Embeddability in $\mathbb{R}^3$ is NP-hard

TL;DR

This paper proves that determining embeddability of 2- or 3-dimensional complexes into three-dimensional space is NP-hard, contrasting with simpler cases, and introduces a reduction from satisfiability problems using advanced topology techniques.

Contribution

It establishes NP-hardness for embeddability problems in 3D topology, using novel reductions and techniques from low-dimensional topology.

Findings

Embeddability in $\\mathbb{R}^3$ is NP-hard for 2- and 3-dimensional complexes.

Deciding if a 3-manifold with boundary tori admits an $\mathbb{S}^3$ filling is NP-hard.

Contrasts with polynomial-time solvable lower-dimensional cases.

Abstract

We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into $\mathbb{R}^3$ is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an $\mathbb{S}^{3}$ filling is NP-hard. The former stands in contrast with the lower dimensional cases which can be solved in linear time,and the latter with a variety of computational problems in 3-manifold topology (for example, unknot or 3-sphere recognition, which are in NP and co-NP assuming the Generalized Riemann Hypothesis). Our reduction encodes a satisfiability instance into the embeddability problem of a 3-manifold with boundary tori, and relies extensively on techniques from low-dimensional topology, most importantly Dehn fillings on link complements.