A short exposition of the Patak-Tancer theorem on non-embeddability of k-complexes in 2k-manifolds
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In 2019 P. Patak and M. Tancer obtained the following higher-dimensional generalization of the Heawood inequality on embeddings of graphs into surfaces. We expose this result in a short well-structured way accessible to non-specialists in the field. Let Δ_n^k be the union of k-dimensional faces of the n-dimensional simplex. Theorem. (a) If Δ_n^k PL embeds into the connected sum of g copies of the Cartesian product S^k× S^k of two k-dimensional spheres, then g≥n-2kk+2. (b) If Δ_n^k PL embeds into a closed (k-1)-connected PL 2k-manifold M, then (-1)^k(χ(M)-2)≥n-2kk+1.
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