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

TL;DR

This paper provides a concise, accessible proof of a theorem on the non-embeddability of certain high-dimensional complexes into specific 2k-manifolds, extending classical embedding inequalities.

Contribution

It offers a simplified proof of the Patak-Tancer theorem, generalizing the Heawood inequality to higher dimensions and complex embeddings.

Findings

Establishes lower bounds on genus for embeddings into connected sums of spheres.

Provides inequalities relating Euler characteristic and embeddability into 2k-manifolds.

Simplifies understanding of high-dimensional embedding obstructions.

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 $g\ge\dfrac{n-2k-1}{k+2}$. (b) If $\Delta_n^k$ PL embeds into a closed $(k-1)$-connected PL $2k$-manifold $M$, then $(-1)^k(\chi(M)-2)\ge\dfrac{n-2k-1}{k+1}$.