Path (topology) explained

X is a continuous function from a closed interval into 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 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

X is a continuous function f : J \to X from a non-empty and non-degenerate interval J \subseteq \R. A 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 of the path and f(b) is called its . A 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 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 or ⁰ 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

See main article: Homotopy.

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 thatf _{t(0)} = x _{0} and f _{t(1)} = x _{1} are fixed.

• the map

F : [ 0 , 1 ] x [ 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( 2 s) & 0 \leq s \leq (1)/(2) \ g( 2 s -1 ) & (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 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.

See also

• • • Path space (disambiguation)

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).