Eliminating Higher-Multiplicity Intersections, III. Codimension 2

We study conditions under which a finite simplicial complex K can be mapped to R without higher-multiplicity intersections. An almost r-embedding is a map f : K → R such that the images of any r pairwise disjoint simplices of K do not have a common point. We show that if r is not a prime power and d ≥ 2r + 1, then there is a counterexample to the topological Tverberg conjecture, i.e., there is an almost r-embedding of the (d+ 1)(r− 1)-simplex in R. This improves on previous constructions of counterexamples (for d ≥ 3r) based on a series of papers by M. Özaydin, M. Gromov, P. Blagojević, F. Frick, G. Ziegler, and the second and fourth present author. The counterexamples are obtained by proving the following algebraic criterion in codimension 2: If r ≥ 3 and if K is a finite 2(r− 1)-complex then there exists an almost r-embedding K → R if and only if there exists a general position PL map f : K → R such that the algebraic intersection number of the f -images of any r pairwise disjoint simplices of K is zero. This result can be restated in terms of cohomological obstructions or equivariant maps, and extends an analogous codimension 3 criterion by the second and fourth author. As an application we classify ornaments f : S t S t S → R up to ornament concordance. It follows from work of M. Freedman, V. Krushkal, and P. Teichner that the analogous criterion for r = 2 is false. We prove a beautiful lemma on singular higher-dimensional Borromean rings, yielding an elementary proof of the counterexample.