Wedge sum

In topology, the wedge sum is a way to connect different shapes at one point. Imagine you have two shapes, each with a special point called a basepoint. The wedge sum joins these shapes by merging their basepoints into one point. This makes a new shape that keeps all the parts of the original shapes but now shares one common point.

The wedge sum can work for any number of shapes. Each shape must have its own basepoint, and all these basepoints are joined together. This operation is both associative and commutative, meaning the order you join the shapes does not change the final result, as long as the shapes look the same.

Although sometimes called the wedge product, this is different from the exterior product, which is another math idea that also uses the word “wedge.” The wedge sum is a useful tool in topology for studying how spaces can be connected.

Examples

The wedge sum of two circles looks like a figure-eight. When we take many circles and join them at one point, we call this a bouquet of circles. We can also join spheres in the same way, calling the result a bouquet of spheres.

One common way to use wedge sums in mathematics is by taking a sphere and joining all points along its middle, called the equator, to a single point. This creates two copies of the sphere connected at that one point.

Categorical description

The wedge sum is a special kind of combination in math. It is like the coproduct in the category of pointed spaces. It is also like the pushout of a simple diagram in the category of topological spaces. Here, { ∙ } means any space with just one point.

Properties

Van Kampen's theorem helps us understand the fundamental group of the wedge sum of two spaces, X and Y, when certain conditions are met. This usually works for simple and organized spaces, like CW complexes. In these cases, the fundamental group of the wedge sum is the combination of the fundamental groups of X and Y, called their free product.