10.1007/s11856-017-1607-7
Goaoc, Xavier
Xavier
Goaoc
Mabillard, Isaac
Isaac
Mabillard
Paták, Pavel
Pavel
Paták
Patakova, Zuzana
Zuzana
Patakova0000-0002-3975-1683
Tancer, Martin
Martin
Tancer0000-0002-1191-6714
Wagner, Uli
Uli
Wagner0000-0002-1494-0568
On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result
Springer
2017
2018-12-11T11:47:29Z
2019-11-14T08:43:24Z
journal_article
https://research-explorer.app.ist.ac.at/record/610
https://research-explorer.app.ist.ac.at/record/610.json
The fact that the complete graph K5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph Kn embeds in a closed surface M (other than the Klein bottle) if and only if (n−3)(n−4) ≤ 6b1(M), where b1(M) is the first Z2-Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of Kn+1) embeds in R2k if and only if n ≤ 2k + 1. Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k − 1)-connected 2k-manifold with kth Z2-Betti number bk only if the following generalized Heawood inequality holds: (k+1 n−k−1) ≤ (k+1 2k+1)bk. This is a common generalization of the case of graphs on surfaces as well as the van Kampen–Flores theorem. In the spirit of Kühnel’s conjecture, we prove that if the k-skeleton of the n-simplex embeds in a compact 2k-manifold with kth Z2-Betti number bk, then n ≤ 2bk(k 2k+2)+2k+4. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k−1)-connected. Our results generalize to maps without q-covered points, in the spirit of Tverberg’s theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.