Before revision:  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 ).
After revision:  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 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 ).