In joint preliminary work with Daryl Cooper and Priyam Patel, we show that each of Thurston’s eight model geometries is “locally combinatorially defined” or LCD. Further, we show this is equivalent to the existence of a compact branched 3-manifold, W(G), for each geometry, G, with the following property: a closed 3-manifold, M, has a G-structure if and only if there is an immersion of M into W(G). In this talk, I will give some background on branched n-manifolds, construct the branched 3-manifold in detail for spherical geometry, and discuss some of the results in the other geometries.