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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) &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.

  • Czech name

  • Czech description

Classification

  • Type

    J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database

  • CEP classification

  • 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

  • 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