# 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}Template:Pad)_{i∈I} is a family of pointed spaces with basepoints {*p*_{i}Template:Pad}. The wedge sum of the family is given by:

where ∼ is the equivalence relation {(*p _{i}*Template:Pad,

*p*Template:Pad) |

_{j}*i,j*∈

*I*Template:Pad}. 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}Template:Pad} are homogeneous.

The wedge sum is again a pointed space, and the binary operation is associative and commutative (up to isomorphism).

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 . Doing so results in two copies of the sphere, joined at the point that was the equator:

Let 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 of a space *X* at the distinguished point can be understood as the composition of and with :

Here, and are understood to be maps, and similarly for , which take a distinguished point to a point . Note that the above defined the wedge sum of two functions, which was possible because , 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

- Smash product
- Hawaiian earring, a topological space resembling, but not the same as, a wedge sum of countably many circles

## References

- Rotman, Joseph.
*An Introduction to Algebraic Topology*, Springer, 2004, p. 153. ISBN 0-387-96678-1