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

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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.


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 (