We consider conditions under which a finite simplicial complex KK can be mapped to RdRd without triple, quadruple, or higher-multiplicity intersections. More precisely, an almost rr-embedding of KK in RdRd is a map f:K→Rdf:K→Rd such that the images of any rr pairwise disjoint simplices of KK do not have a common point. In recent joint work with Mabillard, we proved that a well-known deleted product criterion is sufficient for the existence of almost rr-embeddings of k(r−1)k(r−1)-dimensional complexes in RkrRkr, provided k≥3k≥3. This was extended to codimension k=2k=2 and r≥3r≥3 in joint work with Avvakumov, Mabillard, and Skopenkov. We survey these results and the main ideas, and we discuss how sufficiency of the deleted product criterion, together with results of Özaydin and of Gromov, Blagojevic, Frick, and Ziegler, implies the existence of counterexamples to the topological Tverberg conjecture, i.e., almost rr-embeddings of the (d+1)(r−1)(d+1)(r−1)-simplex in RdRd whenever rr is not a prime power and d≥2rd≥2r.

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