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- W4200631907 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 $kge3$ odd $K$ embeds into $M$ if and only if there are $bullet$ a skew-symmetric $ntimes n$-matrix $A$ with $mathbb Z$-entries whose rank over $mathbb Q$ does not exceed $rk H_k(M;mathbb Z)$, $bullet$ a general position PL map $f:Ktomathbb R^{2k}$, and $bullet$ a collection of orientations on $k$-faces of $K$ such that for any nonadjacent $k$-faces $sigma,tau$ of $K$ the element $A_{sigma,tau}$ equals to the algebraic intersection of $fsigma$ and $ftau$. We prove some analogues of this result including those for $mathbb Z_2$- and $mathbb Z$-embeddability. Our results generalize the Bikeev-Fulek-Kynv cl-Schaefer-Stefankoviv c criteria for the $mathbb Z_2$- and $mathbb Z$-embeddability of graphs to surfaces, and are related to the Harris-Krushkal-Johnson-Pat'ak-Tancer criteria for the embeddability of $k$-complexes into $2k$-manifolds." @default.
- W4200631907 created "2021-12-31" @default.
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- W4200631907 date "2021-12-06" @default.
- W4200631907 modified "2023-09-28" @default.
- W4200631907 title "Embeddings of $k$-complexes in $2k$-manifolds and minimum rank of partial symmetric matrices" @default.
- W4200631907 hasPublicationYear "2021" @default.
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