Bott periodicity theorem

In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott (1957, 1959), which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres.

There are corresponding period-8 phenomena for the matching theories, (real) KO-theory and (quaternionic) KSp-theory, associated to the real orthogonal group and the quaternionic symplectic group, respectively.

The subject of stable homotopy theory was conceived as a simplification, by introducing the suspension (smash product with a circle) operation, and seeing what (roughly speaking) remained of homotopy theory once one was allowed to suspend both sides of an equation as many times as one wished.

What Bott periodicity offered was an insight into some highly non-trivial spaces, with central status in topology because of the connection of their cohomology with characteristic classes, for which all the (unstable) homotopy groups could be calculated.

These spaces are the (infinite, or stable) unitary, orthogonal and symplectic groups U, O and Sp.

In this context, stable refers to taking the union U (also known as the direct limit) of the sequence of inclusions and similarly for O and Sp.

The important connection of Bott periodicity with the stable homotopy groups of spheres

Originally described by George W. Whitehead, it became the subject of the famous Adams conjecture (1963) which was finally resolved in the affirmative by Daniel Quillen (1971).

Bott's original results may be succinctly summarized in: Corollary: The (unstable) homotopy groups of the (infinite) classical groups are periodic: Note: The second and third of these isomorphisms intertwine to give the 8-fold periodicity results: 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,

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.

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

One elegant formulation of Bott periodicity makes use of the observation that there are natural embeddings (as closed subgroups) between the classical groups.

The Bott periodicity results then refine to a sequence of homotopy equivalences: For complex K-theory: For real and quaternionic KO- and KSp-theories: The resulting spaces are homotopy equivalent to the classical reductive symmetric spaces, and are the successive quotients of the terms of the Bott periodicity clock.