Wedge sum

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 "x"0 and "y"0) the wedge sum of "X" and "Y" is the quotient space of the disjoint union of "X" and "Y" by the identification "x"0 ∼ "y"0:
where ∼ is the equivalence closure of the relation {("x"0,"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:
where ∼ is the equivalence closure of the relation {("pi", "pj") | "i,j" ∈ "I"}.
In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints {"p""i"}, unless the spaces {"X""i"} are homogeneous.
The wedge sum is again a pointed space, and the binary operation is associative and commutative (up to homeomorphism).
Sometimes the wedge sum is called the wedge product, but this is not the same concept as the exterior product, which is also often called the wedge product.
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 formula_3. Doing so results in two copies of the sphere, joined at the point that was the equator:
Let formula_5 be the map formula_6, that is, of identifying the equator down to a single point. Then addition of two elements formula_7 of the "n"-dimensional homotopy group formula_8 of a space "X" at the distinguished point formula_9 can be understood as the composition of formula_10 and formula_11 with formula_5:
Here, formula_14 are maps which take a distinguished point formula_15 to the point formula_16 Note that the above uses the wedge sum of two functions, which is possible precisely because they agree at formula_17 the point common to 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".