We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into R^3 is NP-hard. This stands in contrast with the lower dimensional cases which can be solved in linear time,and  a variety of computational problems in \mathbb{R 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.
We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into R^3 is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an \mathbb{S 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.
R
This
Our construction also shows that deciding whether a 3-manifold with boundary tori admits an \mathbb{S
A
the latter with
R
\mathbb{R
3-manifold topology (for example,
A
,
R
(assuming the generalized Riemann hypothesis
assuming the Generalized Riemann Hypothesis