ON GENERALIZED HEAWOOD INEQUALITIES FOR MANIFOLDS: A VAN KAMPEN-FLORES-TYPE NONEMBEDDABILITY RESULT
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F17%3A10368791" target="_blank" >RIV/00216208:11320/17:10368791 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1007/s11856-017-1607-7" target="_blank" >http://dx.doi.org/10.1007/s11856-017-1607-7</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s11856-017-1607-7" target="_blank" >10.1007/s11856-017-1607-7</a>
Alternative languages
Result language
angličtina
Original language name
ON GENERALIZED HEAWOOD INEQUALITIES FOR MANIFOLDS: A VAN KAMPEN-FLORES-TYPE NONEMBEDDABILITY RESULT
Original language description
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 (other than the Klein bottle) if and only if (n-3)(n-4) <= 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 Kn+1) embeds in R-2k if and only if n <= 2k + 1. 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: ((n-k-1)(k+1) ) <= ((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 compact 2k-manifold with kth Z(2)-Betti number bk, then n <= 2b(k)((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.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Center of excellence - Institute for theoretical computer science (CE-ITI)</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2017
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Israel Journal of Mathematics
ISSN
0021-2172
e-ISSN
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Volume of the periodical
222
Issue of the periodical within the volume
2
Country of publishing house
IL - THE STATE OF ISRAEL
Number of pages
26
Pages from-to
841-866
UT code for WoS article
000415195500009
EID of the result in the Scopus database
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