Embeddability in R3 is NP-hard.

We prove that the problem of deciding whether a 2-or 3-dimensional simplicial complex embeds into R3 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 R3 like unknot or 3-sphere recognition which are in NP ∩ 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.