The context of Bott periodicity is that the
homotopy groups of
spheres, which would be expected to play the basic part in
algebraic topology by analogy with
homology theory, have proved elusive (and the theory is complicated). 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. The stable theory was still hard to compute with, in practice.
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. Note that Bott's use of the word stable in the title of his seminal paper refers to these stable
classical groups and not to
stable homotopy groups.
The important connection of Bott periodicity with the
stable homotopy groups of spheres comes via the so-called stable
J-homomorphism from the (unstable) homotopy groups of the (stable) classical groups to these stable homotopy groups . 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:
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.
Geometric model of loop spaces
One elegant formulation of Bott periodicity makes use of the observation that there are natural embeddings (as closed subgroups) between the classical groups. The loop spaces in Bott periodicity are then homotopy equivalent to the
symmetric spaces of successive quotients, with additional discrete factors of Z.
where the division algebras indicate "matrices over that algebra".
Animation of the Bott periodicity clock using a Mod 8 clock face with second hand mnemonics taken from the I-Ching with the real Clifford algebra of signature (p,q) denoted as Clp,q()=Cl(p,q).
As they are 2-periodic/8-periodic, they can be arranged in a circle, where they are called the Bott periodicity clock and Clifford algebra clock.
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. These equivalences immediately yield the Bott periodicity theorems.
^The interpretation and labeling is slightly incorrect, and refers to irreducible symmetric spaces, while these are the more general reductive spaces. For example, SU/Sp is irreducible, while U/Sp is reductive. As these show, the difference can be interpreted as whether or not one includes orientation.