Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the $r$-Metastable Range

TL;DR

This paper extends the deleted product criterion to determine when a finite simplicial complex can be embedded in Euclidean space without higher-multiplicity intersections, generalizing classical embedding results in a new dimension range.

Contribution

It generalizes the classical embeddability criterion to the r-metastable range, providing a sufficient condition for almost r-embeddings of complexes.

Findings

Necessary deleted product condition is sufficient in the r-metastable range.

Extends previous results to more general dimension settings.

Provides new tools for topological combinatorics and embedding problems.

Abstract

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).