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

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

Authors: E. Kogan and A. Skopenkov

Abstract: Let be a -dimensional simplicial complex having faces of dimension and a closed -connected PL -dimensional manifold. We prove that for odd embeds into if and only if there are

a skew-symmetric -matrix with -entries whose rank over does not exceed ,

a general position PL map , and

a collection of orientations on -faces of

such that for any nonadjacent -faces of the element equals to the algebraic intersection of and .

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

See [Kogan&Skopenkov2021].

## References

- [Kogan&Skopenkov2021] E. Kogan and A. Skopenkov,
*Embeddings of -complexes in -manifolds and minimum rank of partial symmetric matrices*, arXiv:2112.06636.