消除高多样性交叉点，II。$r$ -亚稳区的乘积判据
Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the $r$-Metastable Range
摘要：
受拓扑组合学中的Tverberg型问题和嵌入（没有二重点的映射）的经典结果的启发，我们研究了有限单纯复形K是否可以映射到R^d中而没有更高重数的交叉.我们主要讨论几乎r-嵌入存在的条件，即，从K到R^d的映射，在K的任意r个成对不相交单形的集合之间没有r个交点。推广了经典的Haefliger-Weber嵌入性准则，证明了在适当的r-亚稳维数范围内（rd>（r+1）dim K +2），存在几乎r-嵌入的一个著名的必要删积条件是充分的.这显著地扩展了我们以前的论文的一个主要结果（其中处理了d=rk和dim K=（r-1）k的特殊情况，其中k> 3）。
英文摘要：
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 from K to R^d without r-intersection points among any set of r 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 (r d > (r+1) dim K +2). 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> 3).
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