On Generalized Heawood Inequalities for Manifolds: A Van Kampen–Flores-type Nonembeddability Result

Xavier Goaoc (1) , Isaac Mabillard (2) , Pavel Paták (3) , Zuzana Patáková (4) , Martin Tancer (4) , Uli Wagner (2)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 IST Austria - Institute of Science and Technology [Klosterneuburg, Austria]
3 Department of Algebra (MFF-UK)
4 KAM - Department of Applied Mathematics [Prague]

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 K_n embeds in a closed surface M if and only if (n-3)(n-4)≤6b₁(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^2k if and only if n≤2k+2. Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k-1)-connected 2k-manifold with k^th Z₂-Betti number b_k only if the following generalized Heawood inequality holds: binom(n-k-1,k+1)≤binom(2k+1,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 2k-manifold with k^th Z 2-Betti number b_k, then n≤2b_kbinom(2k+2,k)+2k+5. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k-1)-connected. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.

Mots clés: Heawood Inequality; Embeddings; Van Kampen–Flores; Manifolds

Domaines: Géométrie algorithmique [cs.CG]

Xavier Goaoc, Isaac Mabillard, Pavel Paták, Zuzana Patáková, Martin Tancer, et al.. On Generalized Heawood Inequalities for Manifolds: A Van Kampen–Flores-type Nonembeddability Result. 31st International Symposium on Computational Geometry (SoCG’15), Jun 2015, Eindhoven, Netherlands. pp.476-490, ⟨10.4230/LIPIcs.SOCG.2015.476⟩. ⟨hal-01393019⟩