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Motivated by topological Tverberg-type problems and by classical resultsabout embeddings (maps without double points), we study the question whether afinite simplicial complex K can be mapped into R^d without triple, quadruple,or, more generally, r-fold points. Specifically, we are interested in maps ffrom K to R^d that have no r-Tverberg points, i.e., no r-fold points withpreimages in r pairwise disjoint simplices of K, and we seek necessary andsufficient conditions for the existence of such maps.  We present a higher-multiplicity analogue of the completeness of the VanKampen obstruction for embeddability in twice the dimension. Specifically, weshow that under suitable restrictions on the dimensions, a well-known DeletedProduct Criterion (DPC) is not only necessary but also sufficient for theexistence of maps without r-Tverberg points. Our main technical tool is ahigher-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 thelong-standing topological Tverberg conjecture. Unfortunately, our proof of thesufficiency of the DPC requires a "codimension 3" proviso, which is notsatisfied 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 topologicalTverberg conjecture for d at least 3r+1 (r not a prime power). Here, we presenta different construction that yields counterexamples for d at least 3r (r not aprime power).

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