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| {{distinguish|homotopy}}
| | Andrew Simcox is the name his parents gave him and he totally enjoys this title. What me and my family members adore is to climb but I'm thinking on beginning something new. Distributing production has been his profession for some time. Some time ago she chose to live in Alaska and her mothers and fathers live close by.<br><br>My blog - real psychic readings ([http://brazil.amor-amore.com/irboothe brazil.amor-amore.com]) |
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| In [[algebraic topology]], an area of [[mathematics]], a '''homeotopy group''' of a [[topological space]] is a [[homotopy group]] of the group of [[homeomorphism|self-homeomorphism]]s of that space.
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| ==Definition==
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| The [[homotopy group]] [[functor]]s <math>\pi_k</math> assign to each [[path-connected]] topological space <math>X</math> the group <math>\pi_k(X)</math> of [[homotopy class]]es of continuous maps <math>S^k\to X.</math>
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| Another construction on a space <math>X</math> is the [[Homeomorphism group|group of all self-homeomorphisms]] <math>X \to X</math>, denoted <math>{\rm Homeo}(X).</math> If ''X'' is a [[locally compact]], [[locally connected]] [[Hausdorff space]] then a fundamental result of [[R. Arens]] says that <math>{\rm Homeo}(X)</math> will in fact be a [[topological group]] under the [[compact-open topology]].
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| Under the above assumptions, the '''homeotopy''' groups for <math>X</math> are defined to be:
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| :<math>HME_k(X)=\pi_k({\rm Homeo}(X)).</math>
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| Thus <math>HME_0(X)=\pi_0({\rm Homeo}(X))=MCG^*(X)</math> is the '''extended''' [[mapping class group]] for <math>X.</math> In other words, the extended mapping class group is the set of connected components of <math>{\rm Homeo}(X)</math> as specified by the functor <math>\pi_0.</math>
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| ==Example==
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| According to the [[Dehn-Nielsen theorem]], if <math>X</math> is a closed surface then <math>HME_0(X)={\rm Out}(\pi_1(X)),</math> the [[outer automorphism group]] of its [[fundamental group]].
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| ==References==
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| *G.S. McCarty. ''Homeotopy groups''. Trans. A.M.S. 106(1963)293-304.
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| *R. Arens, ''Topologies for homeomorphism groups'', Amer. J. Math. 68 (1946), 593–610.
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| [[Category:Algebraic topology]]
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| [[Category:Homeomorphisms]]
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| {{topology-stub}}
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Latest revision as of 07:46, 1 January 2015
Andrew Simcox is the name his parents gave him and he totally enjoys this title. What me and my family members adore is to climb but I'm thinking on beginning something new. Distributing production has been his profession for some time. Some time ago she chose to live in Alaska and her mothers and fathers live close by.
My blog - real psychic readings (brazil.amor-amore.com)