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:
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 is thus a shadow of Bott periodicity.
Similarly the real reduced topological K-theory groups have an 8-periodicity
a shadow of which is the 8-periodicity in the homotopy groups of the stable orthogonal group
The complex Clifford algebras repeat – up to Morita equivalence – with period 2, .
The real Clifford algebras analogously have period 8, .
Accordingly the basic properties of complex spinor representations are the same for and . Those of the real spinor representations repeat with period 8.
For a list of proofs of Bott periodicity, see
Last revised on May 29, 2017 at 04:30:52. See the history of this page for a list of all contributions to it.