article published yes Xavier Goaoc author Isaac Mabillard author 32BF9DAA-F248-11E8-B48F-1D18A9856A87 Pavel Paták author Zuzana Patakova author 48B57058-F248-11E8-B48F-1D18A9856A870000-0002-3975-1683 Martin Tancer author 38AC689C-F248-11E8-B48F-1D18A9856A870000-0002-1191-6714 Uli Wagner author 36690CA2-F248-11E8-B48F-1D18A9856A870000-0002-1494-0568 UlWa department International IST Postdoc Fellowship Programme project 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. Springer2017 eng 1610.09063 00041519550000910.1007/s11856-017-1607-7 2222841 - 866 https://research-explorer.ista.ac.at/record/1511 Goaoc, Xavier, et al. “On Generalized Heawood Inequalities for Manifolds: A van Kampen–Flores Type Nonembeddability Result.” <i>Israel Journal of Mathematics</i>, vol. 222, no. 2, Springer, 2017, pp. 841–66, doi:<a href="https://doi.org/10.1007/s11856-017-1607-7">10.1007/s11856-017-1607-7</a>. Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result. <i>Israel Journal of Mathematics</i>. 2017;222(2):841-866. doi:<a href="https://doi.org/10.1007/s11856-017-1607-7">10.1007/s11856-017-1607-7</a> X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, and U. Wagner, “On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result,” <i>Israel Journal of Mathematics</i>, vol. 222, no. 2. Springer, pp. 841–866, 2017. X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, U. Wagner, Israel Journal of Mathematics 222 (2017) 841–866. Goaoc, Xavier, Isaac Mabillard, Pavel Paták, Zuzana Patakova, Martin Tancer, and Uli Wagner. “On Generalized Heawood Inequalities for Manifolds: A van Kampen–Flores Type Nonembeddability Result.” <i>Israel Journal of Mathematics</i>. Springer, 2017. <a href="https://doi.org/10.1007/s11856-017-1607-7">https://doi.org/10.1007/s11856-017-1607-7</a>. Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. 2017. On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result. Israel Journal of Mathematics. 222(2), 841–866. Goaoc, X., Mabillard, I., Paták, P., Patakova, Z., Tancer, M., &#38; Wagner, U. (2017). On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result. <i>Israel Journal of Mathematics</i>. Springer. <a href="https://doi.org/10.1007/s11856-017-1607-7">https://doi.org/10.1007/s11856-017-1607-7</a> 6102018-12-11T11:47:29Z2025-09-11T07:35:36Z