  We construct the 3d quantum trace map, a homomorphism from the Kauffman
bracket skein module of an ideally triangulated 3-manifold to its (square root)
quantum gluing module, thereby giving a precise relationship between the two
quantizations of the character variety of ideally triangulated 3-manifolds.
This map, whose existence was conjectured earlier by Agarwal, Gang, Lee, and
Romo, is a natural 3-dimensional analog of the 2d quantum trace map of Bonahon
and Wong. Our construction is based on the study of stated skein modules and
their behavior under splitting, especially into face suspensions.
