In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces (i.e. topological spaces with distinguished basepoints and ) the wedge sum of X and Y is the quotient space of the disjoint union of X and Y by the identification
where is the equivalence closure of the relation
More generally, suppose is a indexed family of pointed spaces with basepoints The wedge sum of the family is given by:
where is the equivalence closure of the relation
In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints unless the spaces are homogeneous.
The wedge sum is again a pointed space, and the binary operation is associative and commutative (up to homeomorphism).
Sometimes the wedge sum is called the wedge product, but this is not the same concept as the exterior product, which is also often called the wedge product.
The wedge sum of two circles is homeomorphic to a figure-eight space. The wedge sum of circles is often called a bouquet of circles, while a wedge product of arbitrary spheres is often called a bouquet of spheres.
A common construction in homotopy is to identify all of the points along the equator of an -sphere . Doing so results in two copies of the sphere, joined at the point that was the equator:
Let be the map that is, of identifying the equator down to a single point. Then addition of two elements of the -dimensional homotopy group of a space at the distinguished point can be understood as the composition of and with :
Here, are maps which take a distinguished point to the point Note that the above uses the wedge sum of two functions, which is possible precisely because they agree at the point common to the wedge sum of the underlying spaces.
The wedge sum can be understood as the coproduct in the . Alternatively, the wedge sum can be seen as the of the diagram in the (where is any one-point space).
Van Kampen's theorem gives certain conditions (which are usually
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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.
In mathematics, a rose (also known as a bouquet of n circles) is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraic topology, where they are closely related to free groups. A rose is a wedge sum of circles. That is, the rose is the quotient space C/S, where C is a disjoint union of circles and S a set consisting of one point from each circle. As a cell complex, a rose has a single vertex, and one edge for each circle.
En mathématiques, et plus particulièrement en théorie des groupes, le produit libre de deux groupes G et H est un nouveau groupe, noté G∗H, qui contient G et H comme sous-groupes, est engendré par les éléments de ces sous-groupes, et constitue le groupe « le plus général » possédant ces propriétés. Le produit libre est le coproduit, ou « somme », dans la catégorie des groupes, c'est-à-dire que la donnée de deux morphismes, de G et H dans un même groupe K, équivaut à celle d'un morphisme de G∗H dans K.
In this thesis, we study two distinct problems.
The first problem consists of studying the linear system of partial differential equations which consists of taking a k-form, and applying the exterior derivative 'd' to it and add the wedge product with a 1- ...