# Loop Space

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.

### Other articles related to "space, loop space, spaces, loop spaces, loops":

Normal Invariant - Different Categories
... a structure of an abelian group since the space is a loop space and in fact an infinite loop space so the normal invariants are a zeroth cohomology group of an extraordinary ... the other categories of manifolds and one has bijections , and, and It is well known that the spaces , and are mutually not homotopy equivalent and hence one obtains three ... He showed that these spaces possess alternative inifinite loop space structures which are in fact better from the following point of view Recall that there is a surgery obstruction map from normal invariants ...
Homotopy Groups Of Spheres - Computational Methods
... it with a fibration involving an Eilenberg-MacLane space ... He used the fact that taking the loop space of a well behaved space shifts all the homotopy groups down by 1, so the nth homotopy group of a space X is the first homotopy group of its (n ... of homotopy groups of X to the calculation of homology groups of its repeated loop spaces ...
Symplectic Floer Homology
... action functional on the (universal cover of the) free loop space of a symplectic manifold ... have cylindrical ends asymptotic to the loops in the mapping torus corresponding to fixed points of the symplectomorphism ... quadratic at infinity, the Floer homology is the singular homology of the free loop space of M (proofs of various versions of this statement are due to Viterbo, Salamon–Weber, Abbondandolo–Schwarz ...

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