Continuous group action

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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^{-1} 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[edit]

See also[edit]