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:
- <math>X\vee Y = (X\amalg Y)\;/ \;\{x_0 \sim y_0\}</math>
More generally, suppose (Xi)i∈I is a family of pointed spaces with basepoints {pi}. The wedge sum of the family is given by:
- <math>\bigvee_i X_i := \coprod_i X_i\;/ \;\{p_i\sim p_j \mid i,j \in I\}</math>
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 {pi} unless the spaces {Xi} are homogeneous.
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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 <math>S^n</math>. Doing so results in two copies of the sphere, joined at the point that was the equator:
- <math>S^n/\sim = S^n \vee S^n </math>
Let <math>\Psi</math> be the map <math>\Psi:S^n\to S^n \vee S^n</math>, that is, of identifying the equator down to a single point. Then addition of two elements <math>f,g\in\pi_n(X,x_0)</math> of the n-dimensional homotopy group <math>\pi_n(X,x_0)</math> of a space X at the distinguished point <math>x_0\in X</math> can be understood as the composition of <math>f</math> and <math>g</math> with <math>\Psi</math>:
- <math>f+g = (f \vee g) \circ \Psi</math>
Here, <math>f</math> and <math>g</math> are understood to be maps, <math>f:S^n\to X</math> and similarly for <math>g</math>, which take a distinguished point <math>s_0\in S^n</math> to a point <math>x_0\in X</math>. Note that the above defined the wedge sum of two functions, which was possible because <math>f(s_0)=g(s_0)=x_0</math>, 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. 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).
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.
