Is the 3-sphere recognition problem NP-complete? It is known that determining whether or not a given triangulated 3-manifold is a 3-sphere
is in NP, via work by
Saul Schleimer in 2004: "Sphere recognition lies in NP"
arXiv:math/0407047v1 [math.GT].
I am wondering if this has been established to be NP-complete in the last five or six years?
Analogous problems, such as the 3-manifold knot genus problem, have been shown NP-complete.
If it's NP-complete, then wouldn't you have proved that no set of (uniformly) polynomial-time computable invariants of 3-manifolds distinguishes 3-spheres from other 3-manifolds. I would be very surprised if this is known.