nLab
loop

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Loops

Definitions

In topology, a (parametrised, oriented) loop in a space XX is a map (a morphism in an appropriate category of spaces, such as a continuous function between topological spaces) to XX from the circle S 1=/S^1 = \mathbb{R}/\mathbb{Z}. Hence a continuous path whose endpoints coincide.

A loop at aa is a loop ff such that f(k)=af(k) = a for any (hence every) integer kk. An unparametrised loop is an equivalence class of loops, such that ff and gg are equivalent if there is an increasing automorphism ϕ\phi of S 1S^1 such that g=fϕg = f \circ \phi. An unoriented loop is an equivalence class of loops such that ff is equivalent to (xf(x))(x \mapsto f(-x)). A Moore loop has domain /n\mathbb{R}/n \mathbb{Z} for some natural number (or possibly any real number) nn. All of these variations can be combined, of course. (A Moore loop at aa has f(kn)=af(k n) = a instead of f(k)=af(k) = a. Also, a Moore loop for n=0n = 0 is simply a point, so possibly there is a better way to define this to avoid making this exception. Finally, there is not much difference between unparametrised loops and unparametrised Moore loops, since we may interpret (tnt)(t \mapsto n t) as a reparametrisation ϕ\phi.)

In graph theory, a loop is an edge whose endpoints are the same vertex. Actually, this is a special case of the above, if we interpret S 1S^1 as the graph with 11 vertex and 11 edge; in this way, the other variations become meaningful. In this context, a Moore loop is called a cycle. (However, as the only directed graph automorphism of S 1S^1 is the identity, parametrisation is trivial for directed graphs and equivalent to orientation for undirected graphs.)

Every loop may be interpreted as a path. Sometimes a loop, say at aa, is defined to be a path from aa to aa. However, this is correct only in certain contexts. In graph theory, it's incorrect, but only because of terminological conventions; the idea is sound. In continuous spaces, it is also correct. However, in smooth spaces, it is not correct, since the derivatives at the endpoints should also agree; the same holds in many other more structured contexts.

Concatenation

Given two Moore loops ff and gg at aa, the concatenation of ff and gg is a Moore loop f;gf ; g or gfg \circ f at aa. If the domain of ff is /m\mathbb{R}/m \mathbb{Z} and the domain of gg is /n\mathbb{R}/n \mathbb{Z}, then the domain of f;gf ; g is /(m+n)\mathbb{R}/(m+n) \mathbb{Z}, and

(f;g)(x){f(x) xm g(m+x) xm. (f ; g)(x) \coloneqq \left \{ \array { f(x) & \quad x \leq m \\ g(m+x) & \quad x \geq m .} \right .

In this way, we get a monoid of Moore loops in XX at aa, with concatenation as multiplication. This monoid may called the Moore loop monoid?.

Often we are more interested in a quotient monoid of the Moore loop monoid. If we use unparametrised loops (in which case we may use loops with domain S 1S^1 if we wish), then we get the unparametrised loop monoid?. If XX is a smooth space, then we may additionally identify loops related through a thin homotopy to get the loop group. Finally, if XX is a continuous space and we identify loops related through any (basepoint-preserving) homotopy, then we get the fundamental group of XX.

See looping and delooping for more.

Revised on July 2, 2017 06:42:07 by Urs Schreiber (88.77.226.246)