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Hardness of almost embedding simplicial complexes in $\mathbb R^d$ (1703.06305v2)

Published 18 Mar 2017 in math.GT and cs.CG

Abstract: A map $f\colon K\to \mathbb Rd$ of a simplicial complex is an almost embedding if $f(\sigma)\cap f(\tau)=\emptyset$ whenever $\sigma,\tau$ are disjoint simplices of $K$. Theorem. Fix integers $d,k\ge2$ such that $d=\frac{3k}2+1$. (a) Assume that $P\ne NP$. Then there exists a finite $k$-dimensional complex $K$ that does not admit an almost embedding in $\mathbb Rd$ but for which there exists an equivariant map $\tilde K\to S{d-1}$. (b) The algorithmic problem of recognition almost embeddability of finite $k$-dimensional complexes in $\mathbb Rd$ is NP hard. The proof is based on the technique from the Matou\v{s}ek-Tancer-Wagner paper (proving an analogous result for embeddings), and on singular versions of the higher-dimensional Borromean rings lemma and a generalized van Kampen--Flores theorem.

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