On generalized Heawood inequalities for manifolds: a van Kampen–Flores-type nonembeddability result

• Published in: Israel journal of mathematics 2017-10, Vol.222 (2), p.841-866
• Main Authors: Goaoc, Xavier, Mabillard, Isaac, Paták, Pavel, Patáková, Zuzana, Tancer, Martin, Wagner, Uli
• Format: Article
• Language: English
• Subjects: Algebra; Analysis; Applications of Mathematics; Computational Geometry; Computer Science; Graphs; Group Theory and Generalizations; Homology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Municipal bonds; Theorems; Theoretical
• ISSN: 0021-2172; 1565-8511
• DOI: 10.1007/s11856-017-1607-7

The fact that the complete graph K 5 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 K n embeds in a closed surface M (other than the Klein bottle) if and only if ( n −3)( n −4) ≤ 6 b 1 ( M ), where b 1 ( M ) is the first Z 2 -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 K n+1 ) embeds in R 2 k if and only if n ≤ 2 k + 1. Two decades ago, Kühnel conjectured that the k -skeleton of the n -simplex embeds in a compact, ( k − 1)-connected 2 k -manifold with k th Z 2 -Betti number b k only if the following generalized Heawood inequality holds: ( k +1 n − k −1 ) ≤ ( k +1 2 k +1 ) b k . 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 2 k -manifold with k th Z 2 -Betti number bk, then n ≤ 2 b k ( k 2 k +2 )+2 k +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.