In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S1 to X, equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an A∞-space. That is, the multiplication is homotopy-coherently associative.
The set of path components of ΩX, i.e. the set of based-homotopy equivalence classes of based loops in X, is a group, the fundamental group π1(X).
The iterated loop spaces of X are formed by applying Ω a number of times.
There is an analogous construction for topological spaces without basepoint. The free loop space of a topological space X is the space of maps from the circle S1 to X with the compact-open topology. The free loop space of X is often denoted by .
As a functor, the free loop space construction is right adjoint to cartesian product with the circle, while the loop space construction is right adjoint to the reduced suspension. This adjunction accounts for much of the importance of loop spaces in stable homotopy theory. (A related phenomenon in computer science is currying, where the cartesian product is adjoint to the hom functor.) Informally this is referred to as Eckmann–Hilton duality.
The loop space is dual to the suspension of the same space; this duality is sometimes called Eckmann–Hilton duality. The basic observation is that
where is the set of homotopy classes of maps ,
and is the suspension of A, and denotes the natural homeomorphism. This homeomorphism is essentially that of currying, modulo the quotients needed to convert the products to reduced products.
In general, does not have a group structure for arbitrary spaces and . However, it can be shown that and do have natural group structures when and are pointed, and the aforementioned isomorphism is of those groups. Thus, setting (the sphere) gives the relationship
This follows since the homotopy group is defined as and the spheres can be obtained via suspensions of each-other, i.
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
En mathématiques, notamment en analyse complexe et en topologie, un lacet est la modélisation d'une « boucle ». C'est un chemin continu et fermé, c'est-à-dire que ses extrémités sont confondues. Par exemple, tout cercle dans le plan euclidien est un lacet. Soit est un espace topologique. Définition 1 : On appelle lacet sur toute application continue telle que . Autrement dit, un lacet sur est un chemin sur dont les deux extrémités (le point initial et le point final) sont identiques.
In algebraic topology, the path space fibration over a based space is a fibration of the form where is the path space of X; i.e., equipped with the compact-open topology. is the fiber of over the base point of X; thus it is the loop space of X. The space consists of all maps from I to X that may not preserve the base points; it is called the free path space of X and the fibration given by, say, , is called the free path space fibration. The path space fibration can be understood to be dual to the mapping cone.
En topologie, un espace pointé est un espace topologique dont on spécifie un point particulier comme étant le point de base. Formellement, il s'agit donc d'un couple (E, x) pour lequel x est un élément de E. Une application pointée entre deux espaces pointés est une application continue préservant les points de base. Les espaces pointés sont les objets d'une catégorie, notée parfois Top, dont les morphismes sont les applications pointées. Cette catégorie admet le point comme objet nul.
We propose an introduction to homotopy theory for topological spaces. We define higher homotopy groups and relate them to homology groups. We introduce (co)fibration sequences, loop spaces, and suspen
On étudie des notions de topologie générale: unions et quotients d'espaces topologiques; on approfondit les notions de revêtements et de groupe fondamental,et d'attachements de cellules et on démontre
This course is an introduction to the non-perturbative bootstrap approach to Conformal Field Theory and to the Gauge/Gravity duality, emphasizing the fruitful interplay between these two ideas.
We investigate the theory of principal bundles from a homotopical point of view. In the first part of the thesis, we prove a classification of principal bundles over a fixed base space, dual to the we
Every principal G-bundle over X is classified up to equivalence by a homotopy class X -> BG, where BG is the classifying space of G. On the other hand, for every nice topological space X Milnor constr
We prove that every elementary (infinity, 1)-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural number object out o