---
_id: '610'
abstract:
- lang: eng
  text: 'The fact that the complete graph K5 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 Kn embeds in a closed surface M (other than the Klein bottle) if and only
    if (n−3)(n−4) ≤ 6b1(M), where b1(M) is the first Z2-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 R2k if and only if
    n ≤ 2k + 1. Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex
    embeds in a compact, (k − 1)-connected 2k-manifold with kth Z2-Betti number bk
    only if the following generalized Heawood inequality holds: (k+1 n−k−1) ≤ (k+1
    2k+1)bk. This is a common generalization of the case of graphs on surfaces as
    well as the van Kampen–Flores theorem. In the spirit of Kühnel’s conjecture, we
    prove that if the k-skeleton of the n-simplex embeds in a compact 2k-manifold
    with kth Z2-Betti number bk, then n ≤ 2bk(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.'
acknowledgement: The work by Z. P. was partially supported by the Israel Science Foundation
  grant ISF-768/12. The work by Z. P. and M. T. was partially supported by the project
  CE-ITI (GACR P202/12/G061) of the Czech Science Foundation and by the ERC Advanced
  Grant No. 267165. Part of the research work of M.T. was conducted at IST Austria,
  supported by an IST Fellowship. The research of P. P. was supported by the ERC Advanced
  grant no. 320924. The work by I. M. and U. W. was supported by the Swiss National
  Science Foundation (grants SNSF-200020-138230 and SNSF-PP00P2-138948). The collaboration
  between U. W. and X. G. was partially supported by the LabEx Bézout (ANR-10-LABX-58).
article_processing_charge: No
arxiv: 1
author:
- first_name: Xavier
  full_name: Goaoc, Xavier
  last_name: Goaoc
- first_name: Isaac
  full_name: Mabillard, Isaac
  id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
  last_name: Mabillard
- first_name: Pavel
  full_name: Paták, Pavel
  last_name: Paták
- first_name: Zuzana
  full_name: Patakova, Zuzana
  id: 48B57058-F248-11E8-B48F-1D18A9856A87
  last_name: Patakova
  orcid: 0000-0002-3975-1683
- first_name: Martin
  full_name: Tancer, Martin
  id: 38AC689C-F248-11E8-B48F-1D18A9856A87
  last_name: Tancer
  orcid: 0000-0002-1191-6714
- first_name: Uli
  full_name: Wagner, Uli
  id: 36690CA2-F248-11E8-B48F-1D18A9856A87
  last_name: Wagner
  orcid: 0000-0002-1494-0568
citation:
  ama: 'Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. On generalized
    Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability
    result. <i>Israel Journal of Mathematics</i>. 2017;222(2):841-866. doi:<a href="https://doi.org/10.1007/s11856-017-1607-7">10.1007/s11856-017-1607-7</a>'
  apa: 'Goaoc, X., Mabillard, I., Paták, P., Patakova, Z., Tancer, M., &#38; Wagner,
    U. (2017). On generalized Heawood inequalities for manifolds: A van Kampen–Flores
    type nonembeddability result. <i>Israel Journal of Mathematics</i>. Springer.
    <a href="https://doi.org/10.1007/s11856-017-1607-7">https://doi.org/10.1007/s11856-017-1607-7</a>'
  chicago: 'Goaoc, Xavier, Isaac Mabillard, Pavel Paták, Zuzana Patakova, Martin Tancer,
    and Uli Wagner. “On Generalized Heawood Inequalities for Manifolds: A van Kampen–Flores
    Type Nonembeddability Result.” <i>Israel Journal of Mathematics</i>. Springer,
    2017. <a href="https://doi.org/10.1007/s11856-017-1607-7">https://doi.org/10.1007/s11856-017-1607-7</a>.'
  ieee: 'X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, and U. Wagner,
    “On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability
    result,” <i>Israel Journal of Mathematics</i>, vol. 222, no. 2. Springer, pp.
    841–866, 2017.'
  ista: 'Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. 2017. On generalized
    Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability
    result. Israel Journal of Mathematics. 222(2), 841–866.'
  mla: 'Goaoc, Xavier, et al. “On Generalized Heawood Inequalities for Manifolds:
    A van Kampen–Flores Type Nonembeddability Result.” <i>Israel Journal of Mathematics</i>,
    vol. 222, no. 2, Springer, 2017, pp. 841–66, doi:<a href="https://doi.org/10.1007/s11856-017-1607-7">10.1007/s11856-017-1607-7</a>.'
  short: X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, U. Wagner, Israel
    Journal of Mathematics 222 (2017) 841–866.
date_created: 2018-12-11T11:47:29Z
date_published: 2017-10-01T00:00:00Z
date_updated: 2025-09-11T07:35:36Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s11856-017-1607-7
ec_funded: 1
external_id:
  arxiv:
  - '1610.09063'
  isi:
  - '000415195500009'
intvolume: '       222'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1610.09063
month: '10'
oa: 1
oa_version: Preprint
page: 841 - 866
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
publication: Israel Journal of Mathematics
publication_status: published
publisher: Springer
publist_id: '7194'
quality_controlled: '1'
related_material:
  record:
  - id: '1511'
    relation: earlier_version
    status: public
scopus_import: '1'
status: public
title: 'On generalized Heawood inequalities for manifolds: A van Kampen–Flores type
  nonembeddability result'
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 222
year: '2017'
...