On generalized Heawood inequalities for manifolds: a van Kampen-Flores-type nonembeddability result (Q1686405)

The non-planarity of the complete graph \(K_5\) has been generalized in two independent directions. On the one hand, the Heawood number of a surface determines the maximum \(n\) such that \(K_n\) embeds in this surface (exception: the Klein bottle). This number is the maximum \(n\) such that \(\binom{n-3}{2}\leq 3b_1(M)\), where \(b_1(M)\) is the first \(\mathbb{Z}_2\)-Betti number of \(M\). On the other hand, the theorem of van Kampen and Flores states that the \(k\)-skeleton of the \(n\)-dimensional simplex does not embed in the ordinary \((2k)\)-dimensional space if \(n\geq 2k+2\). This raises the question for a higher-dimensional Heawood number as a common generalization of these two facts. The reviewer conjectured 25 years ago that the inequality \(\binom{n-k-1}{k+1}\leq \binom{2k+1}{k+1} b_k\) is necessary for the embeddability of the complete \(k\)-skeleton of the \(n\)-dimensional simplex into a compact \((k-1)\)-connected \((2k)\)-manifold with \(k\)th Betti number \(b_k\). Towards this conjecture, the authors prove the weaker inequality \(n\leq 2b_k\binom{2k+2}{k}+ 2k+4\) if the \(k\)-skeleton of the \(n\)-dimensional simplex embeds in a compact \((2k)\)-manifold with \(k\)th \(\mathbb{Z}_2\)-Betti number \(b_k\). This is not assumed to be \((k-1)\)-connected. There is a generalization to maps without \(q\)-covered points.