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

I. Mabillard, U. Wagner, in:, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2016, p. 51.1-51.12.

Download
OA 622.97 KB

Conference Paper | Published | English
Department
Series Title
LIPIcs
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.
Publishing Year
Date Published
2016-06-01
Volume
51
Page
51.1 - 51.12
Conference
SoCG: Symposium on Computational Geometry
Conference Location
Medford, MA, USA
Conference Date
2016-06-14 – 2016-06-17
IST-REx-ID

Cite this

Mabillard I, Wagner U. Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range. In: Vol 51. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH; 2016:51.1-51.12. doi:10.4230/LIPIcs.SoCG.2016.51
Mabillard, I., & Wagner, U. (2016). Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range (Vol. 51, p. 51.1-51.12). Presented at the SoCG: Symposium on Computational Geometry, Medford, MA, USA: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH. https://doi.org/10.4230/LIPIcs.SoCG.2016.51
Mabillard, Isaac, and Uli Wagner. “Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range,” 51:51.1-51.12. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2016. https://doi.org/10.4230/LIPIcs.SoCG.2016.51.
I. Mabillard and U. Wagner, “Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range,” presented at the SoCG: Symposium on Computational Geometry, Medford, MA, USA, 2016, vol. 51, p. 51.1-51.12.
Mabillard I, Wagner U. 2016. Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 51. 51.1-51.12.
Mabillard, Isaac, and Uli Wagner. Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range. Vol. 51, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2016, p. 51.1-51.12, doi:10.4230/LIPIcs.SoCG.2016.51.
All files available under the following license(s):
Creative Commons License:
CC-BYCreative Commons Attribution 4.0 International Public License (CC-BY 4.0)
Main File(s)
Access Level
OA Open Access
Last Uploaded
2018-12-12T10:10:06Z


Export

Marked Publications

Open Data IST Research Explorer

Search this title in

Google Scholar