Eliminating Tverberg points, I. An analogue of the Whitney trick

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.