Definition:Loop (Topology)

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This page is about Loop in the context of Topology. For other uses, see Loop.

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $\gamma: \closedint 0 1 \to S$ be a path in $T$.

Let $\map \gamma 0 = \map \gamma 1$.


Then $\gamma$ is called a loop (in $T$).


Simple Loop

$\gamma$ is a simple loop (in $T$) if and only if:

$\map \gamma {t_1} \ne \map \gamma {t_2}$ for all $t_1 ,t_2 \in \hointr 0 1$ with $t_1 \ne t_2$
$\map \gamma 0 = \map \gamma 1$


Base Point

The base point of $\gamma$ is $\map \gamma 0$.


Set of All Loops

The set of all loops based at $p \in T$ is denoted by $\map \Omega {T, p}$.


Constant Loop

A constant loop $c_p$ is the loop $c_p \in \map \Omega {T, p}$ such that:

$\forall t \in \closedint 0 1 : \map {c_p} t = p$


Null-Homotopic Loop

Suppose $\gamma$ is path-homotopic to a constant loop.


Then $\gamma$ is said to be null-homotopic.


Circle Representative of Loop

Let $\Bbb S^1 \subseteq \C$ be the unit circle in $\C$:

$\Bbb S^1 = \set {z \in \C : \size z = 1}$

Suppose $\omega : \closedint 0 1 \to \Bbb S^1$ such that $\map \omega s = \map \exp {2 \pi i s}$.


Then the unique map $\tilde f : \Bbb S^1 \to T$ such that $\tilde f \circ \omega = f$ is called the circle representative of $f$.


Loop in Topological Manifold

Let $M$ be a topological manifold.

Let $\sigma : \closedint 0 1 \to M$ be a continuous path.

Let $\map \sigma 0 = \map \sigma 1$.


Then $\sigma$ is called a loop.


Also known as

A loop is also referred to as a closed path.

Some sources refer to it as a cycle.


Internationalization

Loop is translated:

In French: lacet
In Dutch: lus  (literally: loop)


Also see

  • Results about loops (in the context of topology) can be found here.


Sources

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