Title: Eliminating Tverberg Points: An Analogue of the Whitney Trick

Abstract: Kuratowski's Theorem gives a criterion to decide the planarity of a graph. I.e., whether a 1-simplicial complex can be embedded in R^2. This problem can be generalised to the embeddability of a n-simplicial complex K into R^{2n}, and this question is solved by van Kampen embeddability criterion: the vanishing of an obstruction cocycle is linked to the existence of an embedding. This readily yields a polynomial-time algorithm for deciding embeddability. The proof of this result is based on the Whitney trick: if two n-balls intersect in R^{2n} in two points of opposite signs, then one can "remove" these two intersections by a "local" isotopy. 

In my talk, I will explain how this trick also works for configurations with more than two balls. In my drawings, "more than two" is going to mean "three". For instance, three balls intersecting in two points of opposite signs can be "untangled".  More generally, the Whitney trick also works for intersection points of higher multiplicity. 

This fact leads to a generalised version of the van Kampen Criterion for the existence of maps K -> R^d without self-intersection of "high multiplicity". In particular, it shows that the problem "Does a complex K mapping into R^d has a "Tverberg-type" theorem?" is decidable in polynomial time -- but we must stress that our techniques only work if the dimension of K is at most d-3.