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.
Famous quotes containing the word space:
“To play is nothing but the imitative substitution of a pleasurable, superfluous and voluntary action for a serious, necessary, imperative and difficult one. At the cradle of play as well as of artistic activity there stood leisure, tedium entailed by increased spiritual mobility, a horror vacui, the need of letting forms no longer imprisoned move freely, of filling empty time with sequences of notes, empty space with sequences of form.”
—Max J. Friedländer (1867–1958)