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 $S^k\times S^k$ of two $k$-dimensional spheres, then $g\ge\dfrac{n-2k}{k+2}$. (b) If $\Delta_n^k$ PL embeds into a closed $(k-1)$-connected PL $2k$-manifold $M$, then $(-1)^k(\chi(M)-2)\ge\dfrac{n-2k}{k+1}$.
翻译：在2019年P. Patak和M. Tancer在将图纸嵌入表面时,对希伍德的不平等进行了以下更高层次的概括化。 我们暴露了这一结果,使外地的非专家能够以结构简便的方式进入现场。 让$Delta_n ⁇ k$成为美元维度简单x的立方面美元组合。Theorem。 (a) 如果$\Delta_n ⁇ k$PL嵌入卡通产品2美元维域的立方元之和中, $S ⁇ k\times S ⁇ k$, 然后$g\ge\dfrac{n-2k ⁇ k+2}。 (b) 如果$\Delta_näk$PLm 嵌入封闭的1美元(k-1美元)联结的PL 2k$-manyx$, 然后$(1)k(M)\\\\\\\\\\\\ g\\dfrac{n-2k+1}。 (b)
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