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

) 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