---
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 ${\ensuremath{\mathbb{R}}}^d$ without higher-multiplicity intersections. We focus on conditions for the existence of *almost $r$-embeddings*, i.e., maps $f\colon K \to {\ensuremath{\mathbb{R}}}^d$ such that $f(\sigma_1)\cap \dots \cap f(\sigma_r) =\emptyset$ whenever $\sigma_1,\dots,\sigma_r$ 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 $r d \geq (r+1) \dim K +3,$ then there exists an almost $r$-embedding $K\to {\ensuremath{\mathbb{R}}}^d$ if and only if there exists an equivariant map ${\ensuremath{{K}^{r}_{\Delta}}} \to_{{\ensuremath{\mathfrak{S}}}_r} S^{d(r-1)-1}$, where ${\ensuremath{{K}^{r}_{\Delta}}}$ is the *deleted $r$-fold product* of $K$ (the subcomplex of the $r$-fold cartesian product whose cells are products of pairwise disjoint simplices of $K$) and ${\ensuremath{\mathfrak{S}}}_r$ denotes the symmetric group. 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 $k\geq 3$), and settles one of the main open questions raised there.

    As a corollary, our result together with recent work of Filakovský and Vokřínek on the homotopy classification of equivariant maps under non-free actions imply that almost $r$-embeddability of simplicial complexes is algorithmically decidable in the $r$-metastable range (in polynomial time if $r$ and $d$ are fixed).
bibliography:
- 'EliminatingMultiplePoints.bib'
---

[Eliminating Higher-Multiplicity Intersections, II.]{} 0.1in [ The Deleted Product Criterion in the $r$-Metastable Range]{}

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`imabillard@ist.ac.at`

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`uli@ist.ac.at`

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[*IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria 0.4in*]{}

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