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 $U = \underset{\longrightarrow}{\lim}_n U(n)$ 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, $Cl_{n}(\mathbb{C}) \simeq_{Morita} Cl_{n+2}(\mathbb{C})$.
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.
Proof of Bott periodicity for topological K-theory, including equivariant K-theory:
Review:
Graeme Segal, Prop. 3.2 in: Equivariant K-theory, Inst. Hautes Etudes Sci. Publ. Math. No. 34 (1968) p. 129-151 (numdam:PMIHES_1968__34__129_0)
Max Karoubi, Bott Periodicity in Topological, Algebraic and Hermitian K-Theory, In: Eric Friedlander, Daniel Grayson (eds.) Handbook of K-Theory, Springer 2005 (doi:10.1007/978-3-540-27855-9_4)
Dai Tamaki, Akira Kono, Section 4.2 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)
For a list of proofs of Bott periodicity, see
Last revised on January 3, 2021 at 02:15:47. See the history of this page for a list of all contributions to it.