---
_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: <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>'
  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>'
  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>.
  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.
  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.” <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>.
  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
corr_author: '1'
date_created: 2018-12-11T11:56:03Z
date_published: 2014-06-08T00:00:00Z
date_updated: 2026-07-29T11:26:06Z
day: '08'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1145/2582112.2582134
file:
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  date_updated: 2020-07-14T12:45:30Z
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fulldoi: https://doi.org/10.1145/2582112.2582134
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:
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    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'
...