Wedge sum

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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 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}. 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.

1 Examples
2 Categorical description
3 Properties
4 See also

[edit] Examples

The wedge sum of two circles is homeomorphic to a figure-eight space. 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.

A common construction in homotopy is to identify all of the points along the equator of an n-sphere $S n$ . 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 $x_0\in X$ . 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.

[edit] Categorical description

The wedge sum can be understood as the coproduct in the category of pointed spaces. 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).

[edit] 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.