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 of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity 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. See for example topological K-theory.

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. They impact the stable homotopy groups of spheres even more tightly than complex K-theory.

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

$U(1)\subset U(2)\subset\cdots\subset U = \bigcup_{k=1}^\infty U(k)$

and similarly for O and Sp. Bott's (now somewhat awkward) 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 $\pi_n^S$ comes via the so called stable J-homomorphism from the (unstable) homotopy groups of the (stable) classical groups to these stable homotopy groups $\pi_n^S$ . Originally described by George 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:

$\pi_k(U)=\pi_{k+2}(U) \,\!$

$\pi_k(O)=\pi_{k+4}(Sp) \,\!$

$\pi_k(Sp)=\pi_{k+4}(O) ,\ \ k=0,1,\dots . \,\!$

Note: The second and third of these isomorphisms intertwine to give the desired 8-fold periodicity results:

$\pi_k(O)=\pi_{k+8}(O) \,\!$

$\pi_k(Sp)=\pi_{k+8}(Sp) ,\ \ k=0,1,\dots . \,\!$

[edit] Further details

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 two-fold loop space, $Ω 2 B U$ 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,

$\Omega^2BU\simeq Z\times BU$

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

$\Omega^2U\simeq U .$

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,

$\Omega^8BO\simeq Z\times BO ;$

or equivalently,

$\Omega^8O\simeq O ,$

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

$\Omega^8BSp\simeq Z\times BSp ;$

or equivalently

$\Omega^8Sp\simeq Sp .$

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.

Bott's original proof used Morse theory; subsequently, many different proofs have been given. One elegant formulation of these results makes use of the following observation that there are natural imbeddings (as closed subgroups)

$O\subset U\subset Sp$ and $Sp\subset U\subset O$ .

The Bott periodicity results then refine to a sequence of homotopy equivalences:

$\Omega U\simeq Z\times BU , \ \ \Omega(Z\times BU)\simeq U ,$ for complex K-theory;

and, for real and quaternionic KO- and KSp-theories:

$\Omega(Z\times BO)\simeq O, \ \ \Omega O\simeq O/U$ ,

$\Omega(O/U)\simeq U/Sp, \ \ \Omega(U/Sp)\simeq Z\times BSp$ ,

$\Omega(Z\times BSp)\simeq Sp, \ \ \Omega Sp\simeq Sp/U$ ,

$\Omega(Sp/U)\simeq U/O, \ \ \Omega(U/O)\simeq Z\times BO$ .

These equivalences immediately yield the Bott periodicity theorems.

[edit] Applications

Bott periodicity says a great deal about th topology of the orthogonal group, in particular the homotopy groups

[edit] References

Bott, R. The Stable Homotopy of the Classical Groups, Ann. Math. 70, 1959, 313–337.
Giffen, C.H. Bott periodicity and the Q-construction, Contemp. Math. 199(1996), 107–124.
Milnor, J. Morse Theory. Princeton University Press, 1969. ISBN 0-691-08008-9.
John Baez "This Week's Finds in Mathematical Physics" week 105

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Categories: Homotopy theory | Topology of Lie groups | Mathematical theorems