{"tmp":{"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","short":"CC BY (4.0)"},"day":"01","external_id":{"isi":["001081646400010"]},"scopus_import":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","doi":"10.1007/s11856-023-2521-9","article_type":"original","volume":256,"publication_status":"published","corr_author":"1","month":"09","quality_controlled":"1","page":"675-717","_id":"14445","publisher":"Springer Nature","date_updated":"2024-10-09T21:07:12Z","language":[{"iso":"eng"}],"status":"public","date_created":"2023-10-22T22:01:14Z","oa_version":"Published Version","isi":1,"title":"Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes","abstract":[{"text":"We prove the following quantitative Borsuk–Ulam-type result (an equivariant analogue of Gromov’s Topological Overlap Theorem): Let X be a free ℤ/2-complex of dimension d with coboundary expansion at least ηk in dimension 0 ≤ k < d. Then for every equivariant map F: X →ℤ/2 ℝd, the fraction of d-simplices σ of X with 0 ∈ F (σ) is at least 2−d Π d−1k=0ηk.\r\n\r\nAs an application, we show that for every sufficiently thick d-dimensional spherical building Y and every map f: Y → ℝ2d, we have f(σ) ∩ f(τ) ≠ ∅ for a constant fraction μd > 0 of pairs {σ, τ} of d-simplices of Y. In particular, such complexes are non-embeddable into ℝ2d, which proves a conjecture of Tancer and Vorwerk for sufficiently thick spherical buildings.\r\n\r\nWe complement these results by upper bounds on the coboundary expansion of two families of simplicial complexes; this indicates some limitations to the bounds one can obtain by straighforward applications of the quantitative Borsuk–Ulam theorem. Specifically, we prove\r\n\r\n• an upper bound of (d + 1)/2d on the normalized (d − 1)-th coboundary expansion constant of complete (d + 1)-partite d-dimensional complexes (under a mild divisibility assumption on the sizes of the parts); and\r\n\r\n• an upper bound of (d + 1)/2d + ε on the normalized (d − 1)-th coboundary expansion of the d-dimensional spherical building associated with GLd+2(Fq) for any ε > 0 and sufficiently large q. This disproves, in a rather strong sense, a conjecture of Lubotzky, Meshulam and Mozes.","lang":"eng"}],"file":[{"content_type":"application/pdf","date_created":"2023-10-31T11:20:31Z","creator":"dernst","success":1,"file_name":"2023_IsraelJourMath_Wagner.pdf","file_id":"14475","date_updated":"2023-10-31T11:20:31Z","file_size":623787,"relation":"main_file","access_level":"open_access","checksum":"fbb05619fe4b650f341cc730425dd9c3"}],"citation":{"ieee":"U. Wagner and P. Wild, “Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes,” <i>Israel Journal of Mathematics</i>, vol. 256, no. 2. Springer Nature, pp. 675–717, 2023.","ama":"Wagner U, Wild P. Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes. <i>Israel Journal of Mathematics</i>. 2023;256(2):675-717. doi:<a href=\"https://doi.org/10.1007/s11856-023-2521-9\">10.1007/s11856-023-2521-9</a>","apa":"Wagner, U., &#38; Wild, P. (2023). Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes. <i>Israel Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11856-023-2521-9\">https://doi.org/10.1007/s11856-023-2521-9</a>","ista":"Wagner U, Wild P. 2023. Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes. Israel Journal of Mathematics. 256(2), 675–717.","short":"U. Wagner, P. Wild, Israel Journal of Mathematics 256 (2023) 675–717.","chicago":"Wagner, Uli, and Pascal Wild. “Coboundary Expansion, Equivariant Overlap, and Crossing Numbers of Simplicial Complexes.” <i>Israel Journal of Mathematics</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s11856-023-2521-9\">https://doi.org/10.1007/s11856-023-2521-9</a>.","mla":"Wagner, Uli, and Pascal Wild. “Coboundary Expansion, Equivariant Overlap, and Crossing Numbers of Simplicial Complexes.” <i>Israel Journal of Mathematics</i>, vol. 256, no. 2, Springer Nature, 2023, pp. 675–717, doi:<a href=\"https://doi.org/10.1007/s11856-023-2521-9\">10.1007/s11856-023-2521-9</a>."},"has_accepted_license":"1","department":[{"_id":"UlWa"}],"issue":"2","article_processing_charge":"Yes (via OA deal)","publication_identifier":{"eissn":["1565-8511"],"issn":["0021-2172"]},"author":[{"first_name":"Uli","last_name":"Wagner","orcid":"0000-0002-1494-0568","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli"},{"full_name":"Wild, Pascal","first_name":"Pascal","last_name":"Wild","id":"4C20D868-F248-11E8-B48F-1D18A9856A87"}],"intvolume":"       256","ddc":["510"],"year":"2023","publication":"Israel Journal of Mathematics","oa":1,"date_published":"2023-09-01T00:00:00Z","file_date_updated":"2023-10-31T11:20:31Z","type":"journal_article"}