License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2016.51
URN: urn:nbn:de:0030-drops-59438
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5943/
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Mabillard, Isaac ; Wagner, Uli

Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range

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Abstract

Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without higher-multiplicity intersections.

We focus on conditions for the existence of almost r-embeddings, i.e., maps f: K -> R^d such that the intersection of f(sigma_1), ..., f(sigma_r) is empty whenever sigma_1,...,sigma_r are pairwise disjoint simplices of K.

Generalizing the classical Haefliger-Weber embeddability criterion, we show that a well-known necessary deleted product condition for the existence of almost r-embeddings is sufficient in a suitable r-metastable range of dimensions: If r d > (r+1) dim K + 2 then there exists an almost r-embedding K-> R^d if and only if there exists an equivariant map of the r-fold deleted product of K to the sphere S^(d(r-1)-1).

This significantly extends one of the main results of our previous paper (which treated the special case where d=rk and dim K=(r-1)k, for some k> 2), and settles an open question raised there.

BibTeX - Entry

@InProceedings{mabillard_et_al:LIPIcs:2016:5943,
  author =	{Isaac Mabillard and Uli Wagner},
  title =	{{Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{51:1--51:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5943},
  URN =		{urn:nbn:de:0030-drops-59438},
  doi =		{10.4230/LIPIcs.SoCG.2016.51},
  annote =	{Keywords: Topological Combinatorics, Tverberg-Type Problems, Simplicial Complexes, Piecewise-Linear Topology, Haefliger-Weber Theorem}
}

Keywords: Topological Combinatorics, Tverberg-Type Problems, Simplicial Complexes, Piecewise-Linear Topology, Haefliger-Weber Theorem
Collection: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 10.06.2016

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