---
res:
bibo_abstract:
- '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.@eng'
bibo_authorlist:
- foaf_Person:
foaf_givenName: Xavier
foaf_name: Goaoc, Xavier
foaf_surname: Goaoc
- foaf_Person:
foaf_givenName: Isaac
foaf_name: Mabillard, Isaac
foaf_surname: Mabillard
foaf_workInfoHomepage: http://www.librecat.org/personId=32BF9DAA-F248-11E8-B48F-1D18A9856A87
- foaf_Person:
foaf_givenName: Pavel
foaf_name: Paták, Pavel
foaf_surname: Paták
- foaf_Person:
foaf_givenName: Zuzana
foaf_name: Patakova, Zuzana
foaf_surname: Patakova
foaf_workInfoHomepage: http://www.librecat.org/personId=48B57058-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-3975-1683
- foaf_Person:
foaf_givenName: Martin
foaf_name: Tancer, Martin
foaf_surname: Tancer
foaf_workInfoHomepage: http://www.librecat.org/personId=38AC689C-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-1191-6714
- foaf_Person:
foaf_givenName: Uli
foaf_name: Wagner, Uli
foaf_surname: Wagner
foaf_workInfoHomepage: http://www.librecat.org/personId=36690CA2-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-1494-0568
bibo_doi: 10.1007/s11856-017-1607-7
bibo_issue: '2'
bibo_volume: 222
dct_date: 2017^xs_gYear
dct_language: eng
dct_publisher: Springer@
dct_title: 'On generalized Heawood inequalities for manifolds: A van Kampen–Flores
type nonembeddability result@'
...