Loop (topology)

Two loops a, b in a torus.

In mathematics, a loop in a topological space X is a continuous function f from the unit interval I = [0,1] to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its terminal point. ^[1]

A loop may also be seen as a continuous map f from the pointed unit circle S¹ into X, because S¹ may be regarded as a quotient of I under the identification of 0 with 1.

The set of all loops in X forms a space called the loop space of X. ^[1]

Definition

Let ${\displaystyle X}$ be a topological space. A loop is a continuous function ${\displaystyle f:[0,1]\to X}$ such that ${\displaystyle f(0)=f(1)}$. If ${\displaystyle f}$ begins and ends at ${\displaystyle x_{0}\in X}$ the loop is said to be based at ${\displaystyle x_{0}}$. A loop is then a path that begins and ends at the same point ${\displaystyle x_{0}}$. ^[2]

The set of homotopy classes of loops based at ${\displaystyle x_{0}}$ together with the operation of path composition, forms the fundamental group of ${\displaystyle X}$ relative to ${\displaystyle x_{0}}$, usually denoted by ${\displaystyle \pi _{1}(X,x_{0})}$. ^[2]

See also

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

References

1. Adams, John Frank (1978), Infinite Loop Spaces, Annals of mathematics studies, vol. 90, Princeton University Press, p. 3, ISBN   9780691082066 .
2. Munkres, James Raymond (2014). Topology (2 ed.). Harlow: Pearson. p. 331. ISBN   978-1-292-02362-5.