Arc (topology)

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

In mathematics, a path in a topological space X {\displaystyle X} {\displaystyle X} is a continuous function from a closed interval into X . {\displaystyle X.} {\displaystyle X.}

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

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

Definition

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

An arc or C0-arc in X {\displaystyle X} {\displaystyle X} is a path in X {\displaystyle X} {\displaystyle X} that is also a topological embedding.

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

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

f : S 1 → X . {\displaystyle f:S^{1}\to X.} {\displaystyle f:S^{1}\to X.}

This is because S 1 {\displaystyle S^{1}} {\displaystyle S^{1}} is the quotient space of I = [ 0 , 1 ] {\displaystyle I=[0,1]} {\displaystyle I=[0,1]} when 0 {\displaystyle 0} {\displaystyle 0} is identified with 1. {\displaystyle 1.} {\displaystyle 1.} The set of all loops in X {\displaystyle X} {\displaystyle X} forms a space called the loop space of X . {\displaystyle X.} {\displaystyle 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 {\displaystyle X} {\displaystyle X} is a family of paths f t : [ 0 , 1 ] → X {\displaystyle f_{t}:[0,1]\to X} {\displaystyle f_{t}:[0,1]\to X} indexed by I = [ 0 , 1 ] {\displaystyle I=[0,1]} {\displaystyle I=[0,1]} such that

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

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

Path composition

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

f g ( s ) = { f ( 2 s ) 0 ≤ s ≤ 1 2 g ( 2 s − 1 ) 1 2 ≤ s ≤ 1. {\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}}} {\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}}}

Clearly path composition is only defined when the terminal point of f {\displaystyle f} {\displaystyle f} coincides with the initial point of g . {\displaystyle g.} {\displaystyle g.} If one considers all loops based at a point x 0 , {\displaystyle x_{0},} {\displaystyle 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, [ ( f g ) h ] = [ f ( g h ) ] . {\displaystyle [(fg)h]=[f(gh)].} {\displaystyle [(fg)h]=[f(gh)].} Path composition defines a group structure on the set of homotopy classes of loops based at a point x 0 {\displaystyle x_{0}} {\displaystyle x_{0}} in X . {\displaystyle X.} {\displaystyle X.} The resultant group is called the fundamental group of X {\displaystyle X} {\displaystyle X} based at x 0 , {\displaystyle x_{0},} {\displaystyle x_{0},} usually denoted π 1 ( X , x 0 ) . {\displaystyle \pi _{1}\left(X,x_{0}\right).} {\displaystyle \pi _{1}\left(X,x_{0}\right).}

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

f g ( s ) = { f ( s ) 0 ≤ s ≤ | f | g ( s − | f | ) | f | ≤ s ≤ | f | + | g | {\displaystyle fg(s)={\begin{cases}f(s)&0\leq s\leq |f|\\g(s-|f|)&|f|\leq s\leq |f|+|g|\end{cases}}} {\displaystyle 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 , {\displaystyle f,} {\displaystyle f,} g {\displaystyle g} {\displaystyle g}, and f g {\displaystyle fg} {\displaystyle fg} all have length 1 {\displaystyle 1} {\displaystyle 1} (the length of the domain of the map), this definition makes | f g | = | f | + | g | . {\displaystyle |fg|=|f|+|g|.} {\displaystyle |fg|=|f|+|g|.} What made associativity fail for the previous definition is that although ( f g ) h {\displaystyle (fg)h} {\displaystyle (fg)h} and f ( g h ) {\displaystyle f(gh)} {\displaystyle f(gh)}have the same length, namely 1 , {\displaystyle 1,} {\displaystyle 1,} the midpoint of ( f g ) h {\displaystyle (fg)h} {\displaystyle (fg)h} occurred between g {\displaystyle g} {\displaystyle g} and h , {\displaystyle h,} {\displaystyle h,} whereas the midpoint of f ( g h ) {\displaystyle f(gh)} {\displaystyle f(gh)} occurred between f {\displaystyle f} {\displaystyle f} and g {\displaystyle g} {\displaystyle g}. With this modified definition ( f g ) h {\displaystyle (fg)h} {\displaystyle (fg)h} and f ( g h ) {\displaystyle f(gh)} {\displaystyle f(gh)} have the same length, namely | f | + | g | + | h | , {\displaystyle |f|+|g|+|h|,} {\displaystyle |f|+|g|+|h|,} and the same midpoint, found at ( | f | + | g | + | h | ) / 2 {\displaystyle \left(|f|+|g|+|h|\right)/2} {\displaystyle \left(|f|+|g|+|h|\right)/2} in both ( f g ) h {\displaystyle (fg)h} {\displaystyle (fg)h} and f ( g h ) {\displaystyle f(gh)} {\displaystyle 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 {\displaystyle X} {\displaystyle X} gives rise to a category where the objects are the points of X {\displaystyle X} {\displaystyle 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 . {\displaystyle X.} {\displaystyle X.} Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point x 0 {\displaystyle x_{0}} {\displaystyle x_{0}} in X {\displaystyle X} {\displaystyle X} is just the fundamental group based at x 0 {\displaystyle x_{0}} {\displaystyle x_{0}}. More generally, one can define the fundamental groupoid on any subset A {\displaystyle A} {\displaystyle A} of X , {\displaystyle X,} {\displaystyle X,} using homotopy classes of paths joining points of A . {\displaystyle A.} {\displaystyle A.} This is convenient for Van Kampen's Theorem.

See also

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).