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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:
More generally, suppose (Xi)i∈I is a family of pointed spaces with basepoints {pi}. The wedge sum of the family is given by:
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.
Sometimes the wedge sum is called the wedge product, but this is not the same thing as the exterior product which is also often called the wedge product.
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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 Sn. Doing so results in two copies of the sphere, joined at the point that was the equator:
Let Ψ be the map 
, that is, of identifying the equator down to a single point. Then addition of two elements 
of the n-dimensional homotopy group πn(X,x0) of a space X at the distinguished point 
can be understood as the composition of f and g with Ψ:
Here, f and g are understood to be maps, 
and similarly for g, which take a distinguished point 
to a point . Note that the above defined the wedge sum of two functions, which was possible because f(s0) = g(s0) = x0, 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.
See also
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