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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SOCG.2015.476
URN: urn:nbn:de:0030-drops-51256
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5125/
Goaoc, Xavier ;
Mabillard, Isaac ;
Paták, Pavel ;
Patáková, Zuzana ;
Tancer, Martin ;
Wagner, Uli
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 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) is at most 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 is less or equal to 2k+2.
Two decades ago, Kuhnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k-1)-connected 2k-manifold with kth Z_2-Betti number b_k only if the following generalized Heawood inequality holds: binom{n-k-1}{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 Kampen--Flores theorem.
In the spirit of Kuhnel's conjecture, we prove that if the k-skeleton of the n-simplex embeds in a 2k-manifold with kth Z_2-Betti 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 (k-1)-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 Kampen-Flores-type Nonembeddability Result}},
booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)},
pages = {476--490},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-83-5},
ISSN = {1868-8969},
year = {2015},
volume = {34},
editor = {Lars Arge and J{\'a}nos Pach},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5125},
URN = {urn:nbn:de:0030-drops-51256},
doi = {10.4230/LIPIcs.SOCG.2015.476},
annote = {Keywords: Heawood Inequality, Embeddings, Van Kampen–Flores, Manifolds}
}
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Keywords: |
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Heawood Inequality, Embeddings, Van Kampen–Flores, Manifolds |
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Collection: |
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31st International Symposium on Computational Geometry (SoCG 2015) |
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Issue Date: |
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2015 |
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Date of publication: |
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12.06.2015 |
