Embeddability in R3 is NP-hard

We prove that the problem of deciding whether a two- or three-dimensional simplicial complex embeds into R3 is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an S3 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 ∩ co- NP. (Membership of the latter problem in co-NP assumes the Generalized Riemann Hypotheses.) 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 of manifolds with boundary tori.