Mabillard, Isaac ;
Wagner, Uli
Eliminating HigherMultiplicity Intersections, II. The Deleted Product Criterion in the rMetastable Range
Abstract
Motivated by Tverbergtype 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 highermultiplicity intersections.
We focus on conditions for the existence of almost rembeddings, 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 HaefligerWeber embeddability criterion, we show that a wellknown necessary deleted product condition for the existence of almost rembeddings is sufficient in a suitable rmetastable range of dimensions: If r d > (r+1) dim K + 2 then there exists an almost rembedding K> R^d if and only if there exists an equivariant map of the rfold deleted product of K to the sphere S^(d(r1)1).
This significantly extends one of the main results of our previous paper (which treated the special case where d=rk and dim K=(r1)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 HigherMultiplicity Intersections, II. The Deleted Product Criterion in the rMetastable Range}},
booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)},
pages = {51:151:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770095},
ISSN = {18688969},
year = {2016},
volume = {51},
editor = {S{\'a}ndor Fekete and Anna Lubiw},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5943},
URN = {urn:nbn:de:0030drops59438},
doi = {10.4230/LIPIcs.SoCG.2016.51},
annote = {Keywords: Topological Combinatorics, TverbergType Problems, Simplicial Complexes, PiecewiseLinear Topology, HaefligerWeber Theorem}
}
2016
Keywords: 

Topological Combinatorics, TverbergType Problems, Simplicial Complexes, PiecewiseLinear Topology, HaefligerWeber Theorem 
Seminar: 

32nd International Symposium on Computational Geometry (SoCG 2016)

Issue date: 

2016 
Date of publication: 

2016 