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

Abstract

In 2019 P. Patak and M. Tancer obtained the following higher-dimensional generalization of the Heawood inequality on embeddings of graphs into surfaces. We present a short well-structured proof 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 $g\ge \frac{n-2k-1}{k+2}$. (b) If ${\Delta }_n^k$ PL embeds into a closed $(k-1)$-connected PL $2k$-manifold $M$, then $(-1{)}^k(\chi (M)-2)\ge \frac{n-2k-1}{k+1}$.

19 pages; exposition improved

References in corpus (2)

• Embedding and knotting of manifolds in Euclidean spaces
• On different reliability standards in current mathematical research