Path (topology)

The points traced by a path from A to B in R². However, different paths can trace the same set of points.

In mathematics, a path in a topological space X is a continuous function f from the unit interval I = [0,1] to X

f : I → X.

The initial point of the path is f(0) and the terminal point is f(1). One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path. Note that a path is not just a subset of X which "looks like" a curve, it also includes a parameterization. For example, the maps f(x) = x and g(x) = x² represent two different paths from 0 to 1 on the real line.

A loop in a space X based at x ∈ X is a path from x to x. A loop may be equally well regarded as a map f : I → X with f(0) = f(1) or as a continuous map from the unit circle S¹ to X

f : S¹ → X.

This is because S¹ may be regarded as a quotient of I under the identification 0 ∼ 1. The set of all loops in X forms a space called the loop space of X.

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 a set of path-connected components. The set of path-connected components of a space X is often denoted π₀(X);.

One can also define paths and loops in pointed spaces, which are important in homotopy theory. If X is a topological space with basepoint x₀, then a path in X is one whose initial point is x₀. Likewise, a loop in X is one that is based at x₀.

Homotopy of paths

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 X is a family of paths f_t : I → X indexed by I such that

• f_t(0) = x₀ and f_t(1) = x₁ are fixed.
• the map F : I × I → X given by F(s, t) = f_t(s) is continuous.

The paths f₀ and f₁ connected by a homotopy are said to 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 f under this relation is called the homotopy class of f, often denoted [f].

Path composition

One can compose paths in a topological space in an obvious manner. Suppose f is a path from x to y and g is a path from y to z. The path fg is defined as the path obtained by first traversing f and then traversing g:

$fg(s) = \begin{cases}f(2s) & 0\leq s \leq \frac{1}{2} \\ g(2s-1) & \frac{1}{2} \leq s \leq 1.\end{cases}$

Clearly path composition is only defined when the terminal point of f coincides with the initial point of g. If one considers all loops based at a point x₀, 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, [(fg)h] = [f(gh)]. Path composition defines a group structure on the set of homotopy classes of loops based at a point x₀ in X. The resultant group is called the fundamental group of X based at x₀, usually denoted π₁(X,x₀).

In situations calling for associativity of path composition "on the nose," a path in X may instead be defined as a continuous map from an interval [0,a] to X for any real a ≥ 0. A path f of this kind has a length |f| defined as a. Path composition is then defined as before with the following modification:

$fg(s) = \begin{cases}f(s) & 0\leq s \leq |f| \\ g(s-|f|) & |f| \leq s \leq |f|+|g|\end{cases}$

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

Fundamental groupoid

There is a categorical picture of paths which is sometimes useful. Any topological space X gives rise to a category where the objects are the points of X 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 X. Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point x₀ in X is just the fundamental group based at X. More generally, one can define the fundamental groupoid on any subset A of X, using homotopy classes of paths joining points of A. This is convenient for the Van Kampen's Theorem.

References

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