In [2], Atiyah, Bott and Shapiro calculated certain groups $A_k$ associated to real Clifford algebra representations, and observed that they were isomorphic to $KO^{-k}(*)$. The same can be said for complex $K$-theory, but a sequence of groups alternating between $\mathbb {Z}$ and $0$ is less impressive. In their paper, they constructed a map $A_k \to KO^{-k}(*)$, and using their knowledge of $KO^{-k}(*)$, they showed this is in fact an isomorphism. It wasn't a particularly exciting proof, since both sides have a ring structure that the map respects, and so they only had to check the map does the right thing to the handful of generators.

Later, in [3], Atiyah and Singer found a good reason why they had to be isomorphic. Roughly, the idea is that $KO^{-k}$ is represented by some group of Clifford algebra representation homomorphisms, and it is not too difficult to show that $\pi _0$ of this this space is isomorphic to $A_k$. The goal of these notes is to work through these results and conclude the Bott periodicity theorem.

If the reader finds the details lacking, they may refer to [3] (which does not provide a very detailed treatment either, but the intersection of details missed out should be minimal).

In these notes, “homotopy equivalence” will mean “weak homotopy equivalence”. However, it follows from results of Milnor [7] that the spaces involved are CW complexes, so it doesn't actually matter.