Loop (topology)

Two loops Template:Mvar, Template:Mvar in a torus.

In mathematics, a loop in a topological space Template:Mvar is a continuous function Template:Mvar from the unit interval Template:Math to Template:Mvar such that Template:Nowrap 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 Template:Mvar from the pointed unit circle Template:Math into Template:Mvar, because Template:Math may be regarded as a quotient of Template:Mvar under the identification of 0 with 1.

The set of all loops in Template:Mvar forms a space called the loop space of Template:Mvar.

Definition

Let $X$ be a topological space. A loop is a continuous function $f:[0,1]\to X$ such that $f(0)=f(1)$. If $f$ begins and ends at $x_0\in X$ the loop is said to be based at $x_0$. A loop is then a path that begins and ends at the same point $x_0$.

The set of homotopy classes of loops based at $x_0$ together with the operation of path composition, forms the fundamental group of $X$ relative to $x_0$, usually denoted by ${\pi }_1(X,x_0)$.

See also

• Free loop
• Loop group
• Loop space
• Loop algebra
• Fundamental group
• Quasigroup

References

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