---
res:
  bibo_abstract:
  - '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.@eng'
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Isaac
      foaf_name: Mabillard, Isaac
      foaf_surname: Mabillard
      foaf_workInfoHomepage: http://www.librecat.org/personId=32BF9DAA-F248-11E8-B48F-1D18A9856A87
  - foaf_Person:
      foaf_givenName: Uli
      foaf_name: Wagner, Uli
      foaf_surname: Wagner
      foaf_workInfoHomepage: http://www.librecat.org/personId=36690CA2-F248-11E8-B48F-1D18A9856A87
    orcid: 0000-0002-1494-0568
  bibo_doi: 10.1145/2582112.2582134
  dct_date: 2014^xs_gYear
  dct_language: eng
  dct_publisher: ACM@
  dct_title: Eliminating Tverberg points, I. An analogue of the Whitney trick@
...