LOOP SPACE

In mathematics, the 'space of loops' or 'loop space' of a topological space ''X'' is the space of continuous maps from the unit circle ''S''¹ to ''X'' together with the compact-open topology.
:$Omega\; X\; =\; mathcal\{C\}(S^1,\; X).$
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. If we consider the quotient of the loop space with respect to the equivalence relation of pointed homotopy, then we obtain a group, the well-known fundamental group.
The 'iterated loop spaces' of ''X'' are formed by applying Ω a number of times.
The loop space construction is right adjoint to the suspension functor, and the version for pointed spaces to the reduced suspension. This accounts for much of the importance of loop spaces in stable homotopy theory.

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See also

References

See also

★ fundamental group

★ path (topology)

★ loop group

References

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