Continuous group action

Let

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

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

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

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

• Greenlees, John; May, Peter (1995). "8. Equivariant stable homotopy theory" (PDF). In James, I.M. (ed.). Handbook of algebraic topology. Elsevier. pp. 277–323. ISBN 978-0-08-053298-1.

• Lie group action

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