On the connectivity of spaces of three-dimensional tilings

We consider domino tilings of three-dimensional cubiculated manifolds with or without boundary, including subsets of Euclidean space and three-dimensional tori. In particular, we are interested in the connected components of the space of tilings of such regions under local moves. Building on the work of the third and fourth authors, we allow two possible local moves, the flip and trit. These moves are considered with respect to two topological invariants, the twist and flux. Our main result proves that, up to refinement: Two tilings are connected by flips and trits if and only if they have the same flux. Two tilings are connected by flips alone if and only if they have the same flux and twist.