Continuous group action explained

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 x X \to X, (g, x) \mapsto g ⋅ 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 ⋅ 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 as 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 ⋅ 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

• Book: Greenlees, John . Peter . May . http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf . 8. Equivariant stable homotopy theory . I.M. . James . Handbook of algebraic topology . 1995 . 277–323 . Elsevier . 978-0-08-053298-1.

See also

• Lie group action