Bott periodicity theorem explained

In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by, 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. The J-homomorphism is a homomorphism from the homotopy groups of orthogonal groups to stable homotopy groups of spheres, which causes the period 8 Bott periodicity to be visible in the stable homotopy groups of spheres.

Statement of result

Bott showed that if

O( inf ty) is defined as the inductive limit of the orthogonal groups, then its homotopy groups are periodic:^[1] \pi _{n} (O( inf ty))\simeq\pi _{n+8} (O( inf ty)) and the first 8 homotopy groups are as follows:\begin{align} \pi _{0} (O( inf ty))&\simeq\Z _{2} \\ \pi _{1} (O( inf ty))&\simeq\Z _{2} \\ \pi _{2} (O( inf ty))&\simeq 0 \\ \pi _{3} (O( inf ty))&\simeq\Z \\ \pi _{4} (O( inf ty))&\simeq 0 \\ \pi _{5} (O( inf ty))&\simeq 0 \\ \pi _{6} (O( inf ty))&\simeq 0 \\ \pi _{7} (O( inf ty))&\simeq\Z \end{align}

Context and significance

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 … \subset U = c up ^{inf ty} _{k =1} U(k)

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.

\pi ^{S} _{n} comes via the so-called stable J-homomorphism from the (unstable) homotopy groups of the (stable) classical groups to these stable homotopy groups \pi ^{S} _{n} . 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:

\begin{align} \pi _{k(U)} & = \pi _{k+2} (U) \\ \pi _{k(O)} & = \pi _{k+4} (\operatorname{Sp}) \\ \pi _{k(\operatorname{Sp})} & = \pi _{k+4} (O) && k =0 , 1 ,\ldots \end{align}

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

\begin{align} \pi _{k(O)} & = \pi _{k+8} (O) \\ \pi _{k(\operatorname{Sp})} & = \pi _{k+8} (\operatorname{Sp}), && k =0 , 1 ,\ldots \end{align}

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,

\Omega ^{2BU} of BU. Here, \Omega 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,$$\backslash Omega^2BU\backslash simeq\; \backslash Z\backslash times\; BU$$is essentially (that is, homotopy equivalent to) the union of a countable number of copies of BU. An equivalent formulation is$$\backslash Omega^2U\backslash simeq\; U\; .$$

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,$$\backslash Omega^8BO\backslash simeq\; \backslash Z\; \backslash times\; BO$$or equivalently,$$\backslash Omega^8O\backslash simeq\; O,$$

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$$\backslash Omega^8\backslash operatorname\backslash simeq\; \backslash Z\; \backslash times\; \backslash operatorname\; ;$$or equivalently$$\backslash Omega^8\; \backslash operatorname\backslash simeq\; \backslash operatorname.$$

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.

Over the complex numbers:

U x U \subset U \subset U x U.

Over the real numbers and quaternions:

O x O \subset O \subset U\subset \operatorname{Sp} \subset \operatorname{Sp} x \operatorname{Sp} \subset \operatorname{Sp} \subset U \subset O \subset O x O.

These sequences corresponds to sequences in Clifford algebras – see classification of Clifford algebras; over the complex numbers:

\Complex ⊕ \Complex \subset \Complex \subset \Complex ⊕ \Complex.

Over the real numbers and quaternions:

\R ⊕ \R \subset \R\subset \Complex\subset H \subset H ⊕ H \subset H \subset \Complex \subset \R \subset \R ⊕ \R,

where the division algebras indicate "matrices over that algebra".

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

For complex K-theory:

\begin{align} \Omega U &\simeq \Z x BU = \Z x U/(U x U)\\ \Omega(\Z x BU) &\simeq U = (U x U)/U \end{align}

For real and quaternionic KO- and KSp-theories:

\begin{align} \Omega(\Z x BO) &\simeq O = (O x O)/O & \Omega(\Z x \operatorname{BSp}) &\simeq \operatorname{Sp} = (\operatorname{Sp} x \operatorname{Sp})/\operatorname{Sp}\\ \Omega O &\simeq O/U & \Omega \operatorname{Sp} &\simeq \operatorname{Sp}/U\\ \Omega(O/U) &\simeq U/\operatorname{Sp} & \Omega(\operatorname{Sp}/U) &\simeq U/O\\ \Omega(U/\operatorname{Sp})&\simeq \Z x \operatorname{BSp} = \Z x \operatorname{Sp}/(\operatorname{Sp} x \operatorname{Sp}) & \Omega(U/O) &\simeq \Z x BO = \Z x O/(O x O) \end{align}

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 specific spaces are,^[2] (for groups, the principal homogeneous space is also listed):

Loop space	Quotient	Cartan's label	Description
\Omega ^{0}	\Z x O/(O x O)	BDI	Real Grassmannian
\Omega ^{1}	O = (O x O)/O		Orthogonal group (real Stiefel manifold)
\Omega ^{2}	O/U	DIII	space of complex structures compatible with a given orthogonal structure
\Omega ^{3}	U/ Sp	AII	space of quaternionic structures compatible with a given complex structure
\Omega ^{4}	\Z x Sp /( Sp x Sp )	CII	Quaternionic Grassmannian
\Omega ^{5}	Sp = ( Sp x Sp )/ Sp		Symplectic group (quaternionic Stiefel manifold)
\Omega ^{6}	Sp /U	CI	complex Lagrangian Grassmannian
\Omega ^{7}	U/O	AI	Lagrangian Grassmannian

Proofs

Bott's original proof used Morse theory, which had used earlier to study the homology of Lie groups. Many different proofs have been given.

Notes

1. Web site: Introduction.
2. 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.

References

• • • • . An expository account of the theorem and the mathematics surrounding it.
  • Book: Milnor, J. . Morse Theory . Princeton University Press . 1969 . 0-691-08008-9 .
• Web site: John . Baez . Week 105 . 21 June 1997 . This Week's Finds in Mathematical Physics .