# Loop space > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Loop_space > Markdown URL: https://mediated.wiki/source/Loop_space.md > Source: https://en.wikipedia.org/wiki/Loop_space > Source revision: 1339183141 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Topological space In [topology](/source/Topology), a branch of [mathematics](/source/Mathematics), the **loop space** Ω*X* of a [pointed](/source/Pointed_space) [topological space](/source/Topological_space) *X* is the space of (based) loops in *X*, i.e. [continuous](/source/Continuous_function_(topology)) pointed maps from the pointed [circle](/source/Circle) *S*1 to *X*, equipped with the [compact-open topology](/source/Compact-open_topology). Two loops can be multiplied by [concatenation](/source/Path_(topology)#Path_composition). With this operation, the loop space is an [*A*∞-space](/source/A-infinity_operad). That is, the multiplication is [homotopy-coherently](/source/Homotopy) [associative](/source/Associative_property). The [set](/source/Set_(mathematics)) of [path components](/source/Path_component) of Ω*X*, i.e. the set of based-homotopy [equivalence classes](/source/Equivalence_class) of based loops in *X*, is a [group](/source/Group_(mathematics)), the [fundamental group](/source/Fundamental_group) *π*1(*X*). The **iterated loop spaces** of *X* are formed by applying Ω a number of times. There is an analogous construction for topological spaces without basepoint. The **free loop space** of a topological space *X* is the space of maps from the circle *S*1 to *X* with the compact-open topology. The free loop space of *X* is often denoted by L X {\displaystyle {\mathcal {L}}X} . As a [functor](/source/Functor), the free loop space construction is [right adjoint](/source/Right_adjoint) to [cartesian product](/source/Cartesian_product) with the circle, while the loop space construction is right adjoint to the [reduced suspension](/source/Reduced_suspension). This adjunction accounts for much of the importance of loop spaces in [stable homotopy theory](/source/Stable_homotopy_theory). (A related phenomenon in [computer science](/source/Computer_science) is [currying](/source/Currying), where the cartesian product is adjoint to the [hom functor](/source/Hom_functor).) Informally this is referred to as [Eckmann–Hilton duality](/source/Eckmann%E2%80%93Hilton_duality). ## Eckmann–Hilton duality The loop space is dual to the [suspension](/source/Suspension_(topology)) of the same space; this duality is sometimes called [Eckmann–Hilton duality](/source/Eckmann%E2%80%93Hilton_duality). The basic observation is that - [ Σ Z , X ] ≊ [ Z , Ω X ] {\displaystyle [\Sigma Z,X]\approxeq [Z,\Omega X]} where [ A , B ] {\displaystyle [A,B]} is the set of homotopy classes of maps A → B {\displaystyle A\rightarrow B} , and Σ A {\displaystyle \Sigma A} is the suspension of A, and ≊ {\displaystyle \approxeq } denotes the [natural](/source/Natural_transformation) [homeomorphism](/source/Homeomorphism). This homeomorphism is essentially that of [currying](/source/Currying), modulo the quotients needed to convert the products to reduced products. In general, [ A , B ] {\displaystyle [A,B]} does not have a group structure for arbitrary spaces A {\displaystyle A} and B {\displaystyle B} . However, it can be shown that [ Σ Z , X ] {\displaystyle [\Sigma Z,X]} and [ Z , Ω X ] {\displaystyle [Z,\Omega X]} do have natural group structures when Z {\displaystyle Z} and X {\displaystyle X} are [pointed](/source/Pointed_space), and the aforementioned isomorphism is of those groups.[1] Thus, setting Z = S k − 1 {\displaystyle Z=S^{k-1}} (the k − 1 {\displaystyle k-1} sphere) gives the relationship - π k ( X ) ≊ π k − 1 ( Ω X ) {\displaystyle \pi _{k}(X)\approxeq \pi _{k-1}(\Omega X)} . This follows since the [homotopy group](/source/Homotopy_group) is defined as π k ( X ) = [ S k , X ] {\displaystyle \pi _{k}(X)=[S^{k},X]} and the spheres can be obtained via suspensions of each-other, i.e. S k = Σ S k − 1 {\displaystyle S^{k}=\Sigma S^{k-1}} .[2] ## See also - [Bott periodicity](/source/Bott_periodicity) - [Eilenberg–MacLane space](/source/Eilenberg%E2%80%93MacLane_space) - [Free loop](/source/Free_loop) - [Fundamental group](/source/Fundamental_group) - [Gray's conjecture](/source/Gray's_conjecture) - [List of topologies](/source/List_of_topologies) - [Loop group](/source/Loop_group) - [Path (topology)](/source/Path_(topology)) - [Spectrum (topology)](/source/Spectrum_(topology)) - [Path space (algebraic topology)](/source/Path_space_(algebraic_topology)) ## References 1. **[^](#cite_ref-may_1-0)** [May, J. P.](/source/J._Peter_May) (1999), [*A Concise Course in Algebraic Topology*](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf) (PDF), U. Chicago Press, Chicago, retrieved 2016-08-27 *(See chapter 8, section 2)* 1. **[^](#cite_ref-2)** [Topospaces wiki – Loop space of a based topological space](http://topospaces.subwiki.org/wiki/Loop_space_of_a_based_topological_space) - [Adams, John Frank](/source/Frank_Adams) (1978), [*Infinite loop spaces*](https://books.google.com/books?id=e2rYkg9lGnsC), Annals of Mathematics Studies, vol. 90, [Princeton University Press](/source/Princeton_University_Press), [ISBN](/source/ISBN_(identifier)) [978-0-691-08207-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-08207-3), [MR](/source/MR_(identifier)) [0505692](https://mathscinet.ams.org/mathscinet-getitem?mr=0505692) - [May, J. Peter](/source/J._Peter_May) (1972), [*The Geometry of Iterated Loop Spaces*](http://www.math.uchicago.edu/~may/BOOKSMaster.html), Lecture Notes in Mathematics, vol. 271, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [doi](/source/Doi_(identifier)):[10.1007/BFb0067491](https://doi.org/10.1007%2FBFb0067491), [ISBN](/source/ISBN_(identifier)) [978-3-540-05904-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-05904-2), [MR](/source/MR_(identifier)) [0420610](https://mathscinet.ams.org/mathscinet-getitem?mr=0420610) --- Adapted from the Wikipedia article [Loop space](https://en.wikipedia.org/wiki/Loop_space) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Loop_space?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.