Intopology, acontinuous group action on atopological spaceX is agroup action of atopological groupG that is continuous: i.e.,

${\displaystyle G\times X\to X,(g,x)\mapsto g\cdot x}$

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

If${\displaystyle f:H\to G}$ is a continuous group homomorphism of topological groups and ifX is aG-space, thenH can act onXby restriction:${\displaystyle h\cdot x=f(h)x}$, makingX aH-space. Oftenf is either an inclusion or a quotient map. In particular, any topological space may be thought of as aG-space via${\displaystyle G\to 1}$ (andG would act trivially.)

Two basic operations are that of taking the space of points fixed by a subgroupH and that of forming a quotient byH. We write${\displaystyle X^H}$ for the set of allx inX such that${\displaystyle hx=x}$. For example, if we write${\displaystyle F(X,Y)}$ for the set of continuous maps from aG-spaceX to anotherG-spaceY, then, with the action${\displaystyle (g\cdot f)(x)=gf(g^{-1}x)}$,${\displaystyle F(X,Y{)}^G}$ consists off such that${\displaystyle f(gx)=gf(x)}$; i.e.,f is anequivariant map. We write${\displaystyle F_G(X,Y)=F(X,Y{)}^G}$. Note, for example, for aG-spaceX and a closed subgroupH,${\displaystyle F_G(G/H,X)=X^H}$.

• Greenlees, John; May, Peter (1995)."8. Equivariant stable homotopy theory"(PDF). In James, I.M. (ed.).Handbook of algebraic topology. Elsevier. pp. 277–323.ISBN 978-0-08-053298-1.

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