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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 $Delta_n^k$ be the union of $k$-dimensional faces of the $n$-dimensional simplex. Theorem. (a) If $Delta_n^k$ PL embeds into the connected sum of $g$ copies of the Cartesian product $S^ktimes S^k$ of two $k$-dimensional spheres, then $ggedfrac{n-2k}{k+2}$. (b) If $Delta_n^k$ PL embeds into a closed $(k-1)$-connected PL $2k$-manifold $M$, then $(-1)^k(chi(M)-2)gedfrac{n-2k}{k+1}$.
We introduce the notion of coupled embeddability, defined for maps on products of topological spaces. We use known results for nonsingular biskew and bilinear maps to generate simple examples and nonexamples of coupled embeddings. We study genericity
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In this paper, we prove a Liouville theorem for holomorphic functions on a class of complete Gauduchon manifolds. This generalizes a result of Yau for complete Kahler manifolds to the complete non-Kahler case.