---
_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
arxiv: 1
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. <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>
  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>
  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>.
  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.
  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.” <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>.
  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: '7806'
abstract:
- lang: eng
  text: "We consider the following decision problem EMBEDk→d in computational topology
    (where k ≤ d are fixed positive integers): Given a finite simplicial complex K
    of dimension k, does there exist a (piecewise-linear) embedding of K into ℝd?\r\nThe
    special case EMBED1→2 is graph planarity, which is decidable in linear time, as
    shown by Hopcroft and Tarjan. In higher dimensions, EMBED2→3 and EMBED3→3 are
    known to be decidable (as well as NP-hard), and recent results of Čadek et al.
    in computational homotopy theory, in combination with the classical Haefliger–Weber
    theorem in geometric topology, imply that EMBEDk→d can be solved in polynomial
    time for any fixed pair (k, d) of dimensions in the so-called metastable range
    .\r\nHere, by contrast, we prove that EMBEDk→d is algorithmically undecidable
    for almost all pairs of dimensions outside the metastable range, namely for .
    This almost completely resolves the decidability vs. undecidability of EMBEDk→d
    in higher dimensions and establishes a sharp dichotomy between polynomial-time
    solvability and undecidability.\r\nOur result complements (and in a wide range
    of dimensions strengthens) earlier results of Matoušek, Tancer, and the second
    author, who showed that EMBEDk→d is undecidable for 4 ≤ k ϵ {d – 1, d}, and NP-hard
    for all remaining pairs (k, d) outside the metastable range and satisfying d ≥
    4."
article_processing_charge: No
author:
- first_name: Marek
  full_name: Filakovský, Marek
  id: 3E8AF77E-F248-11E8-B48F-1D18A9856A87
  last_name: Filakovský
- first_name: Uli
  full_name: Wagner, Uli
  id: 36690CA2-F248-11E8-B48F-1D18A9856A87
  last_name: Wagner
  orcid: 0000-0002-1494-0568
- first_name: Stephan Y
  full_name: Zhechev, Stephan Y
  id: 3AA52972-F248-11E8-B48F-1D18A9856A87
  last_name: Zhechev
citation:
  ama: 'Filakovský M, Wagner U, Zhechev SY. Embeddability of simplicial complexes
    is undecidable. In: <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete
    Algorithms</i>. Vol 2020-January. SIAM; 2020:767-785. doi:<a href="https://doi.org/10.1137/1.9781611975994.47">10.1137/1.9781611975994.47</a>'
  apa: 'Filakovský, M., Wagner, U., &#38; Zhechev, S. Y. (2020). Embeddability of
    simplicial complexes is undecidable. In <i>Proceedings of the Annual ACM-SIAM
    Symposium on Discrete Algorithms</i> (Vol. 2020–January, pp. 767–785). Salt Lake
    City, UT, United States: SIAM. <a href="https://doi.org/10.1137/1.9781611975994.47">https://doi.org/10.1137/1.9781611975994.47</a>'
  chicago: Filakovský, Marek, Uli Wagner, and Stephan Y Zhechev. “Embeddability of
    Simplicial Complexes Is Undecidable.” In <i>Proceedings of the Annual ACM-SIAM
    Symposium on Discrete Algorithms</i>, 2020–January:767–85. SIAM, 2020. <a href="https://doi.org/10.1137/1.9781611975994.47">https://doi.org/10.1137/1.9781611975994.47</a>.
  ieee: M. Filakovský, U. Wagner, and S. Y. Zhechev, “Embeddability of simplicial
    complexes is undecidable,” in <i>Proceedings of the Annual ACM-SIAM Symposium
    on Discrete Algorithms</i>, Salt Lake City, UT, United States, 2020, vol. 2020–January,
    pp. 767–785.
  ista: 'Filakovský M, Wagner U, Zhechev SY. 2020. Embeddability of simplicial complexes
    is undecidable. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms.
    SODA: Symposium on Discrete Algorithms vol. 2020–January, 767–785.'
  mla: Filakovský, Marek, et al. “Embeddability of Simplicial Complexes Is Undecidable.”
    <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms</i>, vol.
    2020–January, SIAM, 2020, pp. 767–85, doi:<a href="https://doi.org/10.1137/1.9781611975994.47">10.1137/1.9781611975994.47</a>.
  short: M. Filakovský, U. Wagner, S.Y. Zhechev, in:, Proceedings of the Annual ACM-SIAM
    Symposium on Discrete Algorithms, SIAM, 2020, pp. 767–785.
conference:
  end_date: 2020-01-08
  location: Salt Lake City, UT, United States
  name: 'SODA: Symposium on Discrete Algorithms'
  start_date: 2020-01-05
date_created: 2020-05-10T22:00:48Z
date_published: 2020-01-01T00:00:00Z
date_updated: 2021-01-12T08:15:38Z
day: '01'
department:
- _id: UlWa
doi: 10.1137/1.9781611975994.47
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1137/1.9781611975994.47
month: '01'
oa: 1
oa_version: Published Version
page: 767-785
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P31312
  name: Algorithms for Embeddings and Homotopy Theory
publication: Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
publication_identifier:
  isbn:
  - '9781611975994'
publication_status: published
publisher: SIAM
quality_controlled: '1'
scopus_import: 1
status: public
title: Embeddability of simplicial complexes is undecidable
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2020-January
year: '2020'
...
---
_id: '7991'
abstract:
- lang: eng
  text: 'We define and study a discrete process that generalizes the convex-layer
    decomposition of a planar point set. Our process, which we call homotopic curve
    shortening (HCS), starts with a closed curve (which might self-intersect) in the
    presence of a set P⊂ ℝ² of point obstacles, and evolves in discrete steps, where
    each step consists of (1) taking shortcuts around the obstacles, and (2) reducing
    the curve to its shortest homotopic equivalent. We find experimentally that, if
    the initial curve is held fixed and P is chosen to be either a very fine regular
    grid or a uniformly random point set, then HCS behaves at the limit like the affine
    curve-shortening flow (ACSF). This connection between HCS and ACSF generalizes
    the link between "grid peeling" and the ACSF observed by Eppstein et al. (2017),
    which applied only to convex curves, and which was studied only for regular grids.
    We prove that HCS satisfies some properties analogous to those of ACSF: HCS is
    invariant under affine transformations, preserves convexity, and does not increase
    the total absolute curvature. Furthermore, the number of self-intersections of
    a curve, or intersections between two curves (appropriately defined), does not
    increase. Finally, if the initial curve is simple, then the number of inflection
    points (appropriately defined) does not increase.'
alternative_title:
- LIPIcs
article_number: 12:1 - 12:15
article_processing_charge: No
arxiv: 1
author:
- first_name: Sergey
  full_name: Avvakumov, Sergey
  id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
  last_name: Avvakumov
- first_name: Gabriel
  full_name: Nivasch, Gabriel
  last_name: Nivasch
citation:
  ama: 'Avvakumov S, Nivasch G. Homotopic curve shortening and the affine curve-shortening
    flow. In: <i>36th International Symposium on Computational Geometry</i>. Vol 164.
    Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:<a href="https://doi.org/10.4230/LIPIcs.SoCG.2020.12">10.4230/LIPIcs.SoCG.2020.12</a>'
  apa: 'Avvakumov, S., &#38; Nivasch, G. (2020). Homotopic curve shortening and the
    affine curve-shortening flow. In <i>36th International Symposium on Computational
    Geometry</i> (Vol. 164). Zürich, Switzerland: Schloss Dagstuhl - Leibniz-Zentrum
    für Informatik. <a href="https://doi.org/10.4230/LIPIcs.SoCG.2020.12">https://doi.org/10.4230/LIPIcs.SoCG.2020.12</a>'
  chicago: Avvakumov, Sergey, and Gabriel Nivasch. “Homotopic Curve Shortening and
    the Affine Curve-Shortening Flow.” In <i>36th International Symposium on Computational
    Geometry</i>, Vol. 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.
    <a href="https://doi.org/10.4230/LIPIcs.SoCG.2020.12">https://doi.org/10.4230/LIPIcs.SoCG.2020.12</a>.
  ieee: S. Avvakumov and G. Nivasch, “Homotopic curve shortening and the affine curve-shortening
    flow,” in <i>36th International Symposium on Computational Geometry</i>, Zürich,
    Switzerland, 2020, vol. 164.
  ista: 'Avvakumov S, Nivasch G. 2020. Homotopic curve shortening and the affine curve-shortening
    flow. 36th International Symposium on Computational Geometry. SoCG: Symposium
    on Computational Geometry, LIPIcs, vol. 164, 12:1-12:15.'
  mla: Avvakumov, Sergey, and Gabriel Nivasch. “Homotopic Curve Shortening and the
    Affine Curve-Shortening Flow.” <i>36th International Symposium on Computational
    Geometry</i>, vol. 164, 12:1-12:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
    2020, doi:<a href="https://doi.org/10.4230/LIPIcs.SoCG.2020.12">10.4230/LIPIcs.SoCG.2020.12</a>.
  short: S. Avvakumov, G. Nivasch, in:, 36th International Symposium on Computational
    Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.
conference:
  end_date: 2020-06-26
  location: Zürich, Switzerland
  name: 'SoCG: Symposium on Computational Geometry'
  start_date: 2020-06-22
date_created: 2020-06-22T09:14:19Z
date_published: 2020-06-01T00:00:00Z
date_updated: 2021-01-12T08:16:23Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.4230/LIPIcs.SoCG.2020.12
external_id:
  arxiv:
  - '1909.00263'
file:
- access_level: open_access
  checksum: 6872df6549142f709fb6354a1b2f2c06
  content_type: application/pdf
  creator: dernst
  date_created: 2020-06-23T11:13:49Z
  date_updated: 2020-07-14T12:48:06Z
  file_id: '8007'
  file_name: 2020_LIPIcsSoCG_Avvakumov.pdf
  file_size: 575896
  relation: main_file
file_date_updated: 2020-07-14T12:48:06Z
has_accepted_license: '1'
intvolume: '       164'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/3.0/
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P31312
  name: Algorithms for Embeddings and Homotopy Theory
publication: 36th International Symposium on Computational Geometry
publication_identifier:
  isbn:
  - '9783959771436'
  issn:
  - '18688969'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: Homotopic curve shortening and the affine curve-shortening flow
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/3.0/legalcode
  name: Creative Commons Attribution 3.0 Unported (CC BY 3.0)
  short: CC BY (3.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 164
year: '2020'
...
---
_id: '6563'
abstract:
- lang: eng
  text: "This paper presents two algorithms. The first decides the existence of a
    pointed homotopy between given simplicial maps \U0001D453,\U0001D454:\U0001D44B→\U0001D44C,
    and the second computes the group [\U0001D6F4\U0001D44B,\U0001D44C]∗ of pointed
    homotopy classes of maps from a suspension; in both cases, the target Y is assumed
    simply connected. More generally, these algorithms work relative to \U0001D434⊆\U0001D44B."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Marek
  full_name: Filakovský, Marek
  id: 3E8AF77E-F248-11E8-B48F-1D18A9856A87
  last_name: Filakovský
- first_name: Lukas
  full_name: Vokřínek, Lukas
  last_name: Vokřínek
citation:
  ama: Filakovský M, Vokřínek L. Are two given maps homotopic? An algorithmic viewpoint.
    <i>Foundations of Computational Mathematics</i>. 2020;20:311-330. doi:<a href="https://doi.org/10.1007/s10208-019-09419-x">10.1007/s10208-019-09419-x</a>
  apa: Filakovský, M., &#38; Vokřínek, L. (2020). Are two given maps homotopic? An
    algorithmic viewpoint. <i>Foundations of Computational Mathematics</i>. Springer
    Nature. <a href="https://doi.org/10.1007/s10208-019-09419-x">https://doi.org/10.1007/s10208-019-09419-x</a>
  chicago: Filakovský, Marek, and Lukas Vokřínek. “Are Two given Maps Homotopic? An
    Algorithmic Viewpoint.” <i>Foundations of Computational Mathematics</i>. Springer
    Nature, 2020. <a href="https://doi.org/10.1007/s10208-019-09419-x">https://doi.org/10.1007/s10208-019-09419-x</a>.
  ieee: M. Filakovský and L. Vokřínek, “Are two given maps homotopic? An algorithmic
    viewpoint,” <i>Foundations of Computational Mathematics</i>, vol. 20. Springer
    Nature, pp. 311–330, 2020.
  ista: Filakovský M, Vokřínek L. 2020. Are two given maps homotopic? An algorithmic
    viewpoint. Foundations of Computational Mathematics. 20, 311–330.
  mla: Filakovský, Marek, and Lukas Vokřínek. “Are Two given Maps Homotopic? An Algorithmic
    Viewpoint.” <i>Foundations of Computational Mathematics</i>, vol. 20, Springer
    Nature, 2020, pp. 311–30, doi:<a href="https://doi.org/10.1007/s10208-019-09419-x">10.1007/s10208-019-09419-x</a>.
  short: M. Filakovský, L. Vokřínek, Foundations of Computational Mathematics 20 (2020)
    311–330.
date_created: 2019-06-16T21:59:14Z
date_published: 2020-04-01T00:00:00Z
date_updated: 2023-08-17T13:50:44Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s10208-019-09419-x
external_id:
  arxiv:
  - '1312.2337'
  isi:
  - '000522437400004'
intvolume: '        20'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1312.2337
month: '04'
oa: 1
oa_version: Preprint
page: 311-330
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P31312
  name: Algorithms for Embeddings and Homotopy Theory
publication: Foundations of Computational Mathematics
publication_identifier:
  eissn:
  - '16153383'
  issn:
  - '16153375'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Are two given maps homotopic? An algorithmic viewpoint
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 20
year: '2020'
...
---
_id: '8182'
abstract:
- lang: eng
  text: "Suppose that $n\\neq p^k$ and $n\\neq 2p^k$ for all $k$ and all primes $p$.
    We prove that for any Hausdorff compactum $X$ with a free action of the symmetric
    group $\\mathfrak S_n$ there exists an $\\mathfrak S_n$-equivariant map $X \\to\r\n{\\mathbb
    R}^n$ whose image avoids the diagonal $\\{(x,x\\dots,x)\\in {\\mathbb R}^n|x\\in
    {\\mathbb R}\\}$.\r\n  Previously, the special cases of this statement for certain
    $X$ were usually proved using the equivartiant obstruction theory. Such calculations
    are difficult and may become infeasible past the first (primary) obstruction.
    We\r\ntake a different approach which allows us to prove the vanishing of all
    obstructions simultaneously. The essential step in the proof is classifying the
    possible degrees of $\\mathfrak S_n$-equivariant maps from the boundary\r\n$\\partial\\Delta^{n-1}$
    of $(n-1)$-simplex to itself.  Existence of equivariant maps between spaces is
    important for many questions arising from discrete mathematics and geometry, such
    as Kneser's conjecture, the Square Peg conjecture, the Splitting Necklace problem,
    and the Topological Tverberg conjecture, etc. We demonstrate the utility of our
    result  applying it to one such question, a specific instance of envy-free division
    problem."
article_number: '1910.12628'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sergey
  full_name: Avvakumov, Sergey
  id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
  last_name: Avvakumov
- first_name: Sergey
  full_name: Kudrya, Sergey
  id: ecf01965-d252-11ea-95a5-8ada5f6c6a67
  last_name: Kudrya
citation:
  ama: Avvakumov S, Kudrya S. Vanishing of all equivariant obstructions and the mapping
    degree. <i>arXiv</i>.
  apa: Avvakumov, S., &#38; Kudrya, S. (n.d.). Vanishing of all equivariant obstructions
    and the mapping degree. <i>arXiv</i>. arXiv.
  chicago: Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions
    and the Mapping Degree.” <i>ArXiv</i>. arXiv, n.d.
  ieee: S. Avvakumov and S. Kudrya, “Vanishing of all equivariant obstructions and
    the mapping degree,” <i>arXiv</i>. arXiv.
  ista: Avvakumov S, Kudrya S. Vanishing of all equivariant obstructions and the mapping
    degree. arXiv, 1910.12628.
  mla: Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions
    and the Mapping Degree.” <i>ArXiv</i>, 1910.12628, arXiv.
  short: S. Avvakumov, S. Kudrya, ArXiv (n.d.).
date_created: 2020-07-30T10:45:08Z
date_published: 2019-10-28T00:00:00Z
date_updated: 2023-09-07T13:12:17Z
day: '28'
department:
- _id: UlWa
external_id:
  arxiv:
  - '1910.12628'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1910.12628
month: '10'
oa: 1
oa_version: Preprint
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P31312
  name: Algorithms for Embeddings and Homotopy Theory
publication: arXiv
publication_status: submitted
publisher: arXiv
related_material:
  record:
  - id: '11446'
    relation: later_version
    status: public
  - id: '8156'
    relation: dissertation_contains
    status: public
status: public
title: Vanishing of all equivariant obstructions and the mapping degree
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2019'
...
---
_id: '8184'
abstract:
- lang: eng
  text: "Denote by ∆N the N-dimensional simplex. A map f : ∆N → Rd is an almost r-embedding
    if fσ1∩. . .∩fσr = ∅ whenever σ1, . . . , σr are pairwise disjoint faces. A counterexample
    to the topological Tverberg conjecture asserts that if r is not a prime power
    and d ≥ 2r + 1, then there is an almost r-embedding ∆(d+1)(r−1) → Rd. This was
    improved by Blagojevi´c–Frick–Ziegler using a simple construction of higher-dimensional
    counterexamples by taking k-fold join power of lower-dimensional ones. We improve
    this further (for d large compared to r): If r is not a prime power and N := (d+
    1)r−r l\r\nd + 2 r + 1 m−2, then there is an almost r-embedding ∆N → Rd. For the
    r-fold van Kampen–Flores conjecture we also produce counterexamples which are
    stronger than previously known. Our proof is based on generalizations of the Mabillard–Wagner
    theorem on construction of almost r-embeddings from equivariant maps, and of the
    Ozaydin theorem on existence of equivariant maps. "
acknowledgement: We would like to thank F. Frick for helpful discussions
article_number: '1908.08731'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sergey
  full_name: Avvakumov, Sergey
  id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
  last_name: Avvakumov
- first_name: R.
  full_name: Karasev, R.
  last_name: Karasev
- first_name: A.
  full_name: Skopenkov, A.
  last_name: Skopenkov
citation:
  ama: Avvakumov S, Karasev R, Skopenkov A. Stronger counterexamples to the topological
    Tverberg conjecture. <i>arXiv</i>.
  apa: Avvakumov, S., Karasev, R., &#38; Skopenkov, A. (n.d.). Stronger counterexamples
    to the topological Tverberg conjecture. <i>arXiv</i>. arXiv.
  chicago: Avvakumov, Sergey, R. Karasev, and A. Skopenkov. “Stronger Counterexamples
    to the Topological Tverberg Conjecture.” <i>ArXiv</i>. arXiv, n.d.
  ieee: S. Avvakumov, R. Karasev, and A. Skopenkov, “Stronger counterexamples to the
    topological Tverberg conjecture,” <i>arXiv</i>. arXiv.
  ista: Avvakumov S, Karasev R, Skopenkov A. Stronger counterexamples to the topological
    Tverberg conjecture. arXiv, 1908.08731.
  mla: Avvakumov, Sergey, et al. “Stronger Counterexamples to the Topological Tverberg
    Conjecture.” <i>ArXiv</i>, 1908.08731, arXiv.
  short: S. Avvakumov, R. Karasev, A. Skopenkov, ArXiv (n.d.).
date_created: 2020-07-30T10:45:34Z
date_published: 2019-08-23T00:00:00Z
date_updated: 2023-09-08T11:20:02Z
day: '23'
department:
- _id: UlWa
external_id:
  arxiv:
  - '1908.08731'
  isi:
  - '000986519600004'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1908.08731
month: '08'
oa: 1
oa_version: Preprint
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P31312
  name: Algorithms for Embeddings and Homotopy Theory
publication: arXiv
publication_status: submitted
publisher: arXiv
related_material:
  record:
  - id: '8156'
    relation: dissertation_contains
    status: public
status: public
title: Stronger counterexamples to the topological Tverberg conjecture
type: preprint
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2019'
...
---
_id: '8185'
abstract:
- lang: eng
  text: "In this paper we study envy-free division problems. The classical approach
    to some of such problems, used by David Gale, reduces to considering continuous
    maps of a simplex to itself and finding sufficient conditions when this map hits
    the center of the simplex. The mere continuity is not sufficient for such a conclusion,
    the usual assumption (for example, in the Knaster--Kuratowski--Mazurkiewicz and
    the Gale theorem) is a certain boundary condition.\r\n  We follow Erel Segal-Halevi,
    Fr\\'ed\\'eric Meunier, and Shira Zerbib, and replace the boundary condition by
    another assumption, which has the economic meaning of possibility for a player
    to prefer an empty part in the segment\r\npartition problem. We solve the problem
    positively when $n$, the number of players that divide the segment, is a prime
    power, and we provide counterexamples for every $n$ which is not a prime power.
    We also provide counterexamples relevant to a wider class of fair or envy-free
    partition problems when $n$ is odd and not a prime power."
article_number: '1907.11183'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sergey
  full_name: Avvakumov, Sergey
  id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
  last_name: Avvakumov
- first_name: Roman
  full_name: Karasev, Roman
  last_name: Karasev
citation:
  ama: Avvakumov S, Karasev R. Envy-free division using mapping degree. <i>arXiv</i>.
    doi:<a href="https://doi.org/10.48550/arXiv.1907.11183">10.48550/arXiv.1907.11183</a>
  apa: Avvakumov, S., &#38; Karasev, R. (n.d.). Envy-free division using mapping degree.
    <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.1907.11183">https://doi.org/10.48550/arXiv.1907.11183</a>
  chicago: Avvakumov, Sergey, and Roman Karasev. “Envy-Free Division Using Mapping
    Degree.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.1907.11183">https://doi.org/10.48550/arXiv.1907.11183</a>.
  ieee: S. Avvakumov and R. Karasev, “Envy-free division using mapping degree,” <i>arXiv</i>.
    .
  ista: Avvakumov S, Karasev R. Envy-free division using mapping degree. arXiv, 1907.11183.
  mla: Avvakumov, Sergey, and Roman Karasev. “Envy-Free Division Using Mapping Degree.”
    <i>ArXiv</i>, 1907.11183, doi:<a href="https://doi.org/10.48550/arXiv.1907.11183">10.48550/arXiv.1907.11183</a>.
  short: S. Avvakumov, R. Karasev, ArXiv (n.d.).
date_created: 2020-07-30T10:45:51Z
date_published: 2019-07-25T00:00:00Z
date_updated: 2023-09-07T13:12:17Z
day: '25'
department:
- _id: UlWa
doi: 10.48550/arXiv.1907.11183
external_id:
  arxiv:
  - '1907.11183'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1907.11183
month: '07'
oa: 1
oa_version: Preprint
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P31312
  name: Algorithms for Embeddings and Homotopy Theory
publication: arXiv
publication_status: submitted
related_material:
  link:
  - relation: later_version
    url: https://doi.org/10.1112/mtk.12059
  record:
  - id: '8156'
    relation: dissertation_contains
    status: public
status: public
title: Envy-free division using mapping degree
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2019'
...