[{"page":"501–534 ","main_file_link":[{"url":"https://arxiv.org/abs/1511.03501","open_access":"1"}],"quality_controlled":"1","doi":"10.1007/s11856-021-2216-z","article_processing_charge":"No","publisher":"Springer Nature","date_updated":"2023-08-14T11:43:55Z","_id":"10220","type":"journal_article","project":[{"_id":"26611F5C-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"P31312","name":"Algorithms for Embeddings and Homotopy Theory"}],"status":"public","publication":"Israel Journal of Mathematics","date_published":"2021-10-30T00:00:00Z","acknowledgement":"Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948), by the Austrian Science Fund (FWF Project P31312-N35), by the Russian Foundation for Basic Research (Grants No. 15-01-06302 and 19-01-00169), by a Simons-IUM Fellowship, and by the D. Zimin Dynasty Foundation Grant. We would like to thank E. Alkin, A. Klyachko, V. Krushkal, S. Melikhov, M. Tancer, P. Teichner and anonymous referees for helpful comments and discussions.","isi":1,"year":"2021","external_id":{"arxiv":["1511.03501"],"isi":["000712942100013"]},"related_material":{"record":[{"id":"8183","status":"public","relation":"earlier_version"},{"id":"9308","status":"public","relation":"earlier_version"}]},"intvolume":"       245","abstract":[{"text":"We study conditions under which a finite simplicial complex K can be mapped to ℝd without higher-multiplicity intersections. An almost r-embedding is a map f: K → ℝd such that the images of any r pairwise disjoint simplices of K do not have a common point. We show that if r is not a prime power and d ≥ 2r + 1, then there is a counterexample to the topological Tverberg conjecture, i.e., there is an almost r-embedding of the (d +1)(r − 1)-simplex in ℝd. This improves on previous constructions of counterexamples (for d ≥ 3r) based on a series of papers by M. Özaydin, M. Gromov, P. Blagojević, F. Frick, G. Ziegler, and the second and fourth present authors.\r\n\r\nThe counterexamples are obtained by proving the following algebraic criterion in codimension 2: If r ≥ 3 and if K is a finite 2(r − 1)-complex, then there exists an almost r-embedding K → ℝ2r if and only if there exists a general position PL map f: K → ℝ2r such that the algebraic intersection number of the f-images of any r pairwise disjoint simplices of K is zero. This result can be restated in terms of a cohomological obstruction and extends an analogous codimension 3 criterion by the second and fourth authors. As another application, we classify ornaments f: S3 ⊔ S3 ⊔ S3 → ℝ5 up to ornament concordance.\r\n\r\nIt follows from work of M. Freedman, V. Krushkal and P. Teichner that the analogous criterion for r = 2 is false. We prove a lemma on singular higher-dimensional Borromean rings, yielding an elementary proof of the counterexample.","lang":"eng"}],"publication_identifier":{"issn":["0021-2172"],"eissn":["1565-8511"]},"publication_status":"published","author":[{"first_name":"Sergey","last_name":"Avvakumov","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","full_name":"Avvakumov, Sergey"},{"first_name":"Isaac","last_name":"Mabillard","full_name":"Mabillard, Isaac","id":"32BF9DAA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Arkadiy B.","full_name":"Skopenkov, Arkadiy B.","last_name":"Skopenkov"},{"first_name":"Uli","orcid":"0000-0002-1494-0568","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli","last_name":"Wagner"}],"scopus_import":"1","day":"30","oa_version":"Preprint","title":"Eliminating higher-multiplicity intersections. III. Codimension 2","volume":245,"article_type":"original","date_created":"2021-11-07T23:01:24Z","oa":1,"language":[{"iso":"eng"}],"citation":{"ista":"Avvakumov S, Mabillard I, Skopenkov AB, Wagner U. 2021. Eliminating higher-multiplicity intersections. III. Codimension 2. Israel Journal of Mathematics. 245, 501–534.","chicago":"Avvakumov, Sergey, Isaac Mabillard, Arkadiy B. Skopenkov, and Uli Wagner. “Eliminating Higher-Multiplicity Intersections. III. Codimension 2.” <i>Israel Journal of Mathematics</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s11856-021-2216-z\">https://doi.org/10.1007/s11856-021-2216-z</a>.","apa":"Avvakumov, S., Mabillard, I., Skopenkov, A. B., &#38; Wagner, U. (2021). Eliminating higher-multiplicity intersections. III. Codimension 2. <i>Israel Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11856-021-2216-z\">https://doi.org/10.1007/s11856-021-2216-z</a>","mla":"Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections. III. Codimension 2.” <i>Israel Journal of Mathematics</i>, vol. 245, Springer Nature, 2021, pp. 501–534, doi:<a href=\"https://doi.org/10.1007/s11856-021-2216-z\">10.1007/s11856-021-2216-z</a>.","ama":"Avvakumov S, Mabillard I, Skopenkov AB, Wagner U. Eliminating higher-multiplicity intersections. III. Codimension 2. <i>Israel Journal of Mathematics</i>. 2021;245:501–534. doi:<a href=\"https://doi.org/10.1007/s11856-021-2216-z\">10.1007/s11856-021-2216-z</a>","ieee":"S. Avvakumov, I. Mabillard, A. B. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity intersections. III. Codimension 2,” <i>Israel Journal of Mathematics</i>, vol. 245. Springer Nature, pp. 501–534, 2021.","short":"S. Avvakumov, I. Mabillard, A.B. Skopenkov, U. Wagner, Israel Journal of Mathematics 245 (2021) 501–534."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","month":"10","arxiv":1,"department":[{"_id":"UlWa"}]},{"date_created":"2021-04-04T22:01:22Z","article_type":"original","volume":75,"oa_version":"Preprint","title":"Eliminating higher-multiplicity intersections, III. Codimension 2","day":"01","scopus_import":"1","author":[{"first_name":"Sergey","last_name":"Avvakumov","full_name":"Avvakumov, Sergey","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Wagner","full_name":"Wagner, Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli","orcid":"0000-0002-1494-0568"},{"last_name":"Mabillard","id":"32BF9DAA-F248-11E8-B48F-1D18A9856A87","full_name":"Mabillard, Isaac","first_name":"Isaac"},{"first_name":"A. B.","last_name":"Skopenkov","full_name":"Skopenkov, A. B."}],"publication_identifier":{"issn":["0036-0279"]},"publication_status":"published","intvolume":"        75","department":[{"_id":"UlWa"}],"arxiv":1,"month":"12","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"ama":"Avvakumov S, Wagner U, Mabillard I, Skopenkov AB. Eliminating higher-multiplicity intersections, III. Codimension 2. <i>Russian Mathematical Surveys</i>. 2020;75(6):1156-1158. doi:<a href=\"https://doi.org/10.1070/RM9943\">10.1070/RM9943</a>","ieee":"S. Avvakumov, U. Wagner, I. Mabillard, and A. B. Skopenkov, “Eliminating higher-multiplicity intersections, III. Codimension 2,” <i>Russian Mathematical Surveys</i>, vol. 75, no. 6. IOP Publishing, pp. 1156–1158, 2020.","short":"S. Avvakumov, U. Wagner, I. Mabillard, A.B. Skopenkov, Russian Mathematical Surveys 75 (2020) 1156–1158.","chicago":"Avvakumov, Sergey, Uli Wagner, Isaac Mabillard, and A. B. Skopenkov. “Eliminating Higher-Multiplicity Intersections, III. Codimension 2.” <i>Russian Mathematical Surveys</i>. IOP Publishing, 2020. <a href=\"https://doi.org/10.1070/RM9943\">https://doi.org/10.1070/RM9943</a>.","ista":"Avvakumov S, Wagner U, Mabillard I, Skopenkov AB. 2020. Eliminating higher-multiplicity intersections, III. Codimension 2. Russian Mathematical Surveys. 75(6), 1156–1158.","apa":"Avvakumov, S., Wagner, U., Mabillard, I., &#38; Skopenkov, A. B. (2020). Eliminating higher-multiplicity intersections, III. Codimension 2. <i>Russian Mathematical Surveys</i>. IOP Publishing. <a href=\"https://doi.org/10.1070/RM9943\">https://doi.org/10.1070/RM9943</a>","mla":"Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections, III. Codimension 2.” <i>Russian Mathematical Surveys</i>, vol. 75, no. 6, IOP Publishing, 2020, pp. 1156–58, doi:<a href=\"https://doi.org/10.1070/RM9943\">10.1070/RM9943</a>."},"issue":"6","language":[{"iso":"eng"}],"oa":1,"type":"journal_article","_id":"9308","date_updated":"2023-08-14T11:43:54Z","publisher":"IOP Publishing","article_processing_charge":"No","doi":"10.1070/RM9943","quality_controlled":"1","main_file_link":[{"url":"https://arxiv.org/abs/1511.03501","open_access":"1"}],"page":"1156-1158","related_material":{"record":[{"id":"8183","relation":"earlier_version","status":"public"},{"status":"public","relation":"later_version","id":"10220"}]},"external_id":{"arxiv":["1511.03501"],"isi":["000625983100001"]},"year":"2020","isi":1,"date_published":"2020-12-01T00:00:00Z","acknowledgement":"This research was carried out with the support of the Russian Foundation for Basic Research(grant no. 19-01-00169)","status":"public","publication":"Russian Mathematical Surveys"},{"month":"10","department":[{"_id":"UlWa"}],"oa":1,"language":[{"iso":"eng"}],"issue":"2","citation":{"ama":"Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result. <i>Israel Journal of Mathematics</i>. 2017;222(2):841-866. doi:<a href=\"https://doi.org/10.1007/s11856-017-1607-7\">10.1007/s11856-017-1607-7</a>","ieee":"X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, and U. Wagner, “On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result,” <i>Israel Journal of Mathematics</i>, vol. 222, no. 2. Springer, pp. 841–866, 2017.","short":"X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, U. Wagner, Israel Journal of Mathematics 222 (2017) 841–866.","chicago":"Goaoc, Xavier, Isaac Mabillard, Pavel Paták, Zuzana Patakova, Martin Tancer, and Uli Wagner. “On Generalized Heawood Inequalities for Manifolds: A van Kampen–Flores Type Nonembeddability Result.” <i>Israel Journal of Mathematics</i>. Springer, 2017. <a href=\"https://doi.org/10.1007/s11856-017-1607-7\">https://doi.org/10.1007/s11856-017-1607-7</a>.","ista":"Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. 2017. On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result. Israel Journal of Mathematics. 222(2), 841–866.","apa":"Goaoc, X., Mabillard, I., Paták, P., Patakova, Z., Tancer, M., &#38; Wagner, U. (2017). On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result. <i>Israel Journal of Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s11856-017-1607-7\">https://doi.org/10.1007/s11856-017-1607-7</a>","mla":"Goaoc, Xavier, et al. “On Generalized Heawood Inequalities for Manifolds: A van Kampen–Flores Type Nonembeddability Result.” <i>Israel Journal of Mathematics</i>, vol. 222, no. 2, Springer, 2017, pp. 841–66, doi:<a href=\"https://doi.org/10.1007/s11856-017-1607-7\">10.1007/s11856-017-1607-7</a>."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"full_name":"Goaoc, Xavier","last_name":"Goaoc","first_name":"Xavier"},{"first_name":"Isaac","full_name":"Mabillard, Isaac","id":"32BF9DAA-F248-11E8-B48F-1D18A9856A87","last_name":"Mabillard"},{"first_name":"Pavel","last_name":"Paták","full_name":"Paták, Pavel"},{"orcid":"0000-0002-3975-1683","first_name":"Zuzana","last_name":"Patakova","id":"48B57058-F248-11E8-B48F-1D18A9856A87","full_name":"Patakova, Zuzana"},{"full_name":"Tancer, Martin","id":"38AC689C-F248-11E8-B48F-1D18A9856A87","last_name":"Tancer","first_name":"Martin","orcid":"0000-0002-1191-6714"},{"last_name":"Wagner","full_name":"Wagner, Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1494-0568","first_name":"Uli"}],"scopus_import":1,"day":"01","title":"On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result","oa_version":"Preprint","volume":222,"date_created":"2018-12-11T11:47:29Z","intvolume":"       222","abstract":[{"lang":"eng","text":"The fact that the complete graph K5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph Kn embeds in a closed surface M (other than the Klein bottle) if and only if (n−3)(n−4) ≤ 6b1(M), where b1(M) is the first Z2-Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of Kn+1) embeds in R2k if and only if n ≤ 2k + 1. Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k − 1)-connected 2k-manifold with kth Z2-Betti number bk only if the following generalized Heawood inequality holds: (k+1 n−k−1) ≤ (k+1 2k+1)bk. This is a common generalization of the case of graphs on surfaces as well as the van Kampen–Flores theorem. In the spirit of Kühnel’s conjecture, we prove that if the k-skeleton of the n-simplex embeds in a compact 2k-manifold with kth Z2-Betti number bk, then n ≤ 2bk(k 2k+2)+2k+4. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k−1)-connected. Our results generalize to maps without q-covered points, in the spirit of Tverberg’s theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition."}],"publication_status":"published","year":"2017","related_material":{"record":[{"id":"1511","status":"public","relation":"earlier_version"}]},"publist_id":"7194","project":[{"name":"International IST Postdoc Fellowship Programme","grant_number":"291734","call_identifier":"FP7","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"publication":"Israel Journal of Mathematics","status":"public","ec_funded":1,"acknowledgement":"The work by Z. P. was partially supported by the Israel Science Foundation grant ISF-768/12. The work by Z. P. and M. T. was partially supported by the project CE-ITI (GACR P202/12/G061) of the Czech Science Foundation and by the ERC Advanced Grant No. 267165. Part of the research work of M.T. was conducted at IST Austria, supported by an IST Fellowship. The research of P. P. was supported by the ERC Advanced grant no. 320924. The work by I. M. and U. W. was supported by the Swiss National Science Foundation (grants SNSF-200020-138230 and SNSF-PP00P2-138948). The collaboration between U. W. and X. G. was partially supported by the LabEx Bézout (ANR-10-LABX-58).","date_published":"2017-10-01T00:00:00Z","doi":"10.1007/s11856-017-1607-7","publisher":"Springer","date_updated":"2023-02-23T10:02:13Z","_id":"610","type":"journal_article","page":"841 - 866","main_file_link":[{"url":"https://arxiv.org/abs/1610.09063","open_access":"1"}],"quality_controlled":"1"},{"citation":{"short":"I. Mabillard, Eliminating Higher-Multiplicity Intersections: An r-Fold Whitney Trick for the Topological Tverberg Conjecture, Institute of Science and Technology Austria, 2016.","ieee":"I. Mabillard, “Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture,” Institute of Science and Technology Austria, 2016.","ama":"Mabillard I. Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture. 2016.","mla":"Mabillard, Isaac. <i>Eliminating Higher-Multiplicity Intersections: An r-Fold Whitney Trick for the Topological Tverberg Conjecture</i>. Institute of Science and Technology Austria, 2016.","apa":"Mabillard, I. (2016). <i>Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture</i>. Institute of Science and Technology Austria.","ista":"Mabillard I. 2016. Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture. Institute of Science and Technology Austria.","chicago":"Mabillard, Isaac. “Eliminating Higher-Multiplicity Intersections: An r-Fold Whitney Trick for the Topological Tverberg Conjecture.” Institute of Science and Technology Austria, 2016."},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","oa":1,"language":[{"iso":"eng"}],"department":[{"_id":"UlWa"}],"file":[{"checksum":"2d140cc924cd1b764544906fc22684ef","relation":"main_file","content_type":"application/pdf","access_level":"closed","file_name":"Thesis_final version_Mabillard_w_signature_page.pdf","file_id":"6809","creator":"dernst","date_updated":"2019-08-13T08:45:27Z","date_created":"2019-08-13T08:45:27Z","file_size":2227916},{"relation":"main_file","checksum":"2d140cc924cd1b764544906fc22684ef","file_name":"2016_Mabillard_Thesis.pdf","success":1,"content_type":"application/pdf","access_level":"open_access","file_id":"9178","file_size":2227916,"date_created":"2021-02-22T11:36:34Z","date_updated":"2021-02-22T11:36:34Z","creator":"dernst"}],"supervisor":[{"full_name":"Wagner, Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","last_name":"Wagner","first_name":"Uli","orcid":"0000-0002-1494-0568"}],"month":"08","file_date_updated":"2021-02-22T11:36:34Z","publication_identifier":{"issn":["2663-337X"]},"publication_status":"published","has_accepted_license":"1","abstract":[{"text":"Motivated by topological Tverberg-type problems  in topological combinatorics and by classical\r\nresults about embeddings (maps without double points), we study the question whether a finite\r\nsimplicial complex K  can be mapped into Rd  without triple, quadruple, or, more generally, r-fold points  (image points with at least r  distinct preimages), for a given multiplicity r ≤ 2. In particular, we are interested in maps f : K → Rd  that have no global r -fold intersection points, i.e., no r -fold points with preimages in r pairwise disjoint  simplices of K , and we seek necessary and sufficient conditions for the existence of such maps.\r\n\r\nWe present higher-multiplicity analogues of several classical results for embeddings, in particular of the completeness of the Van Kampen obstruction  for embeddability of k -dimensional\r\ncomplexes into R2k , k ≥ 3. Speciffically, we show that under suitable restrictions on the dimensions(viz., if dimK  = (r ≥ 1)k  and d  = rk \\ for some k ≥ 3), a well-known deleted product criterion (DPC ) is not only necessary but also sufficient for the existence of maps without global r -fold points. Our main technical tool is a higher-multiplicity version of the classical Whitney trick , by which pairs of isolated r -fold points of opposite sign  can be eliminated by local modiffications of the map, assuming codimension d – dimK ≥ 3.\r\n\r\nAn important guiding idea for our work was that suffciency of the DPC, together with an old\r\nresult of Özaydin's on the existence of equivariant maps, might yield an approach to disproving the remaining open cases of the the long-standing topological Tverberg conjecture , i.e., to construct maps from the N -simplex σN  to Rd  without r-Tverberg points when r not a prime power  and\r\nN  = (d  + 1)(r – 1). Unfortunately, our proof of the sufficiency of the DPC requires codimension d – dimK ≥ 3, which is not satisfied for K  = σN .\r\n\r\nIn 2015, Frick [16] found a very elegant way to overcome this \\codimension 3 obstacle&quot; and\r\nto construct the first counterexamples to the topological Tverberg conjecture for all parameters(d; r ) with d ≥ 3r  + 1 and r  not a prime power, by a reduction1  to a suitable lower-dimensional skeleton, for which the codimension 3 restriction is satisfied and maps without r -Tverberg points exist by Özaydin's result and sufficiency of the DPC.\r\n\r\nIn this thesis, we present a different construction (which does not use the constraint method) that yields counterexamples for d ≥ 3r , r  not a prime power.     ","lang":"eng"}],"date_created":"2018-12-11T11:50:16Z","day":"01","author":[{"first_name":"Isaac","full_name":"Mabillard, Isaac","id":"32BF9DAA-F248-11E8-B48F-1D18A9856A87","last_name":"Mabillard"}],"oa_version":"Published Version","title":"Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture","acknowledgement":"Foremost, I would like to thank Uli Wagner for introducing me to the exciting interface between\r\ntopology and combinatorics, and for our subsequent years of fruitful collaboration.\r\nIn our creative endeavors to eliminate intersection points, we had the chance to be joined later\r\nby Sergey Avvakumov and Arkadiy Skopenkov, which led us to new surprises in dimension 12.\r\nMy stay at EPFL and IST Austria was made very agreeable thanks to all these wonderful\r\npeople: Cyril Becker, Marek Filakovsky, Peter Franek, Radoslav Fulek, Peter Gazi, Kristof Huszar,\r\nMarek Krcal, Zuzana Masarova, Arnaud de Mesmay, Filip Moric, Michal Rybar, Martin Tancer,\r\nand Stephan Zhechev.\r\nFinally, I would like to thank my thesis committee Herbert Edelsbrunner and Roman Karasev\r\nfor their careful reading of the present manuscript and for the many improvements they suggested.","date_published":"2016-08-01T00:00:00Z","degree_awarded":"PhD","status":"public","publist_id":"6237","year":"2016","related_material":{"record":[{"relation":"part_of_dissertation","status":"public","id":"2159"}]},"page":"55","ddc":["500"],"_id":"1123","date_updated":"2023-09-07T11:56:28Z","type":"dissertation","alternative_title":["ISTA Thesis"],"article_processing_charge":"No","publisher":"Institute of Science and Technology Austria"},{"_id":"1381","date_updated":"2021-01-12T06:50:17Z","type":"conference","alternative_title":["LIPIcs"],"doi":"10.4230/LIPIcs.SoCG.2016.51","publisher":"Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH","quality_controlled":"1","page":"51.1 - 51.12","ddc":["510"],"publist_id":"5830","year":"2016","date_published":"2016-06-01T00:00:00Z","conference":{"location":"Medford, MA, USA","start_date":"2016-06-14","end_date":"2016-06-17","name":"SoCG: Symposium on Computational Geometry"},"status":"public","project":[{"name":"Embeddings in Higher Dimensions: Algorithms and Combinatorics","grant_number":"PP00P2_138948","_id":"25FA3206-B435-11E9-9278-68D0E5697425"}],"volume":51,"date_created":"2018-12-11T11:51:41Z","day":"01","scopus_import":1,"author":[{"last_name":"Mabillard","id":"32BF9DAA-F248-11E8-B48F-1D18A9856A87","full_name":"Mabillard, Isaac","first_name":"Isaac"},{"first_name":"Uli","orcid":"0000-0002-1494-0568","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli","last_name":"Wagner"}],"title":"Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range","oa_version":"Published Version","file_date_updated":"2020-07-14T12:44:47Z","publication_status":"published","has_accepted_license":"1","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)"},"license":"https://creativecommons.org/licenses/by/4.0/","abstract":[{"lang":"eng","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."}],"intvolume":"        51","department":[{"_id":"UlWa"}],"file":[{"file_id":"4791","file_size":622969,"date_created":"2018-12-12T10:10:06Z","creator":"system","date_updated":"2020-07-14T12:44:47Z","relation":"main_file","checksum":"92c0c3735fe908f8ded6e484005cb3b1","file_name":"IST-2016-621-v1+1_LIPIcs-SoCG-2016-51.pdf","access_level":"open_access","content_type":"application/pdf"}],"month":"06","citation":{"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>.","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.","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>.","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>","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.","short":"I. Mabillard, U. Wagner, in:, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2016, p. 51.1-51.12."},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","oa":1,"language":[{"iso":"eng"}],"pubrep_id":"621"},{"_id":"8183","date_updated":"2023-09-07T13:12:17Z","date_created":"2020-07-30T10:45:19Z","type":"preprint","day":"15","article_processing_charge":"No","author":[{"full_name":"Avvakumov, Sergey","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","last_name":"Avvakumov","first_name":"Sergey"},{"id":"32BF9DAA-F248-11E8-B48F-1D18A9856A87","full_name":"Mabillard, Isaac","last_name":"Mabillard","first_name":"Isaac"},{"first_name":"A.","last_name":"Skopenkov","full_name":"Skopenkov, A."},{"last_name":"Wagner","full_name":"Wagner, Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli","orcid":"0000-0002-1494-0568"}],"title":"Eliminating higher-multiplicity intersections, III. Codimension 2","oa_version":"Preprint","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1511.03501"}],"publication_status":"submitted","abstract":[{"text":"We study conditions under which a finite simplicial complex $K$ can be mapped to $\\mathbb R^d$ without higher-multiplicity intersections. An almost $r$-embedding is a map $f: K\\to \\mathbb R^d$ such that the images of any $r$\r\npairwise disjoint simplices of $K$ do not have a common point. We show that if $r$ is not a prime power and $d\\geq 2r+1$, then there is a counterexample to the topological Tverberg conjecture, i.e., there is an almost $r$-embedding of\r\nthe $(d+1)(r-1)$-simplex in $\\mathbb R^d$. This improves on previous constructions of counterexamples (for $d\\geq 3r$) based on a series of papers by M. \\\"Ozaydin, M. Gromov, P. Blagojevi\\'c, F. Frick, G. Ziegler, and the second and fourth present authors. The counterexamples are obtained by proving the following algebraic criterion in codimension 2: If $r\\ge3$ and if $K$ is a finite $2(r-1)$-complex then there exists an almost $r$-embedding $K\\to \\mathbb R^{2r}$ if and only if there exists a general position PL map $f:K\\to \\mathbb R^{2r}$ such that the algebraic intersection number of the $f$-images of any $r$ pairwise disjoint simplices of $K$ is zero. This result can be restated in terms of cohomological obstructions or equivariant maps, and extends an analogous codimension 3 criterion by the second and fourth authors. As another application we classify ornaments $f:S^3 \\sqcup S^3\\sqcup S^3\\to \\mathbb R^5$ up to ornament\r\nconcordance. It follows from work of M. Freedman, V. Krushkal and P. Teichner that the analogous criterion for $r=2$ is false. We prove a lemma on singular higher-dimensional Borromean rings, yielding an elementary proof of the counterexample.","lang":"eng"}],"department":[{"_id":"UlWa"}],"article_number":"1511.03501","year":"2015","related_material":{"record":[{"id":"9308","relation":"later_version","status":"public"},{"status":"public","relation":"later_version","id":"10220"},{"status":"public","relation":"dissertation_contains","id":"8156"}]},"month":"11","arxiv":1,"external_id":{"arxiv":["1511.03501"]},"citation":{"mla":"Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections, III. Codimension 2.” <i>ArXiv</i>, 1511.03501.","apa":"Avvakumov, S., Mabillard, I., Skopenkov, A., &#38; Wagner, U. (n.d.). Eliminating higher-multiplicity intersections, III. Codimension 2. <i>arXiv</i>.","ista":"Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity intersections, III. Codimension 2. arXiv, 1511.03501.","chicago":"Avvakumov, Sergey, Isaac Mabillard, A. Skopenkov, and Uli Wagner. “Eliminating Higher-Multiplicity Intersections, III. Codimension 2.” <i>ArXiv</i>, n.d.","short":"S. Avvakumov, I. Mabillard, A. Skopenkov, U. Wagner, ArXiv (n.d.).","ieee":"S. Avvakumov, I. Mabillard, A. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity intersections, III. Codimension 2,” <i>arXiv</i>. .","ama":"Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity intersections, III. Codimension 2. <i>arXiv</i>."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"We would like to thank A. Klyachko, V. Krushkal, S. Melikhov, M. Tancer, P. Teichner and anonymous referees for helpful discussions.","date_published":"2015-11-15T00:00:00Z","oa":1,"publication":"arXiv","language":[{"iso":"eng"}],"status":"public"},{"publication_status":"published","file_date_updated":"2020-07-14T12:44:59Z","has_accepted_license":"1","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)"},"abstract":[{"lang":"eng","text":"The fact that the complete graph K_5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M if and only if (n-3)(n-4) is at most 6b_1(M), where b_1(M) is the first Z_2-Betti number of M. On the other hand, Van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_{n+1}) embeds in R^{2k} if and only if n is less or equal to 2k+2. Two decades ago, Kuhnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k-1)-connected 2k-manifold with kth Z_2-Betti number b_k only if the following generalized Heawood inequality holds: binom{n-k-1}{k+1} is at most binom{2k+1}{k+1} b_k. This is a common generalization of the case of graphs on surfaces as well as the Van Kampen--Flores theorem. In the spirit of Kuhnel's conjecture, we prove that if the k-skeleton of the n-simplex embeds in a 2k-manifold with kth Z_2-Betti number b_k, then n is at most 2b_k binom{2k+2}{k} + 2k + 5. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k-1)-connected. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition."}],"volume":"34 ","date_created":"2018-12-11T11:52:27Z","author":[{"first_name":"Xavier","full_name":"Goaoc, Xavier","last_name":"Goaoc"},{"first_name":"Isaac","last_name":"Mabillard","id":"32BF9DAA-F248-11E8-B48F-1D18A9856A87","full_name":"Mabillard, Isaac"},{"first_name":"Pavel","last_name":"Paták","full_name":"Paták, Pavel"},{"id":"48B57058-F248-11E8-B48F-1D18A9856A87","full_name":"Patakova, Zuzana","last_name":"Patakova","orcid":"0000-0002-3975-1683","first_name":"Zuzana"},{"first_name":"Martin","orcid":"0000-0002-1191-6714","id":"38AC689C-F248-11E8-B48F-1D18A9856A87","full_name":"Tancer, Martin","last_name":"Tancer"},{"orcid":"0000-0002-1494-0568","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli","last_name":"Wagner"}],"day":"11","scopus_import":1,"title":"On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability result","oa_version":"Published Version","citation":{"ama":"Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability result. In: Vol 34. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2015:476-490. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.476\">10.4230/LIPIcs.SOCG.2015.476</a>","ieee":"X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, and U. Wagner, “On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability result,” presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands, 2015, vol. 34, pp. 476–490.","short":"X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, U. Wagner, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 476–490.","ista":"Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. 2015. On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability result. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 34, 476–490.","chicago":"Goaoc, Xavier, Isaac Mabillard, Pavel Paták, Zuzana Patakova, Martin Tancer, and Uli Wagner. “On Generalized Heawood Inequalities for Manifolds: A Van Kampen–Flores-Type Nonembeddability Result,” 34:476–90. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. <a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.476\">https://doi.org/10.4230/LIPIcs.SOCG.2015.476</a>.","apa":"Goaoc, X., Mabillard, I., Paták, P., Patakova, Z., Tancer, M., &#38; Wagner, U. (2015). On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability result (Vol. 34, pp. 476–490). Presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.476\">https://doi.org/10.4230/LIPIcs.SOCG.2015.476</a>","mla":"Goaoc, Xavier, et al. <i>On Generalized Heawood Inequalities for Manifolds: A Van Kampen–Flores-Type Nonembeddability Result</i>. Vol. 34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 476–90, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.476\">10.4230/LIPIcs.SOCG.2015.476</a>."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"pubrep_id":"502","language":[{"iso":"eng"}],"department":[{"_id":"UlWa"}],"file":[{"checksum":"0945811875351796324189312ca29e9e","relation":"main_file","content_type":"application/pdf","access_level":"open_access","file_name":"IST-2016-502-v1+1_42.pdf","file_id":"4871","date_updated":"2020-07-14T12:44:59Z","creator":"system","date_created":"2018-12-12T10:11:18Z","file_size":636735}],"month":"06","quality_controlled":"1","page":"476 - 490","ddc":["510"],"date_updated":"2023-02-23T12:38:00Z","_id":"1511","type":"conference","doi":"10.4230/LIPIcs.SOCG.2015.476","alternative_title":["LIPIcs"],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","ec_funded":1,"date_published":"2015-06-11T00:00:00Z","acknowledgement":"The work by Z. P. was partially supported by the Charles University Grant SVV-2014-260103. The\r\nwork by Z. P. and M. T. was partially supported by the project CE-ITI (GACR P202/12/G061) of\r\nthe Czech Science Foundation and by the ERC Advanced Grant No. 267165. Part of the research\r\nwork of M. T. was conducted at IST Austria, supported by an IST Fellowship. The work by U.W.\r\nwas partially supported by the Swiss National Science Foundation (grants SNSF-200020-138230 and\r\nSNSF-PP00P2-138948).","conference":{"name":"SoCG: Symposium on Computational Geometry","end_date":"2015-06-25","start_date":"2015-06-22","location":"Eindhoven, Netherlands"},"project":[{"grant_number":"291734","name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"status":"public","publist_id":"5666","year":"2015","related_material":{"record":[{"relation":"later_version","status":"public","id":"610"}]}},{"doi":"10.1145/2582112.2582134","publisher":"ACM","date_updated":"2023-09-07T11:56:27Z","_id":"2159","type":"conference","page":"171 - 180","ddc":["510"],"quality_controlled":"1","year":"2014","related_material":{"record":[{"id":"1123","status":"public","relation":"dissertation_contains"}]},"publist_id":"4847","status":"public","publication":"Proceedings of the Annual Symposium on Computational Geometry","date_published":"2014-06-08T00:00:00Z","conference":{"location":"Kyoto, Japan","end_date":"2014-06-11","start_date":"2014-06-08","name":"SoCG: Symposium on Computational Geometry"},"acknowledgement":"Swiss National Science Foundation (Project SNSF-PP00P2-138948)","author":[{"first_name":"Isaac","full_name":"Mabillard, Isaac","id":"32BF9DAA-F248-11E8-B48F-1D18A9856A87","last_name":"Mabillard"},{"orcid":"0000-0002-1494-0568","first_name":"Uli","last_name":"Wagner","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli"}],"scopus_import":1,"day":"08","title":"Eliminating Tverberg points, I. An analogue of the Whitney trick","oa_version":"Submitted Version","date_created":"2018-12-11T11:56:03Z","has_accepted_license":"1","abstract":[{"text":"Motivated by topological Tverberg-type problems, we consider multiple (double, triple, and higher multiplicity) selfintersection points of maps from finite simplicial complexes (compact polyhedra) into ℝd and study conditions under which such multiple points can be eliminated. The most classical case is that of embeddings (i.e., maps without double points) of a κ-dimensional complex K into ℝ2κ. For this problem, the work of van Kampen, Shapiro, and Wu provides an efficiently testable necessary condition for embeddability (namely, vanishing of the van Kampen ob-struction). For κ ≥ 3, the condition is also sufficient, and yields a polynomial-time algorithm for deciding embeddability: One starts with an arbitrary map f : K→ℝ2κ, which generically has finitely many double points; if k ≥ 3 and if the obstruction vanishes then one can successively remove these double points by local modifications of the map f. One of the main tools is the famous Whitney trick that permits eliminating pairs of double points of opposite intersection sign. We are interested in generalizing this approach to intersection points of higher multiplicity. We call a point y 2 ℝd an r-fold Tverberg point of a map f : Kκ →ℝd if y lies in the intersection f(σ1)∩. ∩f(σr) of the images of r pairwise disjoint simplices of K. The analogue of (non-)embeddability that we study is the problem Tverbergκ r→d: Given a κ-dimensional complex K, does it satisfy a Tverberg-type theorem with parameters r and d, i.e., does every map f : K κ → ℝd have an r-fold Tverberg point? Here, we show that for fixed r, κ and d of the form d = rm and k = (r-1)m, m ≥ 3, there is a polynomial-time algorithm for deciding this (based on the vanishing of a cohomological obstruction, as in the case of embeddings). Our main tool is an r-fold analogue of the Whitney trick: Given r pairwise disjoint simplices of K such that the intersection of their images contains two r-fold Tverberg points y+ and y- of opposite intersection sign, we can eliminate y+ and y- by a local isotopy of f. In a subsequent paper, we plan to develop this further and present a generalization of the classical Haeiger-Weber Theorem (which yields a necessary and sufficient condition for embeddability of κ-complexes into ℝd for a wider range of dimensions) to intersection points of higher multiplicity.","lang":"eng"}],"publication_status":"published","file_date_updated":"2020-07-14T12:45:30Z","month":"06","department":[{"_id":"UlWa"}],"file":[{"relation":"main_file","checksum":"2aae223fee8ffeaf57bbabd8d92b6a2c","file_name":"IST-2016-534-v1+1_Eliminating_Tverberg_points_I._An_analogue_of_the_Whitney_trick.pdf","content_type":"application/pdf","access_level":"open_access","file_id":"4735","file_size":914396,"date_created":"2018-12-12T10:09:12Z","date_updated":"2020-07-14T12:45:30Z","creator":"system"}],"oa":1,"language":[{"iso":"eng"}],"pubrep_id":"534","citation":{"mla":"Mabillard, Isaac, and Uli Wagner. “Eliminating Tverberg Points, I. An Analogue of the Whitney Trick.” <i>Proceedings of the Annual Symposium on Computational Geometry</i>, ACM, 2014, pp. 171–80, doi:<a href=\"https://doi.org/10.1145/2582112.2582134\">10.1145/2582112.2582134</a>.","apa":"Mabillard, I., &#38; Wagner, U. (2014). Eliminating Tverberg points, I. An analogue of the Whitney trick. In <i>Proceedings of the Annual Symposium on Computational Geometry</i> (pp. 171–180). Kyoto, Japan: ACM. <a href=\"https://doi.org/10.1145/2582112.2582134\">https://doi.org/10.1145/2582112.2582134</a>","ista":"Mabillard I, Wagner U. 2014. Eliminating Tverberg points, I. An analogue of the Whitney trick. Proceedings of the Annual Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 171–180.","chicago":"Mabillard, Isaac, and Uli Wagner. “Eliminating Tverberg Points, I. An Analogue of the Whitney Trick.” In <i>Proceedings of the Annual Symposium on Computational Geometry</i>, 171–80. ACM, 2014. <a href=\"https://doi.org/10.1145/2582112.2582134\">https://doi.org/10.1145/2582112.2582134</a>.","short":"I. Mabillard, U. Wagner, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 171–180.","ieee":"I. Mabillard and U. Wagner, “Eliminating Tverberg points, I. An analogue of the Whitney trick,” in <i>Proceedings of the Annual Symposium on Computational Geometry</i>, Kyoto, Japan, 2014, pp. 171–180.","ama":"Mabillard I, Wagner U. Eliminating Tverberg points, I. An analogue of the Whitney trick. In: <i>Proceedings of the Annual Symposium on Computational Geometry</i>. ACM; 2014:171-180. doi:<a href=\"https://doi.org/10.1145/2582112.2582134\">10.1145/2582112.2582134</a>"},"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87"}]