Wedge sum

In topology, the wedge sum is a way to connect different spaces at a single 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 creates a new shape that keeps all the parts of the original shapes but now shares a common point.

The wedge sum works for any number of spaces. Each space must have its own basepoint, and all these basepoints are joined together. This operation is both associative and commutative, meaning the order in which you join the spaces does not change the final result, as long as you consider their shapes to be the same.

Although sometimes called the wedge product, this is different from the exterior product, which is another mathematical 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 an 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 can be thought of as a special kind of combination in math. It is like the coproduct in the category of pointed spaces. Another way to see it is as the pushout of a simple diagram in the category of topological spaces. Here, { ∙ } represents any space with just one point.

Properties

Van Kampen's theorem explains how to find the fundamental group of the wedge sum of two spaces, X and Y, under certain conditions. These conditions are usually met for simple and well-organized spaces, like CW complexes. In such cases, the fundamental group of the wedge sum is the combination of the fundamental groups of X and Y, known as their free product.