Author(s):

Mabillard, Isaac; Wagner, Uli

Title: 
Eliminating highermultiplicity intersections, II. The deleted product criterion in the rmetastable range

Title Series: 
LIPIcs

Affiliation 
IST Austria 
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 doublestruck Rd without highermultiplicity intersections. We focus on conditions for the existence of almost rembeddings, i.e., maps f : K → doublestruck Rd such that f(σ1) ∩ ⋯ ∩ f(σr) = ∅ whenever σ1, ..., σr 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 rd ≥ (r + 1) dim K + 3, then there exists an almost rembedding K → doublestruck Rd if and only if there exists an equivariant map (K)Δ r → Sr Sd(r1)1, where (K)Δ r is the deleted rfold product of K, the target Sd(r1)1 is the sphere of dimension d(r  1)  1, and Sr is the symmetric group. 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 ≥ 3), and settles an open question raised there.

Keywords: 
Simplicial complexes; Tverbergtype problems; Topological combinatorics; HaefligerWeber Theorem; Piecewiselinear topology

Conference Title:

SoCG: Symposium on Computational Geometry

Volume: 
51

Conference Dates:

June 14  17, 2016

Conference Location:

Boston, MA, USA

ISBN:

9781450325943

Publisher:

ACM

Date Published:

20160601

Start Page: 
51.1

End Page:

51.12

Copyright Statement: 
CCBY

Sponsor: 
Research supported by the Swiss National Science Foundation (Project SNSFPP00P2138948).

URL: 

DOI: 
10.4230/LIPIcs.SoCG.2016.51

Open access: 
yes (OA journal) 