Embeddings of k-complexes in 2k-manifolds and minimum rank of partial symmetric matrices

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Title: Embeddings of k-complexes in 2k-manifolds and minimum rank of partial symmetric matrices

Authors: E. Kogan and A. Skopenkov

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 \mathbb Z-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 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 f\sigma and f\tau.

We prove some analogues of this result including those for \mathbb Z_2- and \mathbb Z-embeddability. Our results generalize the Bikeev-Fulek-Kyncl-Schaefer-Stefankovic criteria for the \mathbb Z_2- and \mathbb Z-embeddability of graphs to surfaces, and are related to the Harris-Krushkal-Johnson-Patak-Tancer criteria for the embeddability of k-complexes into 2k-manifolds.

See [Kogan&Skopenkov2021].

References

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