A short exposition of the Patak-Tancer theorem on non-embeddability of k-complexes in 2k-manifolds

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 ∆n be the union of k-dimensional faces of the n-dimensional simplex. Theorem. (a) If ∆n PL embeds into the connected sum of g copies of the Cartesian product S × S of two k-dimensional spheres, then g ≥ n− 2k

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