In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
field of
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a free loop is a variant of the notion of a
loop. Whereas a loop has a distinguished point on it, called its basepoint, a free loop lacks such a distinguished point. Formally, let
be a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
. Then a free loop in
is an
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s from the
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
to
. Two loops are equivalent if they differ by a reparameterization of the circle. That is,
if there exists a
homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
such that
Thus, a free loop, as opposed to a based loop used in the definition of the
fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
, is a map from the circle to the space without the basepoint-preserving restriction. Assuming the space is
path-connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties t ...
, free
homotopy
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...
classes of free loops correspond to
conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
es in the fundamental group.
Recently, interest in the space of all free loops
has grown with the advent of
string topology, i.e. the study of new
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
s on the
homology of the free loop space.
See also
*
Loop space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topolog ...
*
Loop (topology)
In mathematics, a loop in a topological space is a continuous function from the unit interval to such that In other words, it is a path whose initial point is equal to its terminal point..
A loop may also be seen as a continuous map from t ...
*
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element pro ...
Further reading
* Brylinski, Jean-Luc: Loop spaces, characteristic classes and geometric quantization. Reprint of the 1993 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008.
* Cohen and Voronov:
Notes on String Topology
Knot theory
Topology
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