Wedge sum - Citizendia

In topology, the wedge sum is a "one-point union" of a family of topological spaces. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Specifically, if X and Y are pointed spaces (i. In Mathematics, a pointed space is a Topological space X with a distinguished basepoint x 0 in X. e. topological spaces with distinguished basepoints x₀ and y₀) the wedge sum of X and Y is the quotient of the disjoint union of X and Y by the identification x₀ ∼ y₀:

$X\vee Y = (X\amalg Y)\;/ \;\{x_0 \sim y_0\}$

More generally, suppose (X_i)_i∈I is a family of pointed spaces with basepoints {p_i}. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying In Set theory, a disjoint union (or discriminated union) is a modified union operation which indexes the elements according to which set they originated The wedge sum of the family is given by:

$\bigvee_i X_i := \coprod_i X_i\;/ \;\{p_i\sim p_j \mid i,j \in I\}$

In other words, the wedge sum is the joining of several spaces at a single point. This definition of course depends on the choice of {p_i} unless the spaces {X_i} are homogeneous. In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group

1 Examples
2 Categorical description
3 Properties
4 See also

Examples

The wedge sum of two circles is homeomorphic to a figure-eight space. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, a rose (also known as a bouquet of circles) is a Topological space obtained by gluing together a collection of circles The wedge sum of n-circles is often called a bouquet of circles, while a wedge product of arbitrary spheres is often called a bouquet of spheres. In Mathematics, a rose (also known as a bouquet of circles) is a Topological space obtained by gluing together a collection of circles

A common construction in homotopy is to identify all of the points along the equator of an n-sphere $S n$ . In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical Doing so results in two copies of the sphere, joined at the point that was the equator:

$S^n/\sim = S^n \vee S^n$

Let $Ψ$ be the map $\Psi:S^n\to S^n \vee S^n$ , that is, of identifying the equator down to a single point. Then addition of two elements $f,g\in\pi_n(X,x_0)$ of the n-dimensional homotopy group $π n (X, x 0)$ of a space X at the distinguished point $x_0\in X$ can be understood as the composition of $f$ and $g$ with $Ψ$ :

$f+g = (f \vee g) \circ \Psi$

Here, $f$ and $g$ are understood to be maps, $f:S^n\to X$ and similarly for $g$ , which take a distinguished point $s_0\in S^n$ to a point $x_0\in X$ . In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional Note that the above defined the wedge sum of two functions, which was possible because $f (s 0) = g (s 0) = x 0$ , which was the point that is equivalenced in the wedge sum of the underlying spaces.

Categorical description

The wedge sum can be understood as the coproduct in the category of pointed spaces. In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological In Mathematics, a pointed space is a Topological space X with a distinguished basepoint x 0 in X. Alternatively, the wedge sum can be seen as the pushout of the diagram X ← {•} → Y in the category of topological spaces (where {•} is any one point space). In Category theory, a branch of Mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square) is the In Mathematics, the category of topological spaces, often denoted Top, is the category whose objects are Topological spaces and whose

Properties

Van Kampen's theorem gives certain conditions (which are usually fulfilled for well-behaved spaces, such as CW complexes) under which the fundamental group of the wedge sum of two spaces X and Y is the free product of the fundamental groups of X and Y. In Mathematics, the Seifert – van Kampen theorem of Algebraic topology, sometimes just called van Kampen's theorem, expresses the Mathematicians (and those in related sciences very frequently speak of whether a mathematical object &mdash a Number, a function, a set, a space In Topology, a CW complex is a type of Topological space introduced by J In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Abstract algebra, the free product of groups constructs a group from two or more given ones

Contents

Examples

Categorical description

Properties

See also