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

Abstract : 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 \emph{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 $(n-3)(n-4)\leq 6b_1(M)$, 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 $\R^{2k}$ if and only if~$n \le 2k+1$. Two decades ago, K\"uhnel conjectured that the $k$-skeleton of the $n$-simplex embeds in a compact, $(k-1)$-connected $2k$-manifold with $k$th $\Z_2$-Betti number $b_k$ only if the following \emph{generalized Heawood inequality} holds: $\binom{n-k-1}{k+1} \le \binom{2k+1}{k+1}b_k$. This is a common generalization of the case of graphs on surfaces as well as the van Kampen--Flores theorem. In the spirit of K\"uhnel's conjecture, we prove that if the $k$-skeleton of the $n$-simplex embeds in a compact $2k$-manifold with $k$th $\Z_2$-Betti number $b_k$, then $n \le 2b_k\binom{2k+2}{k} + 2k + 4$. 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.
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https://hal-upec-upem.archives-ouvertes.fr/hal-01744156
Contributor : Xavier Goaoc Connect in order to contact the contributor
Submitted on : Tuesday, March 27, 2018 - 10:46:19 AM
Last modification on : Tuesday, June 28, 2022 - 10:04:36 AM

Citation

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. Israel Journal of Mathematics, Hebrew University Magnes Press, 2017, 222 (2), pp.841-866. ⟨10.1007/s11856-017-1607-7⟩. ⟨hal-01744156⟩

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