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| {{distinguish|homotopy}}
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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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Revision as of 16:28, 2 March 2014
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