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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.,
| | Nice to meet you, my title is Refugia. My family life in Minnesota and my family members enjoys it. My working day occupation is a librarian. One of the issues he loves most is ice skating but he is having difficulties to find time for it.<br><br>My blog [http://reinodoprazer.com.br/index.php?do=/blog/326/tips-about-how-to-overcome-candidiasis-easily/ http://reinodoprazer.com.br/] |
| :<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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Nice to meet you, my title is Refugia. My family life in Minnesota and my family members enjoys it. My working day occupation is a librarian. One of the issues he loves most is ice skating but he is having difficulties to find time for it.
My blog http://reinodoprazer.com.br/