文章基本信息

标题：Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range
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作者：Isaac Mabillard ; Uli Wagner
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2016
卷号：51
页码：51:1-51:12
DOI：10.4230/LIPIcs.SoCG.2016.51
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：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.
关键词：Topological Combinatorics; Tverberg-Type Problems; Simplicial Complexes; Piecewise-Linear Topology; Haefliger-Weber Theorem