let $K$ be a $k$-dimensional simplicial complex having $n$ faces of dimension $k$, and $M$ a closed $(k-1)$-connected PL $2k$-dimensional manifold. We prove that for $k\ge3$ odd $K$ embeds into $M$ if and only if there are
$\bullet$ a skew-symmetric $n\times n$-matrix $A$ with integer entries, whose rank over $\mathbb Q$ does not exceed $rk H_k(M;\mathbb Z)$,
$\bullet$ a general position PL map $f:K\to\mathbb R^{2k}$, and
$\bullet$ orientations on $k$-faces of $K$ such that for any nonadjacent $k$-faces $σ,τ$ of $K$ the entry $A_{σ,τ}$ equals to the algebraic intersection of $fσ$ and $fτ$.
We prove some analogues of this result (for any parity of $k$), including those for $\mathbb Z_2$- and $\mathbb Z$-embeddability. Our results generalize the Bikeev-Fulek-Kyn\v cl criteria for the $\mathbb Z_2$- and $\mathbb Z$-embeddability of graphs to surfaces, and are related to the Harris-Krushkal-Johnson-Paták-Tancer criteria for the embeddability of $k$-complexes into $2k$-manifolds. The main novelty of this paper is passing from the cohomology condition of Paták-Tancer to the simpler extendability of some intersection function to a low-rank matrix (defined in the paper using the idea of Fulek-Kyn\v cl). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_06636 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Embeddings of $k$-complexes in $2k$-manifolds and minimum rank of partial symmetric matrices Skopenkov, A. Geometric Topology Computational Geometry Algebraic Topology Combinatorics 57Q35, 55S35, 15A83 Let $K$ be a $k$-dimensional simplicial complex having $n$ faces of dimension $k$, and $M$ a closed $(k-1)$-connected PL $2k$-dimensional manifold. We prove that for $k\ge3$ odd $K$ embeds into $M$ if and only if there are $\bullet$ a skew-symmetric $n\times n$-matrix $A$ with integer entries, whose rank over $\mathbb Q$ does not exceed $rk H_k(M;\mathbb Z)$, $\bullet$ a general position PL map $f:K\to\mathbb R^{2k}$, and $\bullet$ orientations on $k$-faces of $K$ such that for any nonadjacent $k$-faces $σ,τ$ of $K$ the entry $A_{σ,τ}$ equals to the algebraic intersection of $fσ$ and $fτ$. We prove some analogues of this result (for any parity of $k$), including those for $\mathbb Z_2$- and $\mathbb Z$-embeddability. Our results generalize the Bikeev-Fulek-Kyn\v cl criteria for the $\mathbb Z_2$- and $\mathbb Z$-embeddability of graphs to surfaces, and are related to the Harris-Krushkal-Johnson-Paták-Tancer criteria for the embeddability of $k$-complexes into $2k$-manifolds. The main novelty of this paper is passing from the cohomology condition of Paták-Tancer to the simpler extendability of some intersection function to a low-rank matrix (defined in the paper using the idea of Fulek-Kyn\v cl). |
| title | Embeddings of $k$-complexes in $2k$-manifolds and minimum rank of partial symmetric matrices |
| topic | Geometric Topology Computational Geometry Algebraic Topology Combinatorics 57Q35, 55S35, 15A83 |
| url | https://arxiv.org/abs/2112.06636 |