1 The groups $\mathrm{U}$ and $\mathrm{O}$

There is an inclusion $\mathrm{U}(n - 1) \hookrightarrow \mathrm{U}(n)$ that sends

$$M \mapsto \begin{pmatrix} M & 0 \\ 0 & 1 \end{pmatrix}.$$

We define $\mathrm{U}$ to be the union (colimit) along all these inclusions. The most basic form of Bott periodicity says

This is a remarkable theorem. The naive way to compute the groups $\pi _k \mathrm{U}(n)$ is to inductively use the fiber sequences

arising from the action of $\mathrm{U}(n + 1)$ on $S^{2n + 1}$. This requires understanding all the unstable homotopy groups of (odd) spheres, which is already immensely complicated, and then piece them together via the long exact sequence. Bott periodicity tells us that in the limit $n \to \infty$, all these cancel out, and we are left with the very simple $2$-periodic homotopy groups.

One can define $\mathrm{O}$ similarly as the union of the $\mathrm{O}(n)$'s, and the resulting homotopy groups are $8$-periodic.

The number theorists in the audience should note (in dismay) that $\mathbb {Z}_2$ refers to the integers mod $2$, not the $2$-adics.