{"year":"2023","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"}],"oa":1,"isi":1,"page":"675-717","day":"01","language":[{"iso":"eng"}],"author":[{"first_name":"Uli","orcid":"0000-0002-1494-0568","last_name":"Wagner","full_name":"Wagner, Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87"},{"id":"4C20D868-F248-11E8-B48F-1D18A9856A87","full_name":"Wild, Pascal","last_name":"Wild","first_name":"Pascal"}],"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","short":"CC BY (4.0)","image":"/images/cc_by.png"},"has_accepted_license":"1","ddc":["510"],"publication_identifier":{"issn":["0021-2172"],"eissn":["1565-8511"]},"article_processing_charge":"Yes (via OA deal)","title":"Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes","intvolume":"       256","citation":{"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>.","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>","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>","short":"U. Wagner, P. Wild, Israel Journal of Mathematics 256 (2023) 675–717.","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.","ista":"Wagner U, Wild P. 2023. Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes. Israel Journal of Mathematics. 256(2), 675–717.","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>."},"scopus_import":"1","quality_controlled":"1","volume":256,"date_published":"2023-09-01T00:00:00Z","external_id":{"isi":["001081646400010"]},"license":"https://creativecommons.org/licenses/by/4.0/","publisher":"Springer Nature","publication_status":"published","month":"09","date_created":"2023-10-22T22:01:14Z","issue":"2","publication":"Israel Journal of Mathematics","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","date_updated":"2023-12-13T13:09:07Z","department":[{"_id":"UlWa"}],"doi":"10.1007/s11856-023-2521-9","oa_version":"Published Version","file_date_updated":"2023-10-31T11:20:31Z","_id":"14445","file":[{"date_created":"2023-10-31T11:20:31Z","file_size":623787,"date_updated":"2023-10-31T11:20:31Z","success":1,"creator":"dernst","access_level":"open_access","relation":"main_file","file_id":"14475","content_type":"application/pdf","file_name":"2023_IsraelJourMath_Wagner.pdf","checksum":"fbb05619fe4b650f341cc730425dd9c3"}],"article_type":"original","status":"public"}