--- _id: '10220' abstract: - lang: eng 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." 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. article_processing_charge: No article_type: original author: - first_name: Sergey full_name: Avvakumov, Sergey id: 3827DAC8-F248-11E8-B48F-1D18A9856A87 last_name: Avvakumov - first_name: Isaac full_name: Mabillard, Isaac id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87 last_name: Mabillard - first_name: Arkadiy B. full_name: Skopenkov, Arkadiy B. last_name: Skopenkov - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 citation: ama: Avvakumov S, Mabillard I, Skopenkov AB, Wagner U. Eliminating higher-multiplicity intersections. III. Codimension 2. Israel Journal of Mathematics. 2021;245:501–534. doi:10.1007/s11856-021-2216-z apa: Avvakumov, S., Mabillard, I., Skopenkov, A. B., & Wagner, U. (2021). Eliminating higher-multiplicity intersections. III. Codimension 2. Israel Journal of Mathematics. Springer Nature. https://doi.org/10.1007/s11856-021-2216-z chicago: Avvakumov, Sergey, Isaac Mabillard, Arkadiy B. Skopenkov, and Uli Wagner. “Eliminating Higher-Multiplicity Intersections. III. Codimension 2.” Israel Journal of Mathematics. Springer Nature, 2021. https://doi.org/10.1007/s11856-021-2216-z. ieee: S. Avvakumov, I. Mabillard, A. B. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity intersections. III. Codimension 2,” Israel Journal of Mathematics, vol. 245. Springer Nature, pp. 501–534, 2021. ista: Avvakumov S, Mabillard I, Skopenkov AB, Wagner U. 2021. Eliminating higher-multiplicity intersections. III. Codimension 2. Israel Journal of Mathematics. 245, 501–534. mla: Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections. III. Codimension 2.” Israel Journal of Mathematics, vol. 245, Springer Nature, 2021, pp. 501–534, doi:10.1007/s11856-021-2216-z. short: S. Avvakumov, I. Mabillard, A.B. Skopenkov, U. Wagner, Israel Journal of Mathematics 245 (2021) 501–534. date_created: 2021-11-07T23:01:24Z date_published: 2021-10-30T00:00:00Z date_updated: 2023-08-14T11:43:55Z day: '30' department: - _id: UlWa doi: 10.1007/s11856-021-2216-z external_id: arxiv: - '1511.03501' isi: - '000712942100013' intvolume: ' 245' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1511.03501 month: '10' oa: 1 oa_version: Preprint page: '501–534 ' project: - _id: 26611F5C-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: P31312 name: Algorithms for Embeddings and Homotopy Theory publication: Israel Journal of Mathematics publication_identifier: eissn: - 1565-8511 issn: - 0021-2172 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '8183' relation: earlier_version status: public - id: '9308' relation: earlier_version status: public scopus_import: '1' status: public title: Eliminating higher-multiplicity intersections. III. Codimension 2 type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 245 year: '2021' ... --- _id: '9308' acknowledgement: This research was carried out with the support of the Russian Foundation for Basic Research(grant no. 19-01-00169) article_processing_charge: No article_type: original author: - first_name: Sergey full_name: Avvakumov, Sergey id: 3827DAC8-F248-11E8-B48F-1D18A9856A87 last_name: Avvakumov - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 - first_name: Isaac full_name: Mabillard, Isaac id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87 last_name: Mabillard - first_name: A. B. full_name: Skopenkov, A. B. last_name: Skopenkov citation: ama: Avvakumov S, Wagner U, Mabillard I, Skopenkov AB. Eliminating higher-multiplicity intersections, III. Codimension 2. Russian Mathematical Surveys. 2020;75(6):1156-1158. doi:10.1070/RM9943 apa: Avvakumov, S., Wagner, U., Mabillard, I., & Skopenkov, A. B. (2020). Eliminating higher-multiplicity intersections, III. Codimension 2. Russian Mathematical Surveys. IOP Publishing. https://doi.org/10.1070/RM9943 chicago: Avvakumov, Sergey, Uli Wagner, Isaac Mabillard, and A. B. Skopenkov. “Eliminating Higher-Multiplicity Intersections, III. Codimension 2.” Russian Mathematical Surveys. IOP Publishing, 2020. https://doi.org/10.1070/RM9943. ieee: S. Avvakumov, U. Wagner, I. Mabillard, and A. B. Skopenkov, “Eliminating higher-multiplicity intersections, III. Codimension 2,” Russian Mathematical Surveys, vol. 75, no. 6. IOP Publishing, pp. 1156–1158, 2020. ista: Avvakumov S, Wagner U, Mabillard I, Skopenkov AB. 2020. Eliminating higher-multiplicity intersections, III. Codimension 2. Russian Mathematical Surveys. 75(6), 1156–1158. mla: Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections, III. Codimension 2.” Russian Mathematical Surveys, vol. 75, no. 6, IOP Publishing, 2020, pp. 1156–58, doi:10.1070/RM9943. short: S. Avvakumov, U. Wagner, I. Mabillard, A.B. Skopenkov, Russian Mathematical Surveys 75 (2020) 1156–1158. date_created: 2021-04-04T22:01:22Z date_published: 2020-12-01T00:00:00Z date_updated: 2023-08-14T11:43:54Z day: '01' department: - _id: UlWa doi: 10.1070/RM9943 external_id: arxiv: - '1511.03501' isi: - '000625983100001' intvolume: ' 75' isi: 1 issue: '6' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1511.03501 month: '12' oa: 1 oa_version: Preprint page: 1156-1158 publication: Russian Mathematical Surveys publication_identifier: issn: - 0036-0279 publication_status: published publisher: IOP Publishing quality_controlled: '1' related_material: record: - id: '8183' relation: earlier_version status: public - id: '10220' relation: later_version status: public scopus_import: '1' status: public title: Eliminating higher-multiplicity intersections, III. Codimension 2 type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 75 year: '2020' ... --- _id: '610' 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.' 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). author: - first_name: Xavier full_name: Goaoc, Xavier last_name: Goaoc - first_name: Isaac full_name: Mabillard, Isaac id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87 last_name: Mabillard - first_name: Pavel full_name: Paták, Pavel last_name: Paták - first_name: Zuzana full_name: Patakova, Zuzana id: 48B57058-F248-11E8-B48F-1D18A9856A87 last_name: Patakova orcid: 0000-0002-3975-1683 - first_name: Martin full_name: Tancer, Martin id: 38AC689C-F248-11E8-B48F-1D18A9856A87 last_name: Tancer orcid: 0000-0002-1191-6714 - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 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. Israel Journal of Mathematics. 2017;222(2):841-866. doi:10.1007/s11856-017-1607-7' apa: '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. Springer. https://doi.org/10.1007/s11856-017-1607-7' 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.” Israel Journal of Mathematics. Springer, 2017. https://doi.org/10.1007/s11856-017-1607-7.' 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,” Israel Journal of Mathematics, vol. 222, no. 2. Springer, pp. 841–866, 2017.' 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.' mla: 'Goaoc, Xavier, et al. “On Generalized Heawood Inequalities for Manifolds: A van Kampen–Flores Type Nonembeddability Result.” Israel Journal of Mathematics, vol. 222, no. 2, Springer, 2017, pp. 841–66, doi:10.1007/s11856-017-1607-7.' short: X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, U. Wagner, Israel Journal of Mathematics 222 (2017) 841–866. date_created: 2018-12-11T11:47:29Z date_published: 2017-10-01T00:00:00Z date_updated: 2023-02-23T10:02:13Z day: '01' department: - _id: UlWa doi: 10.1007/s11856-017-1607-7 ec_funded: 1 intvolume: ' 222' issue: '2' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1610.09063 month: '10' oa: 1 oa_version: Preprint page: 841 - 866 project: - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme publication: Israel Journal of Mathematics publication_status: published publisher: Springer publist_id: '7194' quality_controlled: '1' related_material: record: - id: '1511' relation: earlier_version status: public scopus_import: 1 status: public title: 'On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result' type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 222 year: '2017' ... --- _id: '1381' 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.' alternative_title: - LIPIcs author: - first_name: Isaac full_name: Mabillard, Isaac id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87 last_name: Mabillard - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 citation: 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:10.4230/LIPIcs.SoCG.2016.51' apa: 'Mabillard, I., & 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. https://doi.org/10.4230/LIPIcs.SoCG.2016.51' 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. https://doi.org/10.4230/LIPIcs.SoCG.2016.51. 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.' 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.' mla: Mabillard, Isaac, and Uli Wagner. Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range. Vol. 51, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2016, p. 51.1-51.12, doi:10.4230/LIPIcs.SoCG.2016.51. short: I. Mabillard, U. Wagner, in:, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2016, p. 51.1-51.12. conference: end_date: 2016-06-17 location: Medford, MA, USA name: 'SoCG: Symposium on Computational Geometry' start_date: 2016-06-14 date_created: 2018-12-11T11:51:41Z date_published: 2016-06-01T00:00:00Z date_updated: 2021-01-12T06:50:17Z day: '01' ddc: - '510' department: - _id: UlWa doi: 10.4230/LIPIcs.SoCG.2016.51 file: - access_level: open_access checksum: 92c0c3735fe908f8ded6e484005cb3b1 content_type: application/pdf creator: system date_created: 2018-12-12T10:10:06Z date_updated: 2020-07-14T12:44:47Z file_id: '4791' file_name: IST-2016-621-v1+1_LIPIcs-SoCG-2016-51.pdf file_size: 622969 relation: main_file file_date_updated: 2020-07-14T12:44:47Z has_accepted_license: '1' intvolume: ' 51' language: - iso: eng month: '06' oa: 1 oa_version: Published Version page: 51.1 - 51.12 project: - _id: 25FA3206-B435-11E9-9278-68D0E5697425 grant_number: PP00P2_138948 name: 'Embeddings in Higher Dimensions: Algorithms and Combinatorics' publication_status: published publisher: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH publist_id: '5830' pubrep_id: '621' quality_controlled: '1' scopus_import: 1 status: public title: Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 51 year: '2016' ... --- _id: '1123' abstract: - lang: eng 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" 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. " 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." alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Isaac full_name: Mabillard, Isaac id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87 last_name: Mabillard citation: ama: 'Mabillard I. Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture. 2016.' apa: '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.' ieee: 'I. Mabillard, “Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture,” Institute of Science and Technology Austria, 2016.' ista: 'Mabillard I. 2016. Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture. Institute of Science and Technology Austria.' mla: 'Mabillard, Isaac. Eliminating Higher-Multiplicity Intersections: An r-Fold Whitney Trick for the Topological Tverberg Conjecture. Institute of Science and Technology Austria, 2016.' short: 'I. Mabillard, Eliminating Higher-Multiplicity Intersections: An r-Fold Whitney Trick for the Topological Tverberg Conjecture, Institute of Science and Technology Austria, 2016.' date_created: 2018-12-11T11:50:16Z date_published: 2016-08-01T00:00:00Z date_updated: 2023-09-07T11:56:28Z day: '01' ddc: - '500' degree_awarded: PhD department: - _id: UlWa file: - access_level: closed checksum: 2d140cc924cd1b764544906fc22684ef content_type: application/pdf creator: dernst date_created: 2019-08-13T08:45:27Z date_updated: 2019-08-13T08:45:27Z file_id: '6809' file_name: Thesis_final version_Mabillard_w_signature_page.pdf file_size: 2227916 relation: main_file - access_level: open_access checksum: 2d140cc924cd1b764544906fc22684ef content_type: application/pdf creator: dernst date_created: 2021-02-22T11:36:34Z date_updated: 2021-02-22T11:36:34Z file_id: '9178' file_name: 2016_Mabillard_Thesis.pdf file_size: 2227916 relation: main_file success: 1 file_date_updated: 2021-02-22T11:36:34Z has_accepted_license: '1' language: - iso: eng month: '08' oa: 1 oa_version: Published Version page: '55' publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria publist_id: '6237' related_material: record: - id: '2159' relation: part_of_dissertation status: public status: public supervisor: - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 title: 'Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture' type: dissertation user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2016' ... --- _id: '1511' 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.' 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)." alternative_title: - LIPIcs author: - first_name: Xavier full_name: Goaoc, Xavier last_name: Goaoc - first_name: Isaac full_name: Mabillard, Isaac id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87 last_name: Mabillard - first_name: Pavel full_name: Paták, Pavel last_name: Paták - first_name: Zuzana full_name: Patakova, Zuzana id: 48B57058-F248-11E8-B48F-1D18A9856A87 last_name: Patakova orcid: 0000-0002-3975-1683 - first_name: Martin full_name: Tancer, Martin id: 38AC689C-F248-11E8-B48F-1D18A9856A87 last_name: Tancer orcid: 0000-0002-1191-6714 - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 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:10.4230/LIPIcs.SOCG.2015.476' apa: '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 (Vol. 34, pp. 476–490). Presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SOCG.2015.476' 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. https://doi.org/10.4230/LIPIcs.SOCG.2015.476.' 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.' 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.' mla: 'Goaoc, Xavier, et al. On Generalized Heawood Inequalities for Manifolds: A Van Kampen–Flores-Type Nonembeddability Result. Vol. 34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 476–90, doi:10.4230/LIPIcs.SOCG.2015.476.' short: X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, U. Wagner, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 476–490. conference: end_date: 2015-06-25 location: Eindhoven, Netherlands name: 'SoCG: Symposium on Computational Geometry' start_date: 2015-06-22 date_created: 2018-12-11T11:52:27Z date_published: 2015-06-11T00:00:00Z date_updated: 2023-02-23T12:38:00Z day: '11' ddc: - '510' department: - _id: UlWa doi: 10.4230/LIPIcs.SOCG.2015.476 ec_funded: 1 file: - access_level: open_access checksum: 0945811875351796324189312ca29e9e content_type: application/pdf creator: system date_created: 2018-12-12T10:11:18Z date_updated: 2020-07-14T12:44:59Z file_id: '4871' file_name: IST-2016-502-v1+1_42.pdf file_size: 636735 relation: main_file file_date_updated: 2020-07-14T12:44:59Z has_accepted_license: '1' language: - iso: eng month: '06' oa: 1 oa_version: Published Version page: 476 - 490 project: - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik publist_id: '5666' pubrep_id: '502' quality_controlled: '1' related_material: record: - id: '610' relation: later_version status: public scopus_import: 1 status: public title: 'On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability result' tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: '34 ' year: '2015' ... --- _id: '8183' abstract: - lang: eng 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." acknowledgement: We would like to thank A. Klyachko, V. Krushkal, S. Melikhov, M. Tancer, P. Teichner and anonymous referees for helpful discussions. article_number: '1511.03501' article_processing_charge: No author: - first_name: Sergey full_name: Avvakumov, Sergey id: 3827DAC8-F248-11E8-B48F-1D18A9856A87 last_name: Avvakumov - first_name: Isaac full_name: Mabillard, Isaac id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87 last_name: Mabillard - first_name: A. full_name: Skopenkov, A. last_name: Skopenkov - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 citation: ama: Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity intersections, III. Codimension 2. arXiv. apa: Avvakumov, S., Mabillard, I., Skopenkov, A., & Wagner, U. (n.d.). Eliminating higher-multiplicity intersections, III. Codimension 2. arXiv. chicago: Avvakumov, Sergey, Isaac Mabillard, A. Skopenkov, and Uli Wagner. “Eliminating Higher-Multiplicity Intersections, III. Codimension 2.” ArXiv, n.d. ieee: S. Avvakumov, I. Mabillard, A. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity intersections, III. Codimension 2,” arXiv. . ista: Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity intersections, III. Codimension 2. arXiv, 1511.03501. mla: Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections, III. Codimension 2.” ArXiv, 1511.03501. short: S. Avvakumov, I. Mabillard, A. Skopenkov, U. Wagner, ArXiv (n.d.). date_created: 2020-07-30T10:45:19Z date_published: 2015-11-15T00:00:00Z date_updated: 2023-09-07T13:12:17Z day: '15' department: - _id: UlWa external_id: arxiv: - '1511.03501' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1511.03501 month: '11' oa: 1 oa_version: Preprint publication: arXiv publication_status: submitted related_material: record: - id: '9308' relation: later_version status: public - id: '10220' relation: later_version status: public - id: '8156' relation: dissertation_contains status: public status: public title: Eliminating higher-multiplicity intersections, III. Codimension 2 type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2015' ... --- _id: '2159' abstract: - lang: eng 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.' 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 - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 citation: ama: 'Mabillard I, Wagner U. Eliminating Tverberg points, I. An analogue of the Whitney trick. In: Proceedings of the Annual Symposium on Computational Geometry. ACM; 2014:171-180. doi:10.1145/2582112.2582134' apa: 'Mabillard, I., & Wagner, U. (2014). Eliminating Tverberg points, I. An analogue of the Whitney trick. In Proceedings of the Annual Symposium on Computational Geometry (pp. 171–180). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582134' chicago: Mabillard, Isaac, and Uli Wagner. “Eliminating Tverberg Points, I. An Analogue of the Whitney Trick.” In Proceedings of the Annual Symposium on Computational Geometry, 171–80. ACM, 2014. https://doi.org/10.1145/2582112.2582134. ieee: I. Mabillard and U. Wagner, “Eliminating Tverberg points, I. An analogue of the Whitney trick,” in Proceedings of the Annual Symposium on Computational Geometry, Kyoto, Japan, 2014, pp. 171–180. 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.' mla: Mabillard, Isaac, and Uli Wagner. “Eliminating Tverberg Points, I. An Analogue of the Whitney Trick.” Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 171–80, doi:10.1145/2582112.2582134. short: I. Mabillard, U. Wagner, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 171–180. conference: end_date: 2014-06-11 location: Kyoto, Japan name: 'SoCG: Symposium on Computational Geometry' start_date: 2014-06-08 date_created: 2018-12-11T11:56:03Z date_published: 2014-06-08T00:00:00Z date_updated: 2023-09-07T11:56:27Z day: '08' ddc: - '510' department: - _id: UlWa doi: 10.1145/2582112.2582134 file: - access_level: open_access checksum: 2aae223fee8ffeaf57bbabd8d92b6a2c content_type: application/pdf creator: system date_created: 2018-12-12T10:09:12Z date_updated: 2020-07-14T12:45:30Z file_id: '4735' file_name: IST-2016-534-v1+1_Eliminating_Tverberg_points_I._An_analogue_of_the_Whitney_trick.pdf file_size: 914396 relation: main_file file_date_updated: 2020-07-14T12:45:30Z has_accepted_license: '1' language: - iso: eng month: '06' oa: 1 oa_version: Submitted Version page: 171 - 180 publication: Proceedings of the Annual Symposium on Computational Geometry publication_status: published publisher: ACM publist_id: '4847' pubrep_id: '534' quality_controlled: '1' related_material: record: - id: '1123' relation: dissertation_contains status: public scopus_import: 1 status: public title: Eliminating Tverberg points, I. An analogue of the Whitney trick type: conference user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 year: '2014' ...