Definition of quotient of a topological space by a group action

Question

I was going through the following lecture note on topology as I was trying to understand quotient topology .

http://homepage.math.uiowa.edu/~jsimon/COURSES/M132Fall07/M132Fall07_QuotientSpaces.pdf

Here the example 0.6 ends up with the following line :

When we have a group $G$ acting on a space $X$, there is a "natural" quotient space: for each $x \in X$, let $Gx = \{g x \mid g \in G\}$; view each of these “orbit” sets as a single point in some new space $\hat X$.

First of all I would like to know if the adjective "natural" has got some mathematical meaning as in are there any mathematical objects with name "natural group"?

Now when it says a group acting on a space, I guess it means a group is isomorphic to the group of homeomorphisms of the space $X$ (correct me if I am wrong here). As such I am confused about what does the set $Gx = \{g x \mid g \in G\}$ means, because $G$ is a different group and I could have understood if it were referring to the group comprised of the homeomorphisms of space $X$.

Answer 1

The word natural here is not a precise mathematical term. You can ignore the word.

Given a group $G$ and a set $X$, an action of the group on the set is a function $$ \phi: G\times X \to X. $$ such that ($e$ identity in $G$) for all $g,h\in G$ $$ \phi(e,x) = x\\ \phi(gh,x) = \phi(g, \phi(h,x)). $$ Instead of writing $\phi(g,x)$ we often write $g.x$ or simply $gx$. A way to think about this is that each element $g$ in $G$ corresponds/give you a map $X\to X$. This is a map between sets and so it doesn't really make sense to talk about this being a (group) homomorphism.

The orbit of an element $x$ in $X$ is the subset of $X$ you get by "applying" all elements of $G$. That is, the orbit of $x$ is $$ \{gx: g\in G\}. $$ This is just a subset of $X$.

Answer 2

Let us consider an example: Take the euclidean space $X=\mathbb{R}^2$ and the additive group of real numbers as $G$. We consider the homomorphism $i:\mathbb{R}\rightarrow\text{Hom}(X)$, $$i(\alpha):\mathbb{R}^2\rightarrow\mathbb{R}^2,\ i(\alpha)\begin{pmatrix}x\\y\end{pmatrix} \mapsto \begin{pmatrix}\cos(\alpha)&-\sin(\alpha)\\ \sin(\alpha)&\cos(\alpha)\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} $$ that assigns to a real number $\alpha$ the rotation by $\alpha$. So in this case the group $G$ is not isomorphic to the group of homomorphsims of $X$ but to the subgroup of rotations around $(0,0)^T$. Now we can define a map $$\varphi:\mathbb{R}\times \mathbb{R}^2\rightarrow \mathbb{R}^2,\ \left(\alpha,\begin{pmatrix}x\\y\end{pmatrix}\right)\mapsto i(\alpha)\begin{pmatrix}x\\y\end{pmatrix}.$$ The orbit of $(x,y)^T\in\mathbb{R}^2$ is then defined by $$\varphi\left(\mathbb{R},\begin{pmatrix}x\\y\end{pmatrix}\right) =\left\{ \begin{pmatrix}\cos(\alpha)&-\sin(\alpha)\\ \sin(\alpha)&\cos(\alpha)\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix}\ |\ \alpha\in\mathbb{R} \right\}. $$ So in this example the orbit of $(x,y)^T$ is just the circle around $(0,0)^T$ that contains $(x,y)^T$. If the map $i$ is clear from context, it is normally suppressed. So you simply write $\alpha(x,y)^T$ instead of $i(\alpha)(x,y)^T$ and $\mathbb{R}(x,y)^T$ for the orbit. Note that in the definition of a group action the map $i$ is also allowed to be not injective.

Answer 3

"Natural" does have a technical meaning in mathematics from category theory, in particular, natural transformations, as Eclipse Sun mentions. That's not really how it's being used here. Another time "natural" is often used is for constructions defined by universal properties as such constructions are unique (up to unique isomorphism). (This latter notion does crucially rely on natural transformations.)

There are two ways of viewing what's happening as a universal construction.

First, a more abstract algebra approach. We can define, much as those notes do, an action of a group $G$ on a set $X$ as a group homomorphism from $G$ into the set of automorphisms of $X$, $\text{Aut}(X)$. Note that this picks out a subgroup of the automophisms. Now, $\text{Aut}(X) \subseteq X^X$ the latter being the set of endomorphisms on $X$. This means a group action $\varphi : G \to \text{Aut}(X)$ gives rise to a function $\hat{\varphi} : G\times X \to X$ which will satisfy the equations given in Thomas' answer. Now we can consider the coequalizer between $\hat{\varphi}$ and $\pi_2$, the projection onto the second component. The coequalizer is defined by a universal property. Concretely, this says to quotient $X$ by the equivalence relation $x \sim y \iff \exists g\in G. y = \varphi(g)(x)$. (You should check that this does indeed define an equivalence relation. You should also verify that the equivalence classes are the right thing, i.e. the orbits.) The set of continuous automorphisms is the set of homeomorphisms.

A second, more categorical perspective, is to view $G$ as a category with one object and an arrow for each element of $G$. Composition is multiplication in the group. Now a group action is a functor $G \to \mathbf{Set}$, but this can be generalized to any category instead of $\mathbf{Set}$, in particular, $\mathbf{Top}$, the category of topological spaces and continuous functions. The "natural quotient" in this perspective is just what we get when we take the colimit of this functor, viewed as a diagram. A colimit is another construction defined by a universal property. I would say both of these constructions are very natural in the colloquial sense of the word as well.

Answer 4

Regarding just your last paragraph about group actions, your statement that the group "is isomorphic to the group of homeomorphisms" is in the ballpark but not correct.

An action of a group $G$ on a mathematical object $X$ can be thought of in two equivalent ways (the proof equivalence is easy):

1. A function $G \times X \to X$ satisying some properties: $(gh) \cdot x = g \cdot (h \cdot x)$; and $\text{Id} \cdot x = x$; and for each $g$ the map $x \mapsto g \cdot x$ preserves the mathematical structures on $X$.
2. A homomorphism $G \to \text{Iso}(X)$ where $\text{Iso}(X)$ is the group of bijections of $X$ that preserve the mathematical strucure on $X$. There is no requirement that this homomorphism be one-to-one nor onto, in particular no requirement that it be a group isomorphism.

Usually one can formalize the notion of "preserving mathematical structure" using category theory, and $\text{Iso}(X)$ is the group of all bijections of $X$ which are morphisms of the category and whose inverses are also morphisms.

For instance, if $X$ is a topological space then $\text{Iso}(X)$ is the group of bijective maps which are continuous and whose inverses are also continuous --- in other words, the group $\text{Homeo}(X)$ of self-homeomorphisms of $X$ --- and so an action of a group $G$ on $X$ is just a homomorphism $G \mapsto \text{Homeo}(X)$. Again, this homomorphism need not be one-to-one nor onto, nor in particular an isomorphism.