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- Loop_space abstract "In mathematics, the space of loops or (free) loop space of a topological space X is the space of maps from the unit circle S1 to X together with the compact-open topology.That is, a particular function space.In homotopy theory loop space commonly refers to the same construction applied to pointed spaces, i.e. continuous maps respecting base points. 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, or even as an A∞-space. Concatenation of loops is not strictly associative, but it is associative up to higher homotopies.If we consider the quotient of the based loop space ΩX with respect to the equivalence relation of pointed homotopy, then we obtain a group, the well-known fundamental group π1(X).The iterated loop spaces of X are formed by applying Ω a number of times.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.".
- Loop_space wikiPageExternalLink books?id=e2rYkg9lGnsC.
- Loop_space wikiPageExternalLink BOOKSMaster.html.
- Loop_space wikiPageID "2174332".
- Loop_space wikiPageRevisionID "606726635".
- Loop_space hasPhotoCollection Loop_space.
- Loop_space subject Category:Homotopy_theory.
- Loop_space subject Category:Topological_spaces.
- Loop_space subject Category:Topology.
- Loop_space type Abstraction100002137.
- Loop_space type Attribute100024264.
- Loop_space type MathematicalSpace108001685.
- Loop_space type Set107999699.
- Loop_space type Space100028651.
- Loop_space type TopologicalSpaces.
- Loop_space comment "In mathematics, the space of loops or (free) loop space of a topological space X is the space of maps from the unit circle S1 to X together with the compact-open topology.That is, a particular function space.In homotopy theory loop space commonly refers to the same construction applied to pointed spaces, i.e. continuous maps respecting base points. In this setting there is a natural "concatenation operation" by which two elements of the loop space can be combined.".
- Loop_space label "Espace des lacets".
- Loop_space label "Loop space".
- Loop_space label "Lusruimte".
- Loop_space label "Пространство петель".
- Loop_space label "فضاء العقد".
- Loop_space sameAs Espace_des_lacets.
- Loop_space sameAs Lusruimte.
- Loop_space sameAs m.06sdpb.
- Loop_space sameAs Q2066070.
- Loop_space sameAs Q2066070.
- Loop_space sameAs Loop_space.
- Loop_space wasDerivedFrom Loop_space?oldid=606726635.
- Loop_space isPrimaryTopicOf Loop_space.