nLab
Bott periodicity

Contents

Contents

Idea

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

In topological K-theory

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

K˜ (X)K˜ +2(X). \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=lim nU(n)U = \underset{\longrightarrow}{\lim}_n U(n) is thus a shadow of Bott periodicity.

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

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

K˜ (X)K˜ +8(X) \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

π i(O)=π i(GL )={ 2 | i=0mod8 2 | i=1mod8 0 | i=2mod8 | i=3mod8 0 | i=4mod8 0 | i=5mod8 0 | i=6mod8 | i=7mod8 \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.

In Spin geometry

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

The real Clifford algebras analogously have period 8, Cl n() MoritaCl n+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(d1,1)Spin(d-1,1) and Spin(d+21,1)Spin(d+2-1,1). Those of the real spinor representations repeat with period 8.

References

For a list of proofs of Bott periodicity, see

  • Proofs of Bott periodicity, MO

Last revised on May 29, 2017 at 04:30:52. See the history of this page for a list of all contributions to it.