Bott Periodicity Theorem - Loop Spaces and Classifying Spaces

Loop Spaces and Classifying Spaces

For the theory associated to the infinite unitary group, U, the space BU is the classifying space for stable complex vector bundles (a Grassmannian in infinite dimensions). One formulation of Bott periodicity describes the twofold loop space, Ω2BU of BU. Here, Ω is the loop space functor, right adjoint to suspension and left adjoint to the classifying space construction. Bott periodicity states that this double loop space is essentially BU again; more precisely,

is essentially (that is, homotopy equivalent to) the union of a countable number of copies of BU. An equivalent formulation is

Either of these has the immediate effect of showing why (complex) topological K-theory is a 2-fold periodic theory.

In the corresponding theory for the infinite orthogonal group, O, the space BO is the classifying space for stable real vector bundles. In this case, Bott periodicity states that, for the 8-fold loop space,

or equivalently,

which yields the consequence that KO-theory is an 8-fold periodic theory. Also, for the infinite symplectic group, Sp, the space BSp is the classifying space for stable quaternionic vector bundles, and Bott periodicity states that

or equivalently

Thus both topological real K-theory (also known as KO-theory) and topological quaternionic K-theory (also known as KSp-theory) are 8-fold periodic theories.