I'm currently researching the relationship between Moise's Theorem (Every closed $3$-manifold has a triangulation) and other properties of manifolds. In particular, I'm trying to learn about the fundamental domain of a manifold and I want to know where I can find a formal definition online (or a reference to a popular book that would be in most university libraries, the one book that I heard does have it is "A textbook of topology" by Seifert, but my university does not have the book).

Let $\mathcal{M}$ be a topological space satisfying the conditions to be a topological $n$-manifold (locally euclidean, Hausdorff, second countable). From what I understand, if we have some group action $G(\phi)$ on $\mathcal{M}$, then the fundamental domain of $\mathcal{M}$ is some set of points from $\mathcal{M}$ such that the orbits of that set of points covers all of $\mathcal{M}$ (let me know if this is incorrect).

In the context of closed $3$-manifolds, I also want to have that $\mathcal{M}$ has a maximal smooth atlas $\mathcal{A}$ (for every two coordinate charts in the domain of $\mathcal{M}$, the transition map between these charts is a diffeomorphism). Given that Moise's Theorem permits that there is an essentially unique piece-wise linear structure, how can I learn about the correlation between this linear structure and the fundamental domain of $\mathcal{M}$? I can't come up with any connections, but for motivations to do with Theoretical Physics, I suspect that the structure guaranteed by Moise's Theorem and the fundamental domain are related (I won't digress).

Essentially, my question is two-fold:

1) Where can I find a reference that discusses (In some technical detail) the fundamental domain of a manifold, and maybe in particular a $3$-manifold?

2) Is there any connection between Moise's Theorem for $3$-manifolds and the fundamental domain of a $3$-manifold? That is to say, can some technique such barycentric subdivison or something else generate one from the other? I'm looking for something of that flavour.

anysmooth $n$-manifold has a triangulation. The content of Moise's theorem is that a naked $3$-manifold (without any smooth structure) can be given an essentially unique triangulation. $\endgroup$ – Andy Putman Mar 9 '12 at 20:22