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:
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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[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 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.
[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.
