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
) the wedge sum of X and Y is the quotient space of the disjoint union of X and Y by the identification
is the equivalence closure of the relation
More generally, suppose
is an indexed family of pointed spaces with basepoints
The wedge sum of the family is given by:
is the equivalence closure of the relation
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
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.
The wedge sum of two circles is homeomorphic to a figure-eight space.
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
Doing so results in two copies of the sphere, joined at the point that was the equator:
that is, of identifying the equator down to a single point.
Then addition of two elements
-dimensional homotopy group
at the distinguished point
can be understood as the composition of
are maps which take a distinguished point
Note that the above uses the wedge sum of two functions, which is possible precisely because they agree at
the point common to the wedge sum of the underlying spaces.
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
in the category of topological spaces (where
is any one-point space).
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
is the free product of the fundamental groups of