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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.

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There is no "the" fundamental domain, there are many. Do you want to study a particular way of generating them, or are you interested in something general, and if so, what? –  Ryan Budney Mar 9 '12 at 19:56
In addition to Ryan's comment, if you are assuming that your 3-manifold has a smooth atlas, then there is no need for Moise's theorem -- any smooth $n$-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. –  Andy Putman Mar 9 '12 at 20:22
See, this is exactly why I needed to ask this question, I thought that there was something that was THE canonical fundamental domain. The motivation behind the question comes from research on the topology and geometry of the universe. A lot of theoretical physicists consider the universe to be a closed $3$-manifold and so I was interested to see how we could discern what properties that $3$-manifold would have. If there was a correspondence between the group action defined by "a" fundamental domain and certain group actions in theoretical physics, then that could be a property. –  Samuel Reid Mar 9 '12 at 22:09
You are conflating two very different things. If you want to learn about three-manifolds then there are several references. I suggest Rolfsen's book "Knots and links" and Hempel's book "3-manifolds" as (non-trivial) introductions to the topological side. Weeks' book "The shape of space" is somewhat easier and also gives a very nice introduction to the possible cosmological application of three-manifold topology. Benedetti and Petronio's book "Lectures on hyperbolic geometry" may also be useful to you. –  Sam Nead Mar 9 '12 at 23:51
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1 Answer

up vote 5 down vote accepted

If you are interested in definition, look in John Ratcliffe's book "Foundations of hyperbolic manifolds." He discussed fundamental domains for hyperbolic manifolds in great detail and most of the discussion goes through in the context of smooth/PL manifolds. In general, there is no "canonical" definition of a fundamental domain for a manifold. Here is one possible definition of a fundamental domain for a compact connected triangulated manifold $M$:

A fundamental domain for $M$ is a compact contractible (or, you can even require $P$ to be PL homeomorphic to a ball if you wish) polyhedron $P$ so that $M$ is obtained by gluing faces of $P$ via PL homeomorphisms.

It is easy to see why such a fundamental domain exists: Let $S$ be the dual graph to the triangulation of (connected) $M$ (vertices of $S$ are facets, i.e., top-dimensional faces, while the edges correspond to codimension 1 faces along which facets meet). Now, choose $T$, a maximal subtree in $S$. Hence, $T$ contains all the vertices of $S$. Then glue all the facets of the triangulation along the faces corresponding to the edges of $T$. The result is a fundamental domain in the above sense. (Here you see the connection to a existence of a triangulation which.)

There is a different (and, actually, more popular) school of thought concerning fundamental domains. Namely, fundamental domains are defined for properly discontinuous group actions on "nice" topological spaces $X$. If you have a manifold $M$ then the natural space $X$ in this context is its universal cover of $M$ and the group $G$ is the fundamental group $\pi_1(M)$ acting by covering transformations. Assuming, again, that everything is triangulated, you define a fundamental domain for the action of $G$ on $X$ as a polyhedron $P\subset X$ so that the following hold:

  1. The $G$-orbit of $P$ covers the entire $X$.

  2. $gP \cap P$ is contained in the boundary of $P$ for every $g\in G-1$.

(One has to add more conditions if a metric is present, but we do not need them here.)

Note that such $P$'s may not be contractible. For instance, if $M$ is simply-connected then, necessarily, $M=X=P$.

On the other hand, in the context of hyperbolic manifolds $M$ (and, more generally, manifolds of nonpositive sectional curvature) there is a nice and natural class of fundamental domains in $X$, called Dirichlet domains, which are contractible (since they are star-like). They depend on the choice of a base-point in $X$. You can read more on these in Ratcliffe's book.

In most cases, fundamental domains do not provide much insight into topology of $M$ (once you go beyond dimension $2$), so topologists prefer to describe topology of $M$ through other combinatorial means. For instance, in the context of 3-dimensional manifolds Heegaard splittings (and their generalizations) appear to be most useful. Surgery along links in the 3-sphere is another common way to go.

Addendum in the Riemannian setting: Everything works fine if your manifold is compact. But if it is noncompact, the above definition of a fundamental domain is unsatisfactory. The key reason is that it could happen (even in the hyperbolic case) that if you take the Riemannian metric on $P$ which is the restriction of the metric on $X$ and project it down to the quotient space $P/\sim$ (obtained by gluing faces of $P$) then the resulting metric is incomplete (even if the metric on $M$ is). That's not good! So you need some conditions to ensure that the metric on $P/\sim$ is isometric to the one on $M$. Look in Ratcliffe's book for (somewhat painful) details. The bottom line is: If your manifold is compact, you are in luck (good luck otherwise).

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This is exactly the type of response I was looking for! Would you be able to give a quick explanation of what the conditions would be when a metric is present? If some of my initial inclinations end up turning out to be actual research I would need to talk about manifolds with metrics. –  Samuel Reid Mar 9 '12 at 22:23
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