Hardness of almost embedding simplicial complexes in $\mathbb{R}^d$, II

TL;DR

This paper proves that recognizing almost embeddability of finite simplicial complexes in certain Euclidean spaces is NP-hard, and shows that existing embedding obstructions are incomplete in these cases, extending previous results.

Contribution

It establishes NP-hardness of the recognition problem and demonstrates the incompleteness of embedding obstructions for specific dimensions, generalizing prior work.

Findings

NP-hardness of recognizing almost embeddability in specified dimensions

Embedding obstruction is incomplete for certain complexes in these dimensions

Generalizes previous results for the case d = 3k/2 + 1

Abstract

A map $f: K \to \mathbb{R}^d$ of a simplicial complex is an almost embedding if $f(\sigma) \cap f(\tau) = \varnothing$ whenever $\sigma, \tau$ are disjoint simplices of $K$. Fix integers $d,k \geqslant 2$ such that $k+2 \leqslant d \leqslant\frac{3k}2+1$. Assuming that the "preimage of a cycle is a cycle" we prove $\mathbf{NP}$-hardness of the algorithmic problem of recognition of almost embeddability of finite $k$-dimensional complexes in $\mathbb{R}^d$. Assuming that $\mathbf{P} \ne \mathbf{NP}$ (and that the "preimage of a cycle is a cycle") we prove that the embedding obstruction is incomplete for $k$-dimensional complexes in $\mathbb{R}^d$ using configuration spaces. Our proof generalizes the Skopenkov-Tancer proof of this result for $d = \frac{3k}{2} + 1$.