topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

, a continuous group action on a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...

''X'' is a

group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

of a

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...

''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 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 \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 In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...

. 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

See also