Английская Википедия:Continuous group action

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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.,

<math>G \times X \to X, \quad (g, x) \mapsto g \cdot x</math>

is a continuous map. Together with the group action, X is called a G-space.

If <math>f: H \to G</math> is a continuous group homomorphism of topological groups and if X is a G-space, then H can act on X by restriction: <math>h \cdot x = f(h) x</math>, 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 <math>G \to 1</math> (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 <math>X^H</math> for the set of all x in X such that <math>hx = x</math>. For example, if we write <math>F(X, Y)</math> for the set of continuous maps from a G-space X to another G-space Y, then, with the action <math>(g \cdot f)(x) = g f(g^{-1} x)</math>, <math>F(X, Y)^G</math> consists of f such that <math>f(g x) = g f(x)</math>; i.e., f is an equivariant map. We write <math>F_G(X, Y) = F(X, Y)^G</math>. Note, for example, for a G-space X and a closed subgroup H, <math>F_G(G/H, X) = X^H</math>.

References

• Шаблон:Cite book

See also

• Lie group action

Шаблон:Topology-stub