Homotopy groups are notoriously difficult to compute. For a simple space like the -sphere, already, the higher homotopy groups exhibits no discernible patterns. It was therefore a complete surprise in 1957, when Raoul Bott computed the stable homotopy groups of the classical groups and found a simple periodic pattern for each of the classical groups .
We first explain what is meant by the stable homotopy group. Consider the unitary group . It acts transitively on the unit sphere in , with stabilizer at the point . In this way, the sphere can be identified with the homogeneous space , and there is a fibering with fiber . By the homotopy exact sequence of a fibering, the following sequence is exact:
Since for , it follows immediately that as goes to infinity (in fact for all ), the th homotopy group of the unitary group stabilizes:
This common value is called the th stable homotopy group of the unitary group, denoted .
In the original proof of the periodicity theorem , Bott showed that in the loop space of the special unitary group , the manifold of minimal geodesics is the complex Grassmannian
By Morse theory, the loop space has the homotopy type of a CW complex obtained from the Grassmannian by attaching cells of dimension :
It follows that
It is easily shown that
Using the homotopy exact sequence of the fibering
Putting all this together, for large relative to , we get
Thus, the stable homotopy group of the unitary group is periodic of period :
Applying the same method to the orthogonal group and the symplectic group, Bott showed that their stable homotopy groups are periodic of period .