Bott Periodicity — The groups U\mathrm{U} and O\mathrm{O}

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

There is an inclusion U(n1)U(n)\mathrm{U}(n - 1) \hookrightarrow \mathrm{U}(n) that sends

M(M001). M \mapsto \begin{pmatrix} M & 0 \\ 0 & 1 \end{pmatrix}.

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

Theorem 1 (Complex Bott periodicity)
πkU={Zk odd0k even. \pi _ k \mathrm{U}= \begin{cases} \mathbb {Z}& k\text{ odd}\\ 0 & k\text{ even} \end{cases}.

In particular, the homotopy groups of U\mathrm{U} are 22-periodic.

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

\begin{useimager} 
  \[
    \begin{tikzcd}
      \U(n) \ar[r] & \U(n + 1) \ar[d]\\
      & S^{2n + 1}
    \end{tikzcd}
  \]
\end{useimager}

arising from the action of U(n+1)\mathrm{U}(n + 1) on S2n+1S^{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 nn \to \infty , all these cancel out, and we are left with the very simple 22-periodic homotopy groups.

One can define O\mathrm{O} similarly as the union of the O(n)\mathrm{O}(n)'s, and the resulting homotopy groups are 88-periodic.

Theorem 2 (Real Bott periodicity)

We have

kmod8k \bmod 8

00

11

22

33

44

55

66

77

πkO\pi _ k \mathrm{O}

Z2\mathbb {Z}_2

Z2\mathbb {Z}_2

00

Z\mathbb {Z}

00

00

00

Z\mathbb {Z}

The number theorists in the audience should note (in dismay) that Z2\mathbb {Z}_2 refers to the integers mod 22, not the 22-adics.