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| In [[topology]], a '''continuous group action''' on a topological space ''X'' is a [[group action]] of a group ''G'' that is continuous: i.e.,
| | I am Oscar and I completely dig that name. California is where I've usually been residing and I adore each working day living right here. His spouse doesn't like it the way he does but what he truly likes performing is to do aerobics and he's been doing it for quite a while. Hiring is his profession.<br><br>My weblog ... [http://www.animecontent.com/blog/349119 at home std testing] |
| :<math>G \times X \to X, \quad (g, x) \mapsto g \cdot x</math>
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| is a continuous map. Together with the group action, ''X'' is called a '''''G''-space'''.
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| If <math>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>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 a ''G''-space via <math>G \to 1</math> (and ''G'' would act trivially.)
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| 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>X^H</math> for the set of all ''x'' in ''X'' such that <math>hx = x</math>. For example, if we write <math>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>(g \cdot f)(x) = g f(g^{-1} x)</math>,
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| <math>F(X, Y)^G</math> consists of ''f'' such that <math>f(g x) = g f(x)</math>; i.e., ''f'' is an [[equivariant map]]. We write <math>F_G(X, Y) = F(X, Y)^G</math>. Note, for example, for a ''G''-space ''X'' and a closed subgroup ''H'', <math>F_G(G/H, X) = X^H</math>.
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| == References ==
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| *John Greenlees, Peter May, ''[http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf Equivariant stable homotopy theory]''
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| == See also ==
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| *[[Lie group action]]
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| {{topology-stub}}
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| [[Category:Group actions]]
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| [[Category:Topological groups]]
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I am Oscar and I completely dig that name. California is where I've usually been residing and I adore each working day living right here. His spouse doesn't like it the way he does but what he truly likes performing is to do aerobics and he's been doing it for quite a while. Hiring is his profession.
My weblog ... at home std testing