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

Mabillard, Isaac and Wagner, Uli (2016) Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range. In: SoCG: Symposium on Computational Geometry, June 14 - 17, 2016, Boston, MA, USA.

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Official URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.51

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.

Item Type: Conference or Workshop Item (Paper)
Uncontrolled Keywords: Simplicial complexes, Tverberg-type problems, Topological combinatorics, Haefliger-Weber Theorem, Piecewise-linear topology
Subjects: 500 Science > 510 Mathematics
Research Group: Wagner Group
SWORD Depositor: Sword Import User
Depositing User: Sword Import User
Date Deposited: 21 Jul 2016 07:20
Last Modified: 05 Sep 2017 09:23
URI: https://repository.ist.ac.at/id/eprint/621

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