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