Higher Order π Groups, An Introduction

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 π_n(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, π_n is always abelian. These groups can help characterize certain higher dimensional spaces that are simply connected, and have no fundamental group.