Path (topology)

In mathematics, a path in a topological space $X$ is a continuous function from a closed interval into $X.$

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

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 $X$ is often denoted $\pi _{0}(X).$

One can also define paths and loops in pointed spaces, which are important in homotopy theory. If $X$ is a topological space with basepoint $x_{0},$ then a path in $X$ is one whose initial point is $x_{0}$ . Likewise, a loop in $X$ is one that is based at $x_{0}$ .

Definition

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

An arc or $C$ ⁰-arc in $X$ is a path in $X$ that is also a topological embedding.

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

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

f:S^{1}\to X.

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

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 $f_{t}:[0,1]\to X$ indexed by $I=[0,1]$ such that

$f_{t}(0)=x_{0}$ and $f_{t}(1)=x_{1}$ are fixed.
the map $F:[0,1]\times [0,1]\to X$ given by $F(s,t)=f_{t}(s)$ is continuous.

The paths $f_{0}$ and $f_{1}$ 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 $f$ under this relation is called the homotopy class of $f,$ often denoted $[f].$

Path composition

One can compose paths in a topological space in the following 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 $x_{0},$ 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 $x_{0}$ in $X.$ The resultant group is called the fundamental group of $X$ based at $x_{0},$ usually denoted $\pi _{1}\left(X,x_{0}\right).$

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 $a\geq 0.$ (Such a path is called a Moore path.) 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 $\left(|f|+|g|+|h|\right)/2$ 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 $x_{0}$ in $X$ is just the fundamental group based at $x_{0}$ . 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 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).