Continuous group action

In topology, a continuous group action is a group action adapted to the topological setting: $G$ is a topological group, $X$ is a topological space, and the action map is continuous.

Definition

Let

$\mathrm{\Phi }:G\times X\to X,(g,x)\mapsto g\cdot x$

be a (left) group action of a topological group on a topological space; it is called a continuous group action if the map $\mathrm{\Phi }$ is continuous (with respect to the product topology on $G\times X$). A topological space endowed with a continuous group action is also called a $G$-space.

Properties

A continuous action is automatically an action by homeomorphisms, i.e. the bijections ${\mathrm{\Phi }}_g=\mathrm{\Phi }(g,\cdot ):X\to X$ are homeomorphisms for every $g\in G$, but the converse is not necessarily true. However, if $G$ is discrete, these two concepts coincide.

Given a $G$-space, the quotient map $X\to X/G$ is an open map (with respect to the quotient topology on $X/G$).

Constructions

If $f:H\to G$ is a continuous group homomorphism of topological groups, and if $X$ is a $G$-space, then $H$ acts on $X$ by restriction: $h\cdot x=f(h)x$, making $X$ a $H$-space. Often $f$ is either an inclusion or a quotient map. In particular, any topological space may be thought of as a $G$-space via $G\to 1$ (and $G$ would act trivially).

Two basic operations are that of taking the space of points fixed by a subgroup $H$ and that of forming a quotient by $H$. We write $X^H$ for the set of all $x$ in $X$ such that $hx=x$. For example, if we write $F(X,Y)$ for the set of continuous maps from a $G$-space $X$ to another $G$-space $Y$, then, with the action $(g\cdot f)(x)=gf(g^{-1}x)$, $F(X,Y{)}^G$ consists of $f$ such that $f(gx)=gf(x)$; i.e., $f$ is an equivariant map. We write $F_G(X,Y)=F(X,Y{)}^G$. Note, for example, for a $G$-space $X$ and a closed subgroup $H$, $F_G(G/H,X)=X^H$.

References

• Template:Cite book

See also

• Lie group action