Path (topology)

Short description: Continuous function whose domain is a closed unit interval

The points traced by a path from [math]\displaystyle{ A }[/math] to [math]\displaystyle{ B }[/math] in [math]\displaystyle{ \mathbb{R}^2. }[/math] However, different paths can trace the same set of points.

In mathematics, a path in a topological space [math]\displaystyle{ X }[/math] is a continuous function from the closed unit interval [math]\displaystyle{ [0, 1] }[/math] into [math]\displaystyle{ X. }[/math]

Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there exists a path connecting any two points is said to be path-connected. Any space may be broken up into path-connected components. The set of path-connected components of a space [math]\displaystyle{ X }[/math] is often denoted [math]\displaystyle{ \pi_0(X). }[/math]

One can also define paths and loops in pointed spaces, which are important in homotopy theory. If [math]\displaystyle{ X }[/math] is a topological space with basepoint [math]\displaystyle{ x_0, }[/math] then a path in [math]\displaystyle{ X }[/math] is one whose initial point is [math]\displaystyle{ x_0 }[/math]. Likewise, a loop in [math]\displaystyle{ X }[/math] is one that is based at [math]\displaystyle{ x_0 }[/math].

Definition

A curve in a topological space [math]\displaystyle{ X }[/math] is a continuous function [math]\displaystyle{ f : J \to X }[/math] from a non-empty and non-degenerate interval [math]\displaystyle{ J \subseteq \R. }[/math] A path in [math]\displaystyle{ X }[/math] is a curve [math]\displaystyle{ f : [a, b] \to X }[/math] whose domain [math]\displaystyle{ [a, b] }[/math] is a compact non-degenerate interval (meaning [math]\displaystyle{ a \lt b }[/math] are real numbers), where [math]\displaystyle{ f(a) }[/math] is called the initial point of the path and [math]\displaystyle{ f(b) }[/math] is called its terminal point. A path from [math]\displaystyle{ x }[/math] to [math]\displaystyle{ y }[/math] is a path whose initial point is [math]\displaystyle{ x }[/math] and whose terminal point is [math]\displaystyle{ y. }[/math] Every non-degenerate compact interval [math]\displaystyle{ [a, b] }[/math] is homeomorphic to [math]\displaystyle{ [0, 1], }[/math] which is why a path is sometimes, especially in homotopy theory, defined to be a continuous function [math]\displaystyle{ f : [0, 1] \to X }[/math] from the closed unit interval [math]\displaystyle{ I := [0, 1] }[/math] into [math]\displaystyle{ X. }[/math] An arc or $C$ ⁰-arc in [math]\displaystyle{ X }[/math] is a path in [math]\displaystyle{ X }[/math] that is also a topological embedding.

Importantly, a path is not just a subset of [math]\displaystyle{ X }[/math] that "looks like" a curve, it also includes a parameterization. For example, the maps [math]\displaystyle{ f(x) = x }[/math] and [math]\displaystyle{ g(x) = x^2 }[/math] represent two different paths from 0 to 1 on the real line.

A loop in a space [math]\displaystyle{ X }[/math] based at [math]\displaystyle{ x \in X }[/math] is a path from [math]\displaystyle{ x }[/math] to [math]\displaystyle{ x. }[/math] A loop may be equally well regarded as a map [math]\displaystyle{ f : [0, 1] \to X }[/math] with [math]\displaystyle{ f(0) = f(1) }[/math] or as a continuous map from the unit circle [math]\displaystyle{ S^1 }[/math] to [math]\displaystyle{ X }[/math]

[math]\displaystyle{ f : S^1 \to X. }[/math]

This is because [math]\displaystyle{ S^1 }[/math] is the quotient space of [math]\displaystyle{ I = [0, 1] }[/math] when [math]\displaystyle{ 0 }[/math] is identified with [math]\displaystyle{ 1. }[/math] The set of all loops in [math]\displaystyle{ X }[/math] forms a space called the loop space of [math]\displaystyle{ X. }[/math]

Homotopy of paths

Main page: Homotopy

A homotopy between two paths.

Paths and loops are central subjects of study in the branch of algebraic topology called homotopy theory. A homotopy of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed.

Specifically, a homotopy of paths, or path-homotopy, in [math]\displaystyle{ X }[/math] is a family of paths [math]\displaystyle{ f_t : [0, 1] \to X }[/math] indexed by [math]\displaystyle{ I = [0, 1] }[/math] such that

[math]\displaystyle{ f_t(0) = x_0 }[/math] and [math]\displaystyle{ f_t(1) = x_1 }[/math] are fixed.
the map [math]\displaystyle{ F : [0, 1] \times [0, 1] \to X }[/math] given by [math]\displaystyle{ F(s, t) = f_t(s) }[/math] is continuous.

The paths [math]\displaystyle{ f_0 }[/math] and [math]\displaystyle{ f_1 }[/math] connected by a homotopy are said to be homotopic (or more precisely path-homotopic, to distinguish between the relation defined on all continuous functions between fixed spaces). One can likewise define a homotopy of loops keeping the base point fixed.

The relation of being homotopic is an equivalence relation on paths in a topological space. The equivalence class of a path [math]\displaystyle{ f }[/math] under this relation is called the homotopy class of [math]\displaystyle{ f, }[/math] often denoted [math]\displaystyle{ [f]. }[/math]

Path composition

One can compose paths in a topological space in the following manner. Suppose [math]\displaystyle{ f }[/math] is a path from [math]\displaystyle{ x }[/math] to [math]\displaystyle{ y }[/math] and [math]\displaystyle{ g }[/math] is a path from [math]\displaystyle{ y }[/math] to [math]\displaystyle{ z }[/math]. The path [math]\displaystyle{ fg }[/math] is defined as the path obtained by first traversing [math]\displaystyle{ f }[/math] and then traversing [math]\displaystyle{ g }[/math]:

[math]\displaystyle{ fg(s) = \begin{cases}f(2s) & 0 \leq s \leq \frac{1}{2} \\ g(2s-1) & \frac{1}{2} \leq s \leq 1.\end{cases} }[/math]

Clearly path composition is only defined when the terminal point of [math]\displaystyle{ f }[/math] coincides with the initial point of [math]\displaystyle{ g. }[/math] If one considers all loops based at a point [math]\displaystyle{ x_0, }[/math] then path composition is a binary operation.

Path composition, whenever defined, is not associative due to the difference in parametrization. However it is associative up to path-homotopy. That is, [math]\displaystyle{ [(fg)h] = [f(gh)]. }[/math] Path composition defines a group structure on the set of homotopy classes of loops based at a point [math]\displaystyle{ x_0 }[/math] in [math]\displaystyle{ X. }[/math] The resultant group is called the fundamental group of [math]\displaystyle{ X }[/math] based at [math]\displaystyle{ x_0, }[/math] usually denoted [math]\displaystyle{ \pi_1\left(X, x_0\right). }[/math]

In situations calling for associativity of path composition "on the nose," a path in [math]\displaystyle{ X }[/math] may instead be defined as a continuous map from an interval [math]\displaystyle{ [0, a] }[/math] to [math]\displaystyle{ X }[/math] for any real [math]\displaystyle{ a \geq 0. }[/math] (Such a path is called a Moore path.) A path [math]\displaystyle{ f }[/math] of this kind has a length [math]\displaystyle{ |f| }[/math] defined as [math]\displaystyle{ a. }[/math] Path composition is then defined as before with the following modification:

[math]\displaystyle{ fg(s) = \begin{cases}f(s) & 0 \leq s \leq |f| \\ g(s-|f|) & |f| \leq s \leq |f| + |g|\end{cases} }[/math]

Whereas with the previous definition, [math]\displaystyle{ f, }[/math] [math]\displaystyle{ g }[/math], and [math]\displaystyle{ fg }[/math] all have length [math]\displaystyle{ 1 }[/math] (the length of the domain of the map), this definition makes [math]\displaystyle{ |fg| = |f| + |g|. }[/math] What made associativity fail for the previous definition is that although [math]\displaystyle{ (fg)h }[/math] and [math]\displaystyle{ f(gh) }[/math]have the same length, namely [math]\displaystyle{ 1, }[/math] the midpoint of [math]\displaystyle{ (fg)h }[/math] occurred between [math]\displaystyle{ g }[/math] and [math]\displaystyle{ h, }[/math] whereas the midpoint of [math]\displaystyle{ f(gh) }[/math] occurred between [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math]. With this modified definition [math]\displaystyle{ (fg)h }[/math] and [math]\displaystyle{ f(gh) }[/math] have the same length, namely [math]\displaystyle{ |f| + |g| + |h|, }[/math] and the same midpoint, found at [math]\displaystyle{ \left(|f| + |g| + |h|\right)/2 }[/math] in both [math]\displaystyle{ (fg)h }[/math] and [math]\displaystyle{ f(gh) }[/math]; more generally they have the same parametrization throughout.

Fundamental groupoid

There is a categorical picture of paths which is sometimes useful. Any topological space [math]\displaystyle{ X }[/math] gives rise to a category where the objects are the points of [math]\displaystyle{ X }[/math] and the morphisms are the homotopy classes of paths. Since any morphism in this category is an isomorphism this category is a groupoid, called the fundamental groupoid of [math]\displaystyle{ X. }[/math] Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point [math]\displaystyle{ x_0 }[/math] in [math]\displaystyle{ X }[/math] is just the fundamental group based at [math]\displaystyle{ x_0 }[/math]. More generally, one can define the fundamental groupoid on any subset [math]\displaystyle{ A }[/math] of [math]\displaystyle{ X, }[/math] using homotopy classes of paths joining points of [math]\displaystyle{ A. }[/math] This is convenient for Van Kampen's Theorem.

References

Ronald Brown, Topology and groupoids, Booksurge PLC, (2006).
J. Peter May, A concise course in algebraic topology, University of Chicago Press, (1999).
James Munkres, Topology 2ed, Prentice Hall, (2000).

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