Bott periodicity

*Bott periodicity* is the name of a periodicity phenomenon that appears throughout spin geometry, supersymmetry and K-theory. Incarnations of it include the following:

The complex reduced topological K-theory groups have a degree-2 periodicity:

$\tilde K_{\mathbb{C}}^\bullet(X)
\simeq
\tilde K_{\mathbb{C}}^{\bullet + 2}(X)
\,.$

This isomorphism is induced by external tensor product with the image of the basic line bundle on the 2-sphere in reduced K-theory, called the *Bott element*.

For details see at *topological K-theory* the section *Bott periodicity*.

The periodicity lifts to the classifying spaces and makes the representing spectrum KU of complex K-theory be an even periodic ring spectrum.

In particular the 2-periodicity in the homotopy groups of the stable unitary group $U = \underset{\longrightarrow}{\lim}_n U(n)$ is thus a shadow of Bott periodicity.

$\pi_i(U) = \pi_i(GL_{\mathbb{C}})
=
\left\lbrace
\array{
0 &\vert& i\, \text{even}
\\
\mathbb{Z} &\vert& i \, \text{odd}
}
\right.$

Similarly the real reduced topological K-theory groups have an 8-periodicity

$\tilde K^\bullet_{\mathbb{R}}(X)
\simeq
\tilde K^{\bullet + 8}_{\mathbb{R}}( X )$

a shadow of which is the 8-periodicity in the homotopy groups of the stable orthogonal group

$\pi_i( O )
=
\pi_i(GL_{\mathbb{R}})
=
\left\{
\array{
\mathbb{Z}_2 &\vert& i = 0 \, \text{mod}\, 8
\\
\mathbb{Z}_2 &\vert& i = 1 \, \text{mod}\, 8
\\
0 &\vert& i = 2 \, \text{mod}\, 8
\\
\mathbb{Z} &\vert& i = 3 \, \text{mod}\, 8
\\
0 &\vert& i = 4 \, \text{mod}\, 8
\\
0 &\vert& i = 5 \, \text{mod}\, 8
\\
0 &\vert& i = 6 \, \text{mod}\, 8
\\
\mathbb{Z} &\vert& i = 7 \, \text{mod}\, 8
}
\right.$

The complex Clifford algebras repeat – up to *Morita equivalence* – with period 2, $Cl_{n}(\mathbb{C}) \simeq_{Morita} Cl_{n+2}(\mathbb{C})$.

The real Clifford algebras analogously have period 8, $Cl_n(\mathbb{R}) \simeq_{Morita} Cl_{n+8}(\mathbb{R})$.

Accordingly the basic properties of complex spinor representations are the same for $Spin(d-1,1)$ and $Spin(d+2-1,1)$. Those of the real spinor representations repeat with period 8.

For a list of proofs of Bott periodicity, see

*Proofs of Bott periodicity*, MO

Last revised on December 16, 2019 at 11:56:37. See the history of this page for a list of all contributions to it.