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| In [[mathematics]], the '''space of loops''' or '''(free) loop space''' of a [[topological space]] ''X'' is the space of maps from the [[unit circle]] ''S''<sup>1</sup> to ''X'' together with the [[compact-open topology]].
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| :<math>\Omega X = \mathcal{C}(S^1, X).</math>
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| That is, a particular [[function space]].
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| In [[homotopy theory]] ''loop space'' commonly refers to the same construction applied to [[pointed space]]s, i.e. continuous maps respecting [[base point]]s. In this setting there is a natural "concatenation operation" by which two elements of the loop space can be combined. With this operation, the loop space can be regarded as a [[magma (algebra)|magma]], or even as an [[A-infinity operad|''A''<sub>∞</sub>-space]]. Concatenation of loops is not strictly associative, but it is associative up to higher [[homotopy|homotopies]].
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| If we consider the [[quotient set|quotient]] of the based loop space Ω''X'' with respect to the equivalence relation of pointed homotopy, then we obtain a [[group (mathematics)|group]], the well-known [[fundamental group]] ''π''<sub>1</sub>(''X'').
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| The '''iterated loop spaces''' of ''X'' are formed by applying Ω a number of times.
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| The free loop space construction is [[right adjoint]] to the [[cartesian product]] with the circle, and the version for pointed spaces to the [[reduced suspension]]. This accounts for much of the importance of loop spaces in [[stable homotopy theory]].
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| == Relation between homotopy groups of a space and those of its loop space ==
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| The basic relation between the [[homotopy group]]s is <math>\pi_k(X) \approxeq \pi_{k-1}(\Omega X)</math>.<ref>http://topospaces.subwiki.org/wiki/Loop_space_of_a_based_topological_space</ref>
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| More generally,
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| :<math>[\Sigma Z,X] \approxeq [Z, \Omega X]</math>
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| where, <math>[A,B]</math> is the set of homotopy classes of maps <math>A \rightarrow B</math>,
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| and <math>\Sigma A</math> is the [[Suspension (topology)|suspension]] of A.
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| In general <math>[A, B]</math> does not have a group structure for arbitrary spaces <math>A</math> and <math>B</math>. However, it can be shown that <math>[\Sigma Z,X]</math> and <math>[Z, \Omega X]</math> do have natural group structures when <math>Z</math> and <math>X</math> are [[Pointed space|pointed]], and the aforesaid isomorphism is of those groups.
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| <ref name="may">{{citation |last= May |first=J. P. |authorlink=J. Peter May|title=A Concise Course in Algebraic Topology |year=1999 |publisher=U. Chicago Press, Chicago |url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |accessdate=2008-09-27 |chapter=8}} (chapter 8, section 2)</ref>
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| Note that setting <math>Z = S^{k-1}</math> (the <math>k-1</math> sphere) gives the earlier result.
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| ==See also==
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| *[[fundamental group]]
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| *[[path (topology)]]
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| *[[loop group]]
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| *[[free loop]]
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| *[[quasigroup]]
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| ==References==
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| {{Reflist}}
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| *{{Citation | last1=Adams | first1=John Frank |authorlink=Frank Adams| title=Infinite loop spaces | url=http://books.google.com/books?id=e2rYkg9lGnsC | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | isbn=978-0-691-08207-3; 978-0-691-08206-6 | mr=505692 | year=1978 | volume=90}}
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| *{{Citation | last1=May | first1=J. Peter | author1-link=J. Peter May | title=The Geometry of Iterated Loop Spaces | url=http://www.math.uchicago.edu/~may/BOOKSMaster.html | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-05904-2 | doi=10.1007/BFb0067491 | mr=0420610 | year=1972}}
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| [[Category:Topology]]
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| [[Category:Homotopy theory]]
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| [[Category:Topological spaces]]
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