Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems

Understand

Motivated by topological Tverberg-type problems and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without triple, quadruple, or, more generally, r-fold points.

• Specifically, we are interested in maps f from K to R^d that have no r-Tverberg points, i.e., no r-fold points with preimages in r pairwise disjoint simplices of K, and we seek necessary and sufficient conditions for the existence of such maps.
• We present a higher-multiplicity analogue of the completeness of the Van Kampen obstruction for embeddability in twice the dimension.
• Specifically, we show that under suitable restrictions on the dimensions, a well-known Deleted Product Criterion (DPC) is not only necessary but also sufficient for the existence of maps without r-Tverberg points.