Loop space: Difference between revisions
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More generally, |
More generally, |
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:<math>[\Sigma Z,X] \approxeq [Z, \Omega X]</math> |
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| ⚫ | < |
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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 fr 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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Note that setting <math>Z = S^{k-1}</math> (the <math>k-1</math> sphere) gives the earlier result. |
Note that setting <math>Z = S^{k-1}</math> (the <math>k-1</math> sphere) gives the earlier result. |
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Revision as of 05:02, 28 December 2013
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.
Relation between homotopy groups of a space and those of its loop space
The basic relation between the homotopy groups is .[1]
More generally,
![{\displaystyle [\Sigma Z,X]\approxeq [Z,\Omega X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b991e48782acebd8830fd8cb1450a032fcab4430)
where, is the set of homotopy classes of maps , and is the suspension of A. In general does not have a group structure fr arbitrary spaces and . However, it can be shown that and do have natural group structures when and are pointed, and the aforesaid isomorphism is of those groups. [2]
![{\displaystyle [A,B]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1993067bb075f2ebfa02e78959b7c5bed68e06f4)
![{\displaystyle [\Sigma Z,X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26998c7996ab4873286617962d549ef4112ccf45)
![{\displaystyle [Z,\Omega X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c37e2396275ada45216a5ca9875b93764b2c0ce)
Note that setting (the sphere) gives the earlier result.
See also
References
- ^ http://topospaces.subwiki.org/wiki/Loop_space_of_a_based_topological_space
- ^ May, J. P. (1999), "8", A Concise Course in Algebraic Topology (PDF), U. Chicago Press, Chicago, retrieved 2008-09-27 (chapter 8, section 2)
- Adams, John Frank (1978), Infinite loop spaces, Annals of Mathematics Studies, vol. 90, Princeton University Press, ISBN 978-0-691-08207-3; 978-0-691-08206-6, MR 0505692
{{citation}}: Check|isbn=value: invalid character (help) - May, J. Peter (1972), The Geometry of Iterated Loop Spaces, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0067491, ISBN 978-3-540-05904-2, MR 0420610