Bott Periodicity Theorem - Geometric Model of Loop Spaces

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:

Over the real numbers and quaternions:

O \times O \subset O\subset U\subset \operatorname{Sp} \subset \operatorname{Sp} \times \operatorname{Sp} \subset \operatorname{Sp}\subset U\subset O \subset O \times O. \,

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

Over the real numbers and quaternions:

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 \mathbf{Z}\times BU = \mathbf{Z}\times U/(U \times U)\\ \Omega(Z\times BU)& \simeq U = (U \times U)/U \end{align}

For real and quaternionic KO- and KSp-theories:

\begin{align} \Omega(\mathbf{Z}\times BO) &\simeq O = (O \times O)/O & \Omega(\mathbf{Z}\times \operatorname{BSp}) &\simeq \operatorname{Sp} = (\operatorname{Sp} \times \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 \mathbf{Z}\times \operatorname{BSp} = \mathbf{Z}\times \operatorname{Sp}/(\operatorname{Sp} \times \operatorname{Sp}) & \Omega(U/O) &\simeq \mathbf{Z}\times BO = \mathbf{Z} \times O/(O \times 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, (for groups, the principal homogeneous space is also listed):

Loop space Quotient Cartan's label Description
BDI Real Grassmannian
Orthogonal group (real Stiefel manifold)
DIII space of complex structures compatible with a given orthogonal structure
AII space of quaternionic structures compatible with a given complex structure
CII Quaternionic Grassmannian
Symplectic group (quaternionic Stiefel manifold)
CI complex Lagrangian Grassmannian
AI Lagrangian Grassmannian

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