Bott periodicity theorem
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In mathematics, the Bott periodicity theorem is a result from homotopy theory discovered by Raoul Bott during the latter part of the 1950s, which proved to be of foundational significance for much further research, in particular in K-theory. It can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. There is a corresponding modulo 8 phenomenon, which comes from the matching theory for the real orthogonal group, and which plays a conspicuous role therefore in KO-theory, and other related theories.
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 homotopy groups could be calculated.
In fact for the prime case, the space BU that is the classifying space for complex vector bundles (a Grassmannian in infinite dimensions), one formulation of Bott periodicity describes
- Ω2BU,
where Ω is the loop space functor, right adjoint to suspension. The theorem states that this double loop space is essentially BU again; in fact it is the union of a countable number of copies of BU. This has the immediate effect of showing why topological K-theory is a periodic theory, also.
Bott's original proof used Morse theory; subsequently different kinds of proof have been given.