Not really blogging

A. Makelov

Fundamental domains

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 {X} and group {G} acting on {X}, to establish sufficient conditions for the quotient {D/G} of a fundamental domain {D} (to be defined below) by the induced action of {G} to be homeomorphic to {X/G}.

1. A motivating example

Consider the space {\mathbb{R}^2} in the standard topology, and the (discrete) group {\mathbb{Z}^2} acting by translation: {(m,n)\in\mathbb{Z}^2} acts on {(x,y)\in\mathbb{R}^2} by sending it to {(x+m, y+n)}. Suppose you want to get an understanding of the quotient space {\mathbb{R}^2/\mathbb{Z}^2} with respect to this action. Basically, you want to take the real plane and, for each point, glue to it its orbit under {\mathbb{Z}^2}. Working with {\mathbb{R}^2} can make your head hurt, so here’s a trick: consider the unit square {D=[0,1]^2}. It almost contains a single representative of each orbit, except on the boundary. This gives us the much simpler gluing instructions

torus

The resulting space is the torus. The subspace {D} is known as a fundamental domain for the action of {\mathbb{Z}^2} on {\mathbb{R}^2}.

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:

Definition 1 A topological group {G} is a space and group {G} such that

  1. The composition map {G\times G\to G} is continuous.
  2. The inverse map {G\to G} is continuous.

Definition 2 A topological group {G} acts on a space {X} if {G} acts on the set {X} and the corresponding action {G\times X\to X} is continuous.

To get some practice with these concepts, do these exercises:

Exercise 1 If {f:X\times Y\to Z} is a continuous map, show that for every {x\in X}, the map {f(x,\cdot):Y\to Z} is continuous.

Exercise 2 If {G} acts on {X}, show that for any {g\in G}, the map {f_g:X\to X} given by {x\to gx} is a homeomorphism.

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):

Definition 3 A fundamental domain is a closed subset {D\subset X} such that {X} is the union of translates of {D} under the group action:

\displaystyle \begin{array}{rcl} X= \displaystyle\bigcup_{g\in G}gD \end{array}

and such that {\mathop{Int}(gD\cap g'D)=\emptyset} for any two distinct translates.

What we ideally want to get is {D/G\cong X/G}. 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:

Proposition 4 If :

  1. {X/G} is Hausdorff, and
  2. {D} is compact, or {D/G} is compact,

then {D/G\cong X/G}.

Proof: Observe that if we let {q,p} be the canonical quotient maps from {D,X} to {D/G,X/G} respectively, and {i} be the canonical inclusion {D\to X}, we have that whenever {a,b} are in the same {G}-orbit in {D}, they get mapped to the same element in {X/G}. Consequently, we have the following diagram:diagram

where {f} is continuous and unique by the universal property of the quotient. What does {f} do? Suppose we have an element {a} of a {G} orbit in {D}. Under {p\circ i}, {a} gets mapped to a representative of its {G}-orbit in {X/G}; but {p\circ i = f\circ q}. Hence {f} maps a representative of {a}‘s {G}-orbit in {D/G} to a representative of {a}‘s {G}-orbit in {X/G} (which is exactly the natural map we would expect to get).

Observe that {f} is injective: if {[a],[b]\in D/G} are such that {f([a])=f([b])}, it follows that there are some {a,b\in D} which go to the same element in {X/G}, and thus {a\sim b}, so {[a]=[b]}.

Moreover, {f} is surjective: observe that {p\circ i} is surjective, since {D} contains a representative of each orbit, and thus {f\circ q} is also surjective, which cannot happen if {f} fails to be surjective.

Thus, {f} is a continuous bijection {D/G\to X/G}. Since {D} being compact implies that {D/G} is also compact (being a continuous image under the quotient), and since {X/G} is Hausdorff, {f} is a homeomorphism. \Box

Going back to our example, obviously {[0,1]^2} is compact, and it’s easy to check that {\mathbb{R}^2/\mathbb{Z}^2} is Hausdorff.

Proposition 5 In terms of the notation of the previous proposition, {D/G\cong X/G} if and only if {p\circ i} is a quotient map.

Proof: Observe that {p\circ i} is a surjective continuous map {D\to X/G}, and that {D/G} is the quotient of {D} by the equivalence relation with equivalence classes – the subsets {\{(p\circ i)^{-1}(\{z\}) \ \big| \ z\in X/G\}}. Thus, Corollary 22.3 from Munkres applies to tell us that there is a homeomorphism {D/G\to X/G} \Box

This would hold, for example, if {p} is a closed map, since {i} is a closed map by {D} being closed in {X}, compositions of closed maps are closed, and every closed map is a quotient map.

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