## 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 **S**n,
then the homotopy classes from X into Y,
anchored at x0 → y0, are denoted πn(Y).
When n = 1, **S**n 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.