"Eliminating Multiple Intersections and Tverberg Points"
Motivated by topological Tverberg-type problems and by classical results aboutembeddings (maps without double points), we study conditions under which a finite simplicial complex K can be mapped into d-dimensional Euclidean space without r-fold points (image points with at least r distinct preimages), for a given multiplicity r≥3. In particular, we are interested in maps that have no r-Tverberg points, i.e., no r-fold points with premiages in r pairwise disjoint simplices, and we seek necessary and sufficient conditions for the existence of such maps.
We present higher-multiplicity analogues of several classical results on embeddings, in particular the Whitney trick, the Van Kampen-Shapiro-Wu Theorem and, more generally, the Haefliger-Weber Theorem, which guarantee that under suitable codimension restrictions, a well-known deleted product criterion is not only necessary but also sufficient for the existence of maps without r-Tverberg points. More specifically, if dim K=m and d ≥(k+3)(r+1)/m , then K admits a map into d-space without r-Tverberg point if and only if the r-fold deleted product of K admits an equivariant map into a sphere of dimension d(r-1)-1 (with respect to a suitable action of the symmetric group).
Özaydin showed that if r is not a prime power then the equivariant map in question exists in many interesting cases, including the case that K is the N-simplex,N=(d+1)(r-1). Inspired by this, an important motivation for our work was that sufficiency of the deleted product criterion might thus yield an approach to constructing counterexamples to the topological Tverberg conjecture for these values of r. Unfortunately, our results are not directly applicable to the N-simplex, because of the codimension restrictions they require.
In a recent breakthrough, Frick found an extremely elegant way of sidestepping this difficulty, by using the constraint method of Blagojevic-Frick-Ziegler to reduce the construction of counterexamples to a suitable lower-dimensional skeleton, for which the required codimension retrictions are satisfied and both Özaydin’s and our results apply.
This is joint work with Isaac Mabillard.