Higher Order π Groups, An Introduction


As you recall, a homotopy continuously morphs one function onto another. For instance, f(X) and g(X) are homotopic functions into the space Y if the image of f slides, continuously, within the space Y, until it coincides with the image of g.

Sometimes it is convenient to fix a base point x0 in X, which always maps to y0 in Y. This holds true for f and for g, and for the entire homotopy. As the image of f slides onto the image of g, the image of x0 is anchored at y0.

Let X be the 0 dimensional sphere, which is simply two disconnected points. (The distance between these two points is the diameter of the 0 sphere, but that has no topological significance.) One of these points is x0, which always maps to y0. The other point, x1, maps to any point in Y. Two functions f and g are homotopic iff both f and g map x1 into the same path component of Y. Therefore the homotopy classes of Y, denoted π0(Y), are the path components of Y.

We've seen the π notation before. If X is the n sphere, denoted Sn, then the homotopy classes from X into Y, anchored at x0 → y0, are denoted πn(Y). When n = 1, Sn is the circle. We showed that π1(Y) is the fundamental group of Y. Unlike π0, the homotopy classes of π1 form a group, using path concatenation as the group operator.

Throughout these pages we will explore higher order π sets, when the domain is the 2 sphere, the 3 sphere, and so on. In each case the π set forms a group, but unlike π1, πn is always abelian. These groups can help characterize certain higher dimensional spaces that are simply connected, and have no fundamental group.