ON GENERALIZED HEAWOOD INEQUALITIES FOR MANIFOLDS: A VAN KAMPEN-FLORES-TYPE NONEMBEDDABILITY RESULT

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

ON GENERALIZED HEAWOOD INEQUALITIES FOR MANIFOLDS: A VAN KAMPEN-FLORES-TYPE NONEMBEDDABILITY RESULT

Popis výsledku v původním jazyce

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) &lt;= 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 &lt;= 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) ) &lt;= ((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&apos;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 &lt;= 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&apos;s theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.

Název v anglickém jazyce

ON GENERALIZED HEAWOOD INEQUALITIES FOR MANIFOLDS: A VAN KAMPEN-FLORES-TYPE NONEMBEDDABILITY RESULT

Popis výsledku anglicky

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) &lt;= 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 &lt;= 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) ) &lt;= ((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&apos;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 &lt;= 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&apos;s theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Centrum excelence - Institut teoretické informatiky (CE-ITI)</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2017

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Israel Journal of Mathematics

ISSN

0021-2172

e-ISSN

—

Svazek periodika

222

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

IL - Stát Izrael

Počet stran výsledku

26

Strana od-do

841-866

Kód UT WoS článku

000415195500009

EID výsledku v databázi Scopus

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