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In mathematics, a path in a topological space <math>X</math> is a continuous function from a closed interval into <math>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>X</math> is often denoted <math>\pi_0(X).</math>
One can also define paths and loops in pointed spaces, which are important in homotopy theory. If <math>X</math> is a topological space with basepoint <math>x_0,</math> then a path in <math>X</math> is one whose initial point is <math>x_0</math>. Likewise, a loop in <math>X</math> is one that is based at <math>x_0</math>.
DefinitionEdit
A curve in a topological space <math>X</math> is a continuous function <math>f : J \to X</math> from a non-empty and non-degenerate interval <math>J \subseteq \R.</math> A Template:Em in <math>X</math> is a curve <math>f : [a, b] \to X</math> whose domain <math>[a, b]</math> is a compact non-degenerate interval (meaning <math>a < b</math> are real numbers), where <math>f(a)</math> is called the Template:Em of the path and <math>f(b)</math> is called its Template:Em. A Template:Em is a path whose initial point is <math>x</math> and whose terminal point is <math>y.</math> Every non-degenerate compact interval <math>[a, b]</math> is homeomorphic to <math>[0, 1],</math> which is why a Template:Em is sometimes, especially in homotopy theory, defined to be a continuous function <math>f : [0, 1] \to X</math> from the closed unit interval <math>I := [0, 1]</math> into <math>X.</math>
Template:Anchor An Template:Em or Template:Mvar0Template:Em in <math>X</math> is a path in <math>X</math> that is also a topological embedding.
Importantly, a path is not just a subset of <math>X</math> that "looks like" a curve, it also includes a parameterization. For example, the maps <math>f(x) = x</math> and <math>g(x) = x^2</math> represent two different paths from 0 to 1 on the real line.
A loop in a space <math>X</math> based at <math>x \in X</math> is a path from <math>x</math> to <math>x.</math> A loop may be equally well regarded as a map <math>f : [0, 1] \to X</math> with <math>f(0) = f(1)</math> or as a continuous map from the unit circle <math>S^1</math> to <math>X</math>
- <math>f : S^1 \to X.</math>
This is because <math>S^1</math> is the quotient space of <math>I = [0, 1]</math> when <math>0</math> is identified with <math>1.</math> The set of all loops in <math>X</math> forms a space called the loop space of <math>X.</math>
Homotopy of pathsEdit
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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>X</math> is a family of paths <math>f_t : [0, 1] \to X</math> indexed by <math>I = [0, 1]</math> such that
- <math>f_t(0) = x_0</math> and <math>f_t(1) = x_1</math> are fixed.
- the map <math>F : [0, 1] \times [0, 1] \to X</math> given by <math>F(s, t) = f_t(s)</math> is continuous.
The paths <math>f_0</math> and <math>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>f</math> under this relation is called the homotopy class of <math>f,</math> often denoted <math>[f].</math>
Path compositionEdit
One can compose paths in a topological space in the following manner. Suppose <math>f</math> is a path from <math>x</math> to <math>y</math> and <math>g</math> is a path from <math>y</math> to <math>z</math>. The path <math>fg</math> is defined as the path obtained by first traversing <math>f</math> and then traversing <math>g</math>:
- <math>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>f</math> coincides with the initial point of <math>g.</math> If one considers all loops based at a point <math>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 Template:Em associative up to path-homotopy. That is, <math>[(fg)h] = [f(gh)].</math> Path composition defines a group structure on the set of homotopy classes of loops based at a point <math>x_0</math> in <math>X.</math> The resultant group is called the fundamental group of <math>X</math> based at <math>x_0,</math> usually denoted <math>\pi_1\left(X, x_0\right).</math>
In situations calling for associativity of path composition "on the nose," a path in <math>X</math> may instead be defined as a continuous map from an interval <math>[0, a]</math> to <math>X</math> for any real <math>a \geq 0.</math> (Such a path is called a Moore path.) A path <math>f</math> of this kind has a length <math>|f|</math> defined as <math>a.</math> Path composition is then defined as before with the following modification:
- <math>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>f,</math> <math>g</math>, and <math>fg</math> all have length <math>1</math> (the length of the domain of the map), this definition makes <math>|fg| = |f| + |g|.</math> What made associativity fail for the previous definition is that although <math>(fg)h</math> and <math>f(gh)</math>have the same length, namely <math>1,</math> the midpoint of <math>(fg)h</math> occurred between <math>g</math> and <math>h,</math> whereas the midpoint of <math>f(gh)</math> occurred between <math>f</math> and <math>g</math>. With this modified definition <math>(fg)h</math> and <math>f(gh)</math> have the same length, namely <math>|f| + |g| + |h|,</math> and the same midpoint, found at <math>\left(|f| + |g| + |h|\right)/2</math> in both <math>(fg)h</math> and <math>f(gh)</math>; more generally they have the same parametrization throughout.
Fundamental groupoidEdit
There is a categorical picture of paths which is sometimes useful. Any topological space <math>X</math> gives rise to a category where the objects are the points of <math>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>X.</math> Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point <math>x_0</math> in <math>X</math> is just the fundamental group based at <math>x_0</math>. More generally, one can define the fundamental groupoid on any subset <math>A</math> of <math>X,</math> using homotopy classes of paths joining points of <math>A.</math> This is convenient for Van Kampen's Theorem.
See alsoEdit
ReferencesEdit
- 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).