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

TL;DR

This paper develops a higher-multiplicity Whitney trick to analyze and characterize the existence of maps from simplicial complexes to Euclidean space without r-fold intersection points, advancing the understanding of topological Tverberg problems.

Contribution

It introduces a higher-multiplicity Whitney trick and establishes the sufficiency of the Deleted Product Criterion under certain conditions, extending the tools for Tverberg-type problems.

Findings

Proves the sufficiency of the Deleted Product Criterion for maps without r-Tverberg points under specific dimension restrictions.

Develops a higher-multiplicity Whitney trick as a key technical tool.

Provides new counterexamples to the topological Tverberg conjecture for certain dimensions.

Abstract

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. Our main technical tool is a higher-multiplicity version of the classical Whitney trick. An important guiding idea for our work was that sufficiency of the DPC, together with an old result of Ozaydin on the existence of equivariant maps, might yield an approach to disproving the remaining open cases of the long-standing topological Tverberg conjecture. Unfortunately, our proof of the sufficiency of the DPC requires a "codimension 3" proviso, which is not satisfied for when K is the N-simplex. Recently, Frick found an extremely elegant way to overcome this last "codimension 3" obstacle and to construct counterexamples to the topological Tverberg conjecture for d at least 3r+1 (r not a prime power). Here, we present a different construction that yields counterexamples for d at least 3r (r not a prime power).