# Generalization of Moise's theorem

I am looking for a generalization of Moise's theorem, which the few professors that I asked treat as a "known geometric fact" but none could find a reference to an article proving it.

The claim is like this: Let $M$ be a compact 3 manifold (Riemannian but I do not think that helps), and let $X,Y$ be compact 2 manifolds (Riemannian aswell) that intersect transversally, then there is a triangulation of $M$ such that $X,Y$ are sub-complexes.

I looked and I believe that this is correct. The fact that there are two surfaces doesn't matter. You can take any finite collection of immersions of compact surfaces and take the union of the maps. There will be a triangulation of $M$ where the resulting image is a subcomplex. – Sam Nead Apr 6 '11 at 8:29
In fact it helps immensely that $M$, $X$, and $Y$ are all Riemannian, so much so that the question is both true and not at all a generalization of Moise's theorem. Instead, you are looking for a smooth triangulation of $M$ that supports $X$ and $Y$. A much better citation is to Goresky's theorem that any smooth stratification of a smooth manifold is supported by a smooth triangulation. This theorem is not specific to dimension 3; and stratifications in this sense are much more general than transversely intersecting submanifolds. On the other hand, the special case of a transverse intersection was known long before. Actually all of Goresky's theorem was sort-of previously known; the merit of the paper was to clean up and unify previous knowledge.
In general, in geometric topology, the hard direction is to go from continuous structures, to piecewise linear structures like triangulations, to smooth structures. Moise's theorem is about triangulating a topological manifold with no smoothness in sight --- so if the manifold is secretly smooth, the simplices of the produced triangulation could well be extreme fractals. On the other hand, Moise's theorem can be adapted to a variation of your question, in which $X$ and $Y$ are both collared and intersect transversely. This is in a way what you requested, but since your manifolds are smooth, not really what you want. Note that if $X$ and $Y$ are not collared, they could have Alexander horns, and then the supporting triangulation would not exist.