Goaoc, Xavier ;
Mabillard, Isaac ;
Paták, Pavel ;
Patáková, Zuzana ;
Tancer, Martin ;
Wagner, Uli
On Generalized Heawood Inequalities for Manifolds: A Van KampenFlorestype 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 Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M if and only if (n3)(n4) is at most 6b_1(M), where b_1(M) is the first Z_2Betti number of M. On the other hand, Van Kampen and Flores proved that the kskeleton of the ndimensional simplex (the higherdimensional analogue of K_{n+1}) embeds in R^{2k} if and only if n is less or equal to 2k+2.
Two decades ago, Kuhnel conjectured that the kskeleton of the nsimplex embeds in a compact, (k1)connected 2kmanifold with kth Z_2Betti number b_k only if the following generalized Heawood inequality holds: binom{nk1}{k+1} is at most binom{2k+1}{k+1} b_k. This is a common generalization of the case of graphs on surfaces as well as the Van KampenFlores theorem.
In the spirit of Kuhnel's conjecture, we prove that if the kskeleton of the nsimplex embeds in a 2kmanifold with kth Z_2Betti number b_k, then n is at most 2b_k binom{2k+2}{k} + 2k + 5. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k1)connected. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.
BibTeX  Entry
@InProceedings{goaoc_et_al:LIPIcs:2015:5125,
author = {Xavier Goaoc and Isaac Mabillard and Pavel Pat{\'a}k and Zuzana Pat{\'a}kov{\'a} and Martin Tancer and Uli Wagner},
title = {{On Generalized Heawood Inequalities for Manifolds: A Van KampenFlorestype Nonembeddability Result}},
booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)},
pages = {476490},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897835},
ISSN = {18688969},
year = {2015},
volume = {34},
editor = {Lars Arge and J{\'a}nos Pach},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5125},
URN = {urn:nbn:de:0030drops51256},
doi = {10.4230/LIPIcs.SOCG.2015.476},
annote = {Keywords: Heawood Inequality, Embeddings, Van Kampen–Flores, Manifolds}
}
2015
Keywords: 

Heawood Inequality, Embeddings, Van Kampen–Flores, Manifolds 
Seminar: 

31st International Symposium on Computational Geometry (SoCG 2015)

Issue date: 

2015 
Date of publication: 

2015 