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

Abstract : 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 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 Kn+1) embeds in R2k 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 kth Z2-Betti number bk only if the following generalized Heawood inequality holds: binom(n-k-1,k+1)≤binom(2k+1,k+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 2k-manifold with kth Z 2-Betti number bk, then n≤2bkbinom(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.
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Lars Arge; János Pach. 31st International Symposium on Computational Geometry (SoCG’15), Jun 2015, Eindhoven, Netherlands. Proceedings of 31st International Symposium on Computational Geometry, pp.476-490, LIPIcs. 〈10.4230/LIPIcs.SOCG.2015.476〉
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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. Lars Arge; János Pach. 31st International Symposium on Computational Geometry (SoCG’15), Jun 2015, Eindhoven, Netherlands. Proceedings of 31st International Symposium on Computational Geometry, pp.476-490, LIPIcs. 〈10.4230/LIPIcs.SOCG.2015.476〉. 〈hal-01393019〉

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