{"project":[{"grant_number":"PP00P2_138948","name":"Embeddings in Higher Dimensions: Algorithms and Combinatorics","_id":"25FA3206-B435-11E9-9278-68D0E5697425"}],"date_published":"2016-06-01T00:00:00Z","day":"01","status":"public","alternative_title":["LIPIcs"],"pubrep_id":"621","quality_controlled":"1","file_date_updated":"2020-07-14T12:44:47Z","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"publisher":"Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH","publist_id":"5830","abstract":[{"text":"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.","lang":"eng"}],"conference":{"location":"Medford, MA, USA","start_date":"2016-06-14","end_date":"2016-06-17","name":"SoCG: Symposium on Computational Geometry"},"citation":{"apa":"Mabillard, I., &#38; 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. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2016.51\">https://doi.org/10.4230/LIPIcs.SoCG.2016.51</a>","short":"I. Mabillard, U. Wagner, in:, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2016, p. 51.1-51.12.","mla":"Mabillard, Isaac, and Uli Wagner. <i>Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range</i>. Vol. 51, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2016, p. 51.1-51.12, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2016.51\">10.4230/LIPIcs.SoCG.2016.51</a>.","ista":"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.","ieee":"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.","ama":"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:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2016.51\">10.4230/LIPIcs.SoCG.2016.51</a>","chicago":"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. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2016.51\">https://doi.org/10.4230/LIPIcs.SoCG.2016.51</a>."},"doi":"10.4230/LIPIcs.SoCG.2016.51","scopus_import":1,"author":[{"id":"32BF9DAA-F248-11E8-B48F-1D18A9856A87","first_name":"Isaac","last_name":"Mabillard","full_name":"Mabillard, Isaac"},{"orcid":"0000-0002-1494-0568","full_name":"Wagner, Uli","last_name":"Wagner","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87"}],"ddc":["510"],"type":"conference","year":"2016","month":"06","date_updated":"2021-01-12T06:50:17Z","page":"51.1 - 51.12","volume":51,"date_created":"2018-12-11T11:51:41Z","department":[{"_id":"UlWa"}],"title":"Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range","_id":"1381","intvolume":"        51","file":[{"access_level":"open_access","file_size":622969,"checksum":"92c0c3735fe908f8ded6e484005cb3b1","content_type":"application/pdf","date_created":"2018-12-12T10:10:06Z","file_id":"4791","file_name":"IST-2016-621-v1+1_LIPIcs-SoCG-2016-51.pdf","relation":"main_file","creator":"system","date_updated":"2020-07-14T12:44:47Z"}],"has_accepted_license":"1","publication_status":"published","oa_version":"Published Version","oa":1,"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"}}