Eliminating higher-multiplicity intersections. III. Codimension 2

We study conditions under which a finite simplicial complex K can be mapped to ?d without higher-multiplicity intersections. An almost r-embedding is map f: ? such that the images of any r pairwise disjoint simplices do not have common point. show if prime power and d ? 2r + 1, then there counterexample topological Tverberg conjecture, i.e., an (d +1)(r ? 1)-simplex in ?d. This improves on previous constructions counterexamples (for 3r) based series papers by M. Özaydin, Gromov, P. Blagojevi?, F. Frick, G. Ziegler, second fourth present authors. The are obtained proving following algebraic criterion codimension 2: If 3 2(r 1)-complex, exists ?2r only general position PL intersection number f-images zero. result restated terms cohomological obstruction extends analogous As another application, we classify ornaments S3 ? ?5 up ornament concordance. It follows from work Freedman, V. Krushkal Teichner for = 2 false. prove lemma singular higher-dimensional Borromean rings, yielding elementary proof counterexample.