Combinatorics Seminar

When: Sunday, June 14, 10am
Where: Schreiber 309
Speaker: Uli Wagner, IST Austria
Title: Eliminating Multiple Intersections and Tverberg Points


In this talk, we will survey some recent developments at the interface between combinatorics and topology regarding higher-multiplicity intersections and a new approach to topological Tverberg-type problems.

The general theme of the talk will be the question whether a given k-dimensional simplicial complex (in other words a (k+1)-uniform hypergraph) can be mapped (topologically, i.e., with curvy simplices) into R^d without self-intersections of a given multiplicity r: e.g., for r=2, we forbid double points, i.e., we insist on embeddings (higher-dimensional analogues of graph planarity); for r=3, we do not care about double points, but we forbid triple points (e.g., three triangles intersecting in a common point in R^3), etc. In particular, we are interested in maps that have no r-Tverberg points, i.e., no r-fold intersections between r pairwise vertex-disjoint simplices.

We present higher-multiplicity analogues of several classical notions and results concerning embeddings, e.g., of the Whitney trick and of the Van Kampen obstruction. The main result is that in a certain range of dimensions (more precisely, for d \geq \frac{r+1}{r}k+3), a certain well-known necessary condition (the "deleted product obstruction", or DPO for short) is also sufficient for the existence of a map f:K -> R^d without r-Tverberg points.

One main motivation for our work was to propose an approach to disproving the topological Tverberg conjecture for non-prime-power multiplicities r, i.e., to construct, for such r, maps from the N-simplex into d-space, N=(d+1)(r-1) that do not have r-Tverberg points, by combining sufficiency of the DPO with an old result of Ozaydin's on the existence of equivariant maps. Unfortunately, there remained one difficulty to completing this approach: our results critically rely on an assumption of "codimension" d-k\geq 3, which is not satisfied for K= the N-simplex. However, very recently, Frick found an extremely elegant way to sidestep this "codimension 3 obstacle", by applying the so-called constraint method of Blagojevic-Frick-Ziegler to reduce the problem to a setting where the codimension restrictions are satisfied. This leads to the first counterexamples for r=6 and d=19.

This is presenting joint work with Isaac Mabillard, and reporting on follow-up work by Florian Frick.