The 3D index of an ideal triangulation and angle structures

The 3D index of Dimofte-Gaiotto-Gukov a partially defined function on the set of ideal triangulations of 3-manifolds with $r$ torii boundary components. For a fixed $2r$ tuple of integers, the index takes values in the set of $q$-series with integer coefficients. Our goal is to give an axiomatic definition of the tetrahedron index, and a proof that the domain of the 3D index consists precisely of the set of ideal triangulations that support an index structure. The latter is a generalization of a strict angle structure. We also prove that the 3D index is invariant under 3-2 moves, but not in general under 2-3 moves.