Embeddings of k-complexes in 2k-manifolds and minimum rank of partial symmetric matrices
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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 rk H_k(M;ℤ), ∙ a general position PL map f:K→ℝ^2k, and ∙ a collection of orientations on k-faces of K such that for any nonadjacent k-faces σ,τ of K the element A_σ,τ equals to the algebraic intersection of fσ and fτ. We prove some analogues of this result including those for ℤ_2- and ℤ-embeddability. Our results generalize the Bikeev-Fulek-Kynčl-Schaefer-Stefankovič criteria for the ℤ_2- and 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.
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