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continuous group action : ウィキペディア英語版
continuous group action
In topology, a continuous group action on a topological space ''X'' is a group action of a topological group ''G'' that is continuous: i.e.,
:G \times X \to X, \quad (g, x) \mapsto g \cdot x
is a continuous map. Together with the group action, ''X'' is called a ''G''-space.
If f: H \to G is a continuous group homomorphism of topological groups and if ''X'' is a ''G''-space, then ''H'' can act 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 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) = g f(g^ x),
F(X, Y)^G consists of ''f'' such that f(g x) = g f(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 ==

*John Greenlees, Peter May, ''(Equivariant stable homotopy theory )''

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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