Whitney Trick and Counterexamples to the Topological Tverberg Conjecture

Let's assume that $r$ embedded balls intersect in $\mathbf{R}^d$ transversally 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 of 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 $\mathbf{R}^d$, one can always find “a large number” of disjoint cells of the $N$-simplex that intersect in the image in $\mathbf{R}^d.$ Due to the codimension requirements of our current techniques, we can only build counterexamples for d at least $12$. So what happens in lower dimensions remains a mystery...
(Joint work with A. Avvakumov, A. Skopenkov, and U. Wagner)