January 8, 2014Posted by on
This post is about the area of math known as general topology, and I’ll thus assume some basic background (for example, chapter 2 of the textbook by Munkres should be enough). The post was produces with a little modified version of Luca Trevisan’s latex2wp, and the source .tex can be found here. There is a bit of a hope that next time I’ll be able to make the math look better.
What made me write this was the Internet’s apparent lack of a simple but rigorous introduction to fundamental domains and their usefulness in taking quotients by group actions. The goal is, for a space and group acting on , to establish sufficient conditions for the quotient of a fundamental domain (to be defined below) by the induced action of to be homeomorphic to .
1. A motivating example
Consider the space in the standard topology, and the (discrete) group acting by translation: acts on by sending it to . Suppose you want to get an understanding of the quotient space with respect to this action. Basically, you want to take the real plane and, for each point, glue to it its orbit under . Working with can make your head hurt, so here’s a trick: consider the unit square . It almost contains a single representative of each orbit, except on the boundary. This gives us the much simpler gluing instructions
The resulting space is the torus. The subspace is known as a fundamental domain for the action of on .
2. Group actions on topological spaces: a crash course
In topology, one often considers quotients of topological spaces by group actions (for example, in the beautiful theory of covering spaces, which you don’t really need to know to get this post). This is a natural extension of group actions on sets which takes into account the continuity of the topological spaces:
- The composition map is continuous.
- The inverse map is continuous.
To get some practice with these concepts, do these exercises:
3. Fundamental domains
One way to define a fundamental domain formally is the following (though we won’t use the full strength of this definition):
and such that for any two distinct translates.
What we ideally want to get is . I know of no result stating that, and maybe there’s a counterexample. The following two propositions detail what I do know in terms of sufficient conditions:
- is Hausdorff, and
- is compact, or is compact,
Proof: Observe that if we let be the canonical quotient maps from to respectively, and be the canonical inclusion , we have that whenever are in the same -orbit in , they get mapped to the same element in . Consequently, we have the following diagram:
where is continuous and unique by the universal property of the quotient. What does do? Suppose we have an element of a orbit in . Under , gets mapped to a representative of its -orbit in ; but . Hence maps a representative of ‘s -orbit in to a representative of ‘s -orbit in (which is exactly the natural map we would expect to get).
Observe that is injective: if are such that , it follows that there are some which go to the same element in , and thus , so .
Moreover, is surjective: observe that is surjective, since contains a representative of each orbit, and thus is also surjective, which cannot happen if fails to be surjective.
Thus, is a continuous bijection . Since being compact implies that is also compact (being a continuous image under the quotient), and since is Hausdorff, is a homeomorphism.
Going back to our example, obviously is compact, and it’s easy to check that is Hausdorff.
Proof: Observe that is a surjective continuous map , and that is the quotient of by the equivalence relation with equivalence classes – the subsets . Thus, Corollary 22.3 from Munkres applies to tell us that there is a homeomorphism
This would hold, for example, if is a closed map, since is a closed map by being closed in , compositions of closed maps are closed, and every closed map is a quotient map.