Hypercycle (geometry) - Revision history

en>Tomruen at 22:38, 22 November 2014

2014-11-22T22:38:31Z

← Older revision		Revision as of 00:38, 23 November 2014
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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~~.<br><br>My ~~weblog~~ ... [http://~~www~~.~~animecontent~~.~~com/blog~~/~~349119 at home std testing~~]		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/]

en>ArnoldReinhold: /* Construction */ add caption

2014-02-18T13:04:50Z

Construction: add caption

← Older revision		Revision as of 15:04, 18 February 2014
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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>~~
	~~is a continuous map~~. ~~Together with the group action, ''X''~~ is ~~called a '''~~'~~'G''-space'''.~~

	~~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.)~~

	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>,~~
	~~<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>~~.

	~~== References ==~~
	*John Greenlees, Peter May, ''[http://www.~~math.uchicago~~.~~edu/~may~~/~~PAPERS~~/~~Newthird.pdf Equivariant stable homotopy theory]''~~

	~~== See also ==~~
	*[[Lie group action]]




	~~{{topology-stub}}~~

	~~[[Category:Group actions]]~~
	~~[[Category:Topological groups]~~]

en>John of Reading: Typo/general fixing, replaced: the have → they have using AWB

2013-12-04T12:15:50Z

Typo/general fixing, replaced: the have → they have using AWB

← Older revision		Revision as of 14:15, 4 December 2013
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	~~Nice~~ to ~~meet you~~, ~~I am Marvella Shryock~~. ~~My day occupation~~ is a ~~librarian~~. ~~His family lives in South Dakota but his spouse desires them~~ to ~~transfer~~. ~~What I adore doing~~ is to ~~gather badges but I~~'~~ve been~~ taking ~~on new things recently~~.<br><br>~~Review my site [http:~~//~~www~~.~~buzzbit~~.~~net~~/~~blog~~/~~288042~~ http://www.~~buzzbit~~.~~net~~]		In [[topology]], a '''continuous group action''' on a topological space ''X'' is a [[group action]] of a group ''G'' that is continuous: i.e.,
			:<math>G \times X \to X, \quad (g, x) \mapsto g \cdot x</math>
			is a continuous map. Together with the group action, ''X'' is called a '''''G''-space'''.

			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.)

			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>,
			<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>.

			== References ==
			*John Greenlees, Peter May, ''[http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf Equivariant stable homotopy theory]''

			== See also ==
			*[[Lie group action]]




			{{topology-stub}}

			[[Category:Group actions]]
			[[Category:Topological groups]]

en>Tkuvho at 15:27, 14 February 2012

2012-02-14T15:27:48Z

New page

Nice to meet you, I am Marvella Shryock. My day occupation is a librarian. His family lives in South Dakota but his spouse desires them to transfer. What I adore doing is to gather badges but I've been taking on new things recently.<br><br>Review my site [http://www.buzzbit.net/blog/288042 http://www.buzzbit.net]