Loop space

In mathematics, the space of loops or (free) loop space of a topological space X is the space of loops 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.

See also

• fundamental group
• path (topology)
• loop group
• free loop

References

• 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, MR505692, http://books.google.com/books?id=e2rYkg9lGnsC 
• 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, MR0420610, http://www.math.uchicago.edu/~may/BOOKSMaster.html