In topology, a continuous group action is a group action adapted to the topological setting: G {\displaystyle G} is a topological group, X {\displaystyle X} is a topological space, and the action map is continuous.
Definition Let
Φ : G × X → X , ( g , x ) ↦ g ⋅ x {\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 Φ {\displaystyle \Phi } is continuous (with respect to the product topology on G × X {\displaystyle G\times X} ). A topological space endowed with a continuous group action is also called a G {\displaystyle G} -space.
Properties A continuous action is automatically an action by homeomorphisms, i.e. the bijections Φ g = Φ ( g , ⋅ ) : X → X {\displaystyle \Phi _{g}=\Phi (g,\cdot ):X\to X} are homeomorphisms for every g ∈ G {\displaystyle g\in G} , but the converse is not necessarily true. However, if G {\displaystyle G} is discrete, these two concepts coincide. Given a G {\displaystyle G} -space, the quotient map X → X / G {\displaystyle X\to X/G} is an open map (with respect to the quotient topology on X / G {\displaystyle X/G} ).
Constructions If f : H → G {\displaystyle f:H\to G} is a continuous group homomorphism of topological groups, and if X {\displaystyle X} is a G {\displaystyle G} -space, then H {\displaystyle H} acts on X {\displaystyle X} by restriction: h ⋅ x = f ( h ) x {\displaystyle h\cdot x=f(h)x} , making X {\displaystyle X} a H {\displaystyle H} -space. Often f {\displaystyle f} is either an inclusion or a quotient map. In particular, any topological space may be thought of as a G {\displaystyle G} -space via G → 1 {\displaystyle G\to 1} (and G {\displaystyle G} would act trivially). Two basic operations are that of taking the space of points fixed by a subgroup H {\displaystyle H} and that of forming a quotient by H {\displaystyle H} . We write X H {\displaystyle X^{H}} for the set of all x {\displaystyle x} in X {\displaystyle X} such that h x = x {\displaystyle hx=x} . For example, if we write F ( X , Y ) {\displaystyle F(X,Y)} for the set of continuous maps from a G {\displaystyle G} -space X {\displaystyle X} to another G {\displaystyle G} -space Y {\displaystyle Y} , then, with the action ( g ⋅ f ) ( x ) = g f ( g − 1 x ) {\displaystyle (g\cdot f)(x)=gf(g^{-1}x)} ,
F ( X , Y ) G {\displaystyle F(X,Y)^{G}} consists of f {\displaystyle f} such that f ( g x ) = g f ( x ) {\displaystyle f(gx)=gf(x)} ; i.e., f {\displaystyle f} is an equivariant map. We write F G ( X , Y ) = F ( X , Y ) G {\displaystyle F_{G}(X,Y)=F(X,Y)^{G}} . Note, for example, for a G {\displaystyle G} -space X {\displaystyle X} and a closed subgroup H {\displaystyle H} , F G ( G / H , X ) = X H {\displaystyle F_{G}(G/H,X)=X^{H}} .
References 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.
See also Lie group action