Bott periodicity in nLab

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˜ ℂ •(X)≃K˜ ℂ •+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 ℂ)={0 | ieven ℤ | iodd \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

K˜ ℝ •(X)≃K˜ ℝ •+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 ℝ)={ℤ 2 | i=0mod8 ℤ 2 | i=1mod8 0 | i=2mod8 ℤ | i=3mod8 0 | i=4mod8 0 | i=5mod8 0 | i=6mod8 ℤ | 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(ℂ)≃ MoritaCl n+2(ℂ)Cl_{n}(\mathbb{C}) \simeq_{Morita} Cl_{n+2}(\mathbb{C}).

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

• ten-fold way

• Hopf fibration

• periodic cohomology theory

• supersymmetry and division algebras.

References

The original statement as periodicity of stable homotopy groups of classical Lie groups:

• Raoul Bott, The Stable Homotopy of the Classical Groups, Proceedings of the National Academy of Sciences of the United States of America 43 10 (1957) 933-935 [jstor:89403]

• Eldon Dyer, Richard Lashof, A topological Proof of the Bott Periodicity Theorems, Annali di Matematica Pura ed Applicata 54 (1961) 231–254 [doi:10.1007/BF02415354]

Proof of Bott periodicity for topological K-theory, including equivariant K-theory:

• Michael Atiyah, Bott periodicity and the index of elliptic operators, The Quarterly Journal of Mathematics, 19 1 (1968) 113–140 [doi:10.1093/qmath/19.1.113]

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)

• Dale Husemöller, Michael Joachim, Branislav Jurčo, Martin Schottenloher, Section 15 of: Basic Bundle Theory and K-Cohomology Invariants, Springer Lecture Notes in Physics 726, 2008, (pdf, doi:10.1007/978-3-540-74956-1)

For a list of proofs of Bott periodicity, see

• Proofs of Bott periodicity, MO