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.,
- [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ X^H }[/math] for the set of all x in X such that [math]\displaystyle{ hx = x }[/math]. For example, if we write [math]\displaystyle{ 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]\displaystyle{ (g \cdot f)(x) = g f(g^{-1} x) }[/math], [math]\displaystyle{ F(X, Y)^G }[/math] consists of f such that [math]\displaystyle{ f(g x) = g f(x) }[/math]; i.e., f is an equivariant map. We write [math]\displaystyle{ F_G(X, Y) = F(X, Y)^G }[/math]. Note, for example, for a G-space X and a closed subgroup H, [math]\displaystyle{ F_G(G/H, X) = X^H }[/math].
References
- Greenlees, John; May, Peter (1995). "8. Equivariant stable homotopy theory". in James, I.M.. Handbook of algebraic topology. Elsevier. pp. 277–323. ISBN 978-0-08-053298-1. http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf.
See also
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