Loop space

Encyclopedia

In mathematics

, the

from the unit circle

.

That is, a particular function space

.

In homotopy theory

s, 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

.

If we consider the quotient of the based loop space Ω

, the well-known fundamental group

The

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

.

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the

**space of loops**or**(free) loop space**of a topological spaceTopological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

*X*is the space of loopsLoop (topology)

In mathematics, a loop in a topological space X is a path f from the unit interval I = [0,1] to X such that f = f...

from the unit circle

Unit circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

*S*^{1}to*X*together with the compact-open topologyCompact-open topology

In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis...

.

That is, a particular function space

Function space

In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

.

In homotopy theory

*loop space*commonly refers to the same construction applied to pointed spacePointed space

In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...

s, 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

Magma (algebra)

In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....

, or even as an

*A*_{∞}-space. Concatenation of loops is not strictly associative, but it is associative up to higher homotopiesHomotopy

In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

.

If we consider the quotient of the based loop space Ω

*X*with respect to the equivalence relation of pointed homotopy, then we obtain a groupGroup (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

, the well-known fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

*π*_{1}(*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

Cartesian product

In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

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

Stable homotopy theory

In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...

.