A higher-multiplicity whitney trick and counterexamples to the topological tverberg conjecture

Let’s assume that r balls intersect in R^d in general position and that their intersection consists of two points of opposite intersection signs. I’ll describe a generalization of the classical Whitney Trick to this situation: our goal is to eliminate the pair of intersection points, by means of ambient isotopies having «small» support. A neat application and our original motivation to prove this «generalized Whitney trick» is the construction of counterexamples to the Topological Tverberg Conjecture, which asserts that for any continuous map from the N-simplex to R^d, one can always find «a large number» of disjoint cells of the N-simplex that intersect in the image in R^d. Our Whitney trick requires a technical «codimension assumption» to work, so in order to use it and obtain counterexamples to the topological Tverberg, one needs to overcome this dimensional barrier. Two methods have been found so far: the first, due to Frick, uses a combinatorial trick found by Gromov (and rediscovered by Blagojevic-Frick-Ziegler), yielding counterexamples in dimension d>18. The second method uses a restricted class of «prismatic maps» and yields counterexamples in dimension d>11. What happens in lower dimensions remains a mystery… (Joint work with S. Avvakumov, A. Skopenkov, and U. Wagner)