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

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 ≥ 3 odd K embeds into M if and only if there are • a skew-symmetric n × n-matrix A with Z-entries whose rank over Q does not exceed rkHk(M ;Z), • a general position PL map f : K → R2k, and • a collection of orientations on k-faces of K such that for any nonadjacent k-faces σ, τ of K the element Aσ,τ equals to the algebraic intersection of fσ and fτ . We prove some analogues of this result including those for Z2and Z-embeddability. Our results generalize the Bikeev-Fulek-Kynčl-Schaefer-Stefankovič criteria for the Z2and Z-embeddability of graphs to surfaces, and are related to the Harris-KrushkalJohnson-Paták-Tancer criteria for the embeddability of k-complexes into 2k-manifolds.

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