3.2 and Bott Periodicity

We already touched upon Bott periodicity when we mentioned that the Clifford algebra is isomorphic to the algebra of matrices with entries lying in . This is but one of many related `period-8' phenomena that go by the name of Bott periodicity. The appearance of the number 8 here is no coincidence: all these phenomena are related to the octonions! Since this marvelous fact is somewhat under-appreciated, it seems worthwhile to say a bit about it. Here we shall focus on those aspects that are related to and the canonical octonionic line bundle over this space.

Let us start with K-theory. This is a way of gaining information about a topological space by studying the vector bundles over it. If the space has holes in it, there will be nontrivial vector bundles that have `twists' as we go around these holes. The simplest example is the `Möbius strip' bundle over , a 1-dimensional real vector bundle which has a twist as we go around the circle. In fact, this is just the canonical line bundle . The canonical line bundles and provide higher-dimensional analogues of this example.

K-theory tells us to study the vector bundles over a topological space
by constructing an abelian group as follows. First, take the set
consisting of all isomorphism classes of real vector bundles over .
Our ability to take direct sums of vector bundles gives this set an
`addition' operation making it into a commutative monoid. Next, adjoin
formal `additive inverses' for all the elements of this set, obtaining
an abelian group. This group is called , the **real K-theory**
of . Alternatively we could start with complex vector bundles and
get a group called , but here we will be interested in real vector
bundles.

Any real vector bundle over gives an element , and these
elements generate this group. If we pick a point in , there is an
obvious homomorphism
sending to the
dimension of the fiber of at this point. Since the dimension is a
rather obvious and boring invariant of vector bundles, it is nice to
work with the kernel of this homomorphism, which is called the **reduced** real K-theory of and denoted
. This is
an invariant of pointed spaces, i.e. spaces equipped with a designated
point or **basepoint**.

Any sphere becomes a pointed space if we take the north pole as
basepoint. The reduced real K-theory of the first eight spheres
looks like this:

where, as one might guess,

- generates .
- generates .
- generates .
- generates .

This fact gives us the list of homotopy groups of which appears in the Introduction. It also means that to prove Bott periodicity for these homotopy groups:

it suffices to prove Bott periodicity for real K-theory:

Why do we have Bott periodicity in real K-theory? It turns out
that there is a graded ring with

The product in this ring comes from our ability to take `smash products' of spheres and also of real vector bundles over these spheres. Multiplying by gives an isomorphism

In other words, the canonical octonionic line bundle over generates Bott periodicity!

There is much more to say about this fact and how it relates to Bott periodicity for Clifford algebras, but alas, this would take us too far afield. We recommend that the interested reader turn to some introductory texts on K-theory, for example the one by Dale Husemoller [52]. Unfortunately, all the books I know downplay the role of the octonions. To spot it, one must bear in mind the relation between the octonions and Clifford algebras, discussed in Section 2.3 above.

© 2001 John Baez