# Hardness of almost embedding simplicial complexes in $\mathbb R^d$

## TL;DR

This paper proves that determining almost embeddability of simplicial complexes in certain dimensions is NP-hard, highlighting computational complexity barriers in topological embedding problems.

## Contribution

It establishes the NP-hardness of recognizing almost embeddability for finite complexes in specific dimensions, extending previous embedding complexity results.

## Key findings

- Existence of complexes not almost embeddable but with equivariant maps
- NP-hardness of recognition problem for almost embeddability
- Extension of techniques from embedding to almost embedding context

## Abstract

A map $f\colon K\to \mathbb R^d$ 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 $\mathbb R^d$ 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 $\mathbb R^d$ 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.