Transformation Group | Itinerarium Mentis

Basic Notions

Definition(Group action) . Let $G$ be a topological group and $X$ a topological space. A left action of $G$ on $X$ is a continuous map $$ \rho:G\times X\to X $$ s.t. $\rho(g,\rho(h,x)) = \rho(gh,x)$ for all $g,h\in G$ and $x\in X$, and $\rho(e,x)=x$ for $x\in X$ and $e$ the neutral element.

Group actions are used to study the symmetry of a space.

Definition($G$-space/transformation group) . A left $G$-space or a transformation group is a pair $(X,\rho)$ consisting of a space $X$ together with a left action $\rho$ of $G$ on $X$.

If $\rho(g,\rho(h,x)) = \rho (hg,x)$ the word 'left' should be replaced by 'right'. Notice that for $\rho$ a left action, $\sigma(g,x) = \rho(g^{-1},x)$ is a right action. It is helpful to write a right action as $\rho: X\times G\to X$ instead of $G\times X\to X$. In the following $\rho(g,x)$ will be denoted by $gx$ and $\rho(x,g)$ by $xg$.

The left translation $L_g: X\to X$ given by $x\mapsto gx$. This is a homeomorphism of $X$ with inverse $L_{g^{-1}}$. THere is a homomorphism $G\to \operatorname{Aut}(X): g\mapsto L_g$ from $G$ into the group of homeomorphisms (automorphisms of topological space) of $X$.

An action is effective if the kernel of $G\to \operatorname{Aut}(X)$ is trivial (the map is injective).
An action is trivial if the kernel is $G$ itself.
An action is free if $gx=x$ always implies $g=e$ (trivial stablizers aka. isotropy groups for all $x\in X$).
An action if transitive if $X$ consists of a single orbit, this means for all $x,y\in X$, there is $g\in G$ s.t. $y=gx$.
A subset $F\subset X$ of a $G$-space $X$ is called a fundamental domain of $X$ if the natural projection $F\subset X \to X/G$ is bijective, i.e. it contains exactly one point from each orbit.
Let $G_x$ denote the isotropy group of $X$ at $x$. Put $\operatorname{Iso}(X)$ the set of isotropy groups of $X$. From $G_{gx} = g G_x g^{-1}$ it follows that on a same orbit the isotropy groups are conjugate to each other and $\operatorname{Iso}(X)$ consists of complete conjugacy classes of isotropy groups.

Orbits and fixed points

The orbits and fixed points are primary objects of study.

Let $H\subset G$ be a subgroup.

Definition(Orbit bundle) . Let $X\_{H} = \\{ x\in X | G_x = H\\}$, i.e. the collection of points in $X$ of which the isotropy group is $H$. The $(H)$-orbit bundle $X_{(H)}$ of $X$, where $(H)$ is the conjugacy class of $H$, is defined as $$ X_{(H)} = \{x\in X | G_x \sim H\}. $$ where "$\sim$" is the conjugate relation.

The space $X$ is the disjoint union of its orbit bundles. Orbit bundles are $G$-subspaces, since for $x\in X_{(H)}$, $G_{gx} = g H g^{-1}$ and thus $gx \in X_{g H g^{-1}} \subset X_{(H)}$.

The orbit bundle $X_{\{e\}}$ is the largest subspace of $X$ where the action is free, whose complement $X\subset X_{\{e\}}$ is often called the singular set $X_s$ of $X$, it is the union of the fixed point sets $X^H$ with $H$ nontrivial.

Definition(Fixed point set) . The $H$-fixed point set of $X$ is the set $$ X^H = \{ x\in X| hx = x, \forall h\in H\}. $$

One has $X_H \subset X^H$ since the points in $X^H$ can have larger isotropy groups. The complement $X^H\setminus X_H$ is denoted by $$ X^{\gt H} = \bigcup_K X^K, \forall H\subset K, K\neq H $$

Categorical structure

$G$-spaces and $G$-(equivariant) maps form a category of $G$-spaces.

Definition(Equivariant maps) . A map $f:X\to Y$ of $G$-spaces is called a $G$-(equivariant) map if $\forall g\in G$ and $\forall x\in X$, $f(gx) = g f(x)$.

A $G$-map $f:X\to Y$ induces a map between orbit spaces $$ f/G : X/G \to Y/G: \operatorname{Orbit}_G (x) \mapsto \operatorname{Orbit}_G (f(x)). $$ This is continuous if $f$ is continuous.

For an equivariant map $f$, $G_x \subset G_{fx}$. If this is an equiality for all $x\in X$, then $f$ is called isovariant.

The category of $G$-spaces is complete. Let $\{X_j\}_{j\in J}$ be a family of $G$-spaces, on the topological product $\prod_{j\in J} X_j$, there is a $G$-action, called the diagonal action: $$ (g,\{x_j\}) \mapsto \{gx_j\} $$ which makes the topological product a categorical product.

Homotopy

Definition($G$-homotopic maps) . Two $G$-maps $f_0,f_1: X\to Y$ are $G$-homotopic if there is a continuous $G$-map, called a $G$-homotopy from $f_0$ to $f_1$, $$ F: X\times [0,1]\to Y $$ s.t. $F(x,0) = f_0(x), F(x,1)=f_1(x)$ where $[0,1]$ is equipped with the trivial $G$-action and $X\times [0,1]$ is equipped with the diagonal action.

Each map $f_t: x\mapsto F(x,t)$ is a $G$-map. Being $G$-homotopic is an equivalence relation, hence there is a category of $G$-spaces with morphisms $[X,Y]_G$, the $G$-homotopy classes of $G$-maps $X\to Y$.