nLab loop (topology)

This entry is about loops in topology. For loops in the sense of algebra see at loop (algebra), also at quasigroup and Moufang loop.

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Definitions

In topology, a (parametrised, oriented) loop in a space $X$ is a map (a morphism in an appropriate category of spaces, such as a continuous function between topological spaces) to $X$ from the circle $S^1 = \mathbb{R}/\mathbb{Z}$. Hence a continuous path whose endpoints coincide.

A loop at $a$ is a loop $f$ such that $f(k) = a$ for any (hence every) integer $k$. An unparametrised loop is an equivalence class of loops, such that $f$ and $g$ are equivalent if there is an increasing automorphism $\phi$ of $S^1$ such that $g = f \circ \phi$. An unoriented loop is an equivalence class of loops such that $f$ is equivalent to $(x \mapsto f(-x))$. A Moore loop has domain $\mathbb{R}/n \mathbb{Z}$ for some natural number (or possibly any real number) $n$. All of these variations can be combined, of course. (A Moore loop at $a$ has $f(k n) = a$ instead of $f(k) = a$. Also, a Moore loop for $n = 0$ is simply a point, so possibly there is a better way to define this to avoid making this exception. Finally, there is not much difference between unparametrised loops and unparametrised Moore loops, since we may interpret $(t \mapsto n t)$ as a reparametrisation $\phi$.)

In graph theory, a loop is an edge whose endpoints are the same vertex. Actually, this is a special case of the above, if we interpret $S^1$ as the graph with $1$ vertex and $1$ edge; in this way, the other variations become meaningful. In this context, a Moore loop is called a cycle. (However, as the only directed graph automorphism of $S^1$ is the identity, parametrisation is trivial for directed graphs and equivalent to orientation for undirected graphs.)

Every loop may be interpreted as a path. Sometimes a loop, say at $a$, is defined to be a path from $a$ to $a$. However, this is correct only in certain contexts. In graph theory, it's incorrect, but only because of terminological conventions; the idea is sound. In continuous spaces, it is also correct. However, in smooth spaces, it is not correct, since the derivatives at the endpoints should also agree; the same holds in many other more structured contexts.

Concatenation

Given two Moore loops $f$ and $g$ at $a$, the concatenation of $f$ and $g$ is a Moore loop $f ; g$ or $g \circ f$ at $a$. If the domain of $f$ is $\mathbb{R}/m \mathbb{Z}$ and the domain of $g$ is $\mathbb{R}/n \mathbb{Z}$, then the domain of $f ; g$ is $\mathbb{R}/(m+n) \mathbb{Z}$, and

$$(f ; g)(x) \coloneqq \left \{ \array { f(x) & \quad x \leq m \\ g(m+x) & \quad x \geq m .} \right .$$

In this way, we get a monoid of Moore loops in $X$ at $a$, with concatenation as multiplication. This monoid may called the Moore loop monoid?.

Often we are more interested in a quotient monoid of the Moore loop monoid. If we use unparametrised loops (in which case we may use loops with domain $S^1$ if we wish), then we get the unparametrised loop monoid?. If $X$ is a smooth space, then we may additionally identify loops related through a thin homotopy to get the loop group. Finally, if $X$ is a continuous space and we identify loops related through any (basepoint-preserving) homotopy, then we get the fundamental group of $X$.

See looping and delooping for more.

Examples

• Pochhammer loop

Related entries

• loop

• path,

• loop space, free loop space

  loop space object, loop space type

• fundamental group

• loop group

• loop braid group