Motivated by Tverberg-type problems in topological combinatorics 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 higher-multiplicity intersections. We focus on conditions for the existence of almost r-embeddings, i.e., maps f: K -> R^d such that the intersection of f(sigma_1), ..., f(sigma_r) is empty whenever sigma_1,...,sigma_r are pairwise disjoint simplices of K. Generalizing the classical Haefliger-Weber embeddability criterion, we show that a well-known necessary deleted product condition for the existence of almost r-embeddings is sufficient in a suitable r-metastable range of dimensions: If r d > (r+1) dim K + 2 then there exists an almost r-embedding K-> R^d if and only if there exists an equivariant map of the r-fold deleted product of K to the sphere S^(d(r-1)-1). This significantly extends one of the main results of our previous paper (which treated the special case where d=rk and dim K=(r-1)k, for some k> 2), and settles an open question raised there.
@InProceedings{mabillard_et_al:LIPIcs.SoCG.2016.51, author = {Mabillard, Isaac and Wagner, Uli}, title = {{Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {51:1--51:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.51}, URN = {urn:nbn:de:0030-drops-59438}, doi = {10.4230/LIPIcs.SoCG.2016.51}, annote = {Keywords: Topological Combinatorics, Tverberg-Type Problems, Simplicial Complexes, Piecewise-Linear Topology, Haefliger-Weber Theorem} }
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