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.
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
- May, J. P. (1999), A Concise Course in Algebraic Topology, U. Chicago Press, Chicago, retrieved 2008-09-27 (chapter 8, section 2)
- Adams, John Frank (1978), Infinite loop spaces, Annals of Mathematics Studies 90, Princeton University Press, ISBN 978-0-691-08207-3; 978-0-691-08206-6 Check
|isbn=value (help), MR 505692
- 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