Abstract: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 $\sigma,\tau$ of $K$ the entry $A_{\sigma,\tau}$ equals to the algebraic intersection of $f\sigma$ and $f\tau$.
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).
| Comments: | 25 pages, 2 figures, exposition improved |
| Subjects: | Geometric Topology (math.GT); Computational Geometry (cs.CG); Algebraic Topology (math.AT); Combinatorics (math.CO) |
| MSC classes: | 57Q35, 55S35, 15A83 |
| Cite as: | arXiv:2112.06636 [math.GT] |
| (or arXiv:2112.06636v5 [math.GT] for this version) | |
| https://doi.org/10.48550/arXiv.2112.06636 arXiv-issued DOI via DataCite |
From: Arkadiy Skopenkov [view email]
[v1]
Mon, 6 Dec 2021 20:11:30 UTC (67 KB)
[v2]
Thu, 3 Mar 2022 09:21:55 UTC (69 KB)
[v3]
Sun, 28 Jul 2024 09:32:13 UTC (75 KB)
[v4]
Mon, 14 Oct 2024 09:39:37 UTC (85 KB)
[v5]
Sun, 24 May 2026 10:01:23 UTC (102 KB)
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