Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range Conference Paper


Author(s): Mabillard, Isaac; Wagner, Uli
Title: Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range
Title Series: LIPIcs
Affiliation IST Austria
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 double-struck Rd without higher-multiplicity intersections. We focus on conditions for the existence of almost r-embeddings, i.e., maps f : K → double-struck Rd such that f(σ1) ∩ ⋯ ∩ f(σr) = ∅ whenever σ1, ..., σr 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 rd ≥ (r + 1) dim K + 3, then there exists an almost r-embedding K → double-struck Rd if and only if there exists an equivariant map (K)Δ r → Sr Sd(r-1)-1, where (K)Δ r is the deleted r-fold product of K, the target Sd(r-1)-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; Tverberg-type problems; Topological combinatorics; Haefliger-Weber Theorem; Piecewise-linear topology
Conference Title: SoCG: Symposium on Computational Geometry
Volume: 51
Conference Dates: June 14 - 17, 2016
Conference Location: Boston, MA, USA
ISBN: 978-1-4503-2594-3
Publisher: ACM  
Date Published: 2016-06-01
Start Page: 51.1
End Page: 51.12
Copyright Statement: CC-BY
Sponsor: Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948).
URL:
DOI: 10.4230/LIPIcs.SoCG.2016.51
Open access: yes (OA journal)
IST Austria Authors
  1. Uli Wagner
    43 Wagner
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