Editing Bott periodicity theorem

{{short description|Describes a periodicity in the homotopy groups of classical groups}}
{{Use American English|date=January 2019}}

In [[mathematics]], the '''Bott periodicity theorem''' describes a periodicity in the [[homotopy group]]s of [[classical group]]s, discovered by {{harvs|txt|authorlink=Raoul Bott|first=Raoul |last=Bott|year1=1957|year2=1959}}, which proved to be of foundational significance for much further research, in particular in [[K-theory]] of stable complex [[vector bundle]]s, 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 number|real]]) [[KO-theory]] and ([[quaternion]]ic) [[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 <math>O(\infty)</math> is defined as the [[Direct limit of groups|inductive limit]] of the [[orthogonal groups]], then its [[homotopy groups]] are periodic:<ref>{{cite web |url=http://math.ucr.edu/home/baez/octonions/node1.html |title = Introduction}}</ref>
:<math>\pi_{n}(O(\infty))\simeq\pi_{n+8}(O(\infty))</math> 
and the first 8 homotopy groups are as follows:
:<math>\begin{align}
\pi_{0}(O(\infty))&\simeq\Z_2 \\
\pi_{1}(O(\infty))&\simeq\Z_2 \\
\pi_{2}(O(\infty))&\simeq 0 \\
\pi_{3}(O(\infty))&\simeq\Z \\
\pi_{4}(O(\infty))&\simeq 0 \\
\pi_{5}(O(\infty))&\simeq 0 \\
\pi_{6}(O(\infty))&\simeq 0 \\
\pi_{7}(O(\infty))&\simeq\Z
\end{align}</math>

==Context and significance==
The context of Bott periodicity is that the [[homotopy group]]s of [[sphere]]s, 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 (mathematics)|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 class]]es, 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&nbsp;Sp.  In this context, ''stable'' refers to taking the union ''U'' (also known as the [[direct limit]]) of the sequence of inclusions

:<math>U(1)\subset U(2)\subset\cdots\subset U = \bigcup_{k=1}^\infty U(k)</math>

and similarly for ''O'' and&nbsp;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 theory|stable homotopy]] groups.

The important connection of Bott periodicity with the [[stable homotopy groups of spheres]] <math>\pi_n^S</math> comes via the so-called stable [[J-homomorphism|''J''-homomorphism]] from the (unstable) homotopy groups of the (stable) classical groups to these stable homotopy groups <math>\pi_n^S</math>.  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:

:<math>\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}</math>

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

:<math>\begin{align}
\pi_k(O) &=\pi_{k+8}(O) \\
\pi_k(\operatorname{Sp}) &=\pi_{k+8}(\operatorname{Sp}), && k=0,1,\ldots 
\end{align}</math>

==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 bundle]]s (a [[Grassmannian]] in infinite dimensions).  One formulation of Bott periodicity describes the twofold loop space, <math>\Omega^2BU</math> of ''BU''.  Here, <math>\Omega</math> is the [[loop space]] [[functor]], [[right adjoint]] to [[Suspension (topology)|suspension]] and [[left adjoint]] to the [[classifying space]] construction. Bott periodicity states that this double loop space is essentially ''BU'' again; more precisely,
<math display="block">\Omega^2BU\simeq \Z\times BU</math>
is essentially (that is, [[homotopy equivalence|homotopy equivalent]] to) the union of a countable number of copies of ''BU''.  An equivalent formulation is
<math display="block">\Omega^2U\simeq U .</math>

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 bundle]]s. In this case, Bott periodicity states that, for the 8-fold loop space,
<math display="block">\Omega^8BO\simeq \Z \times BO </math>
or equivalently,
<math display="block">\Omega^8O\simeq O ,</math>

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 bundle]]s, and Bott periodicity states that
<math display="block">\Omega^8\operatorname{BSp}\simeq \Z \times \operatorname{BSp} ;</math>
or equivalently
<math display="block">\Omega^8 \operatorname{Sp}\simeq \operatorname{Sp}.</math>

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 space]]s of successive quotients, with additional discrete factors of '''Z'''.

Over the [[complex number]]s:

:<math> U \times U \subset U \subset U \times U. </math>

Over the real numbers and quaternions:

:<math>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.</math>

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

:<math>\Complex \oplus \Complex \subset \Complex \subset \Complex \oplus \Complex. </math>

Over the real numbers and quaternions:

:<math>\R \oplus \R \subset \R\subset \Complex\subset \mathbb{H} \subset \mathbb{H} \oplus \mathbb{H} \subset \mathbb{H} \subset \Complex \subset \R \subset \R \oplus \R,</math>

where the [[Division algebra|division algebras]] indicate "matrices over that algebra".

[[File:BottPeriodicityClock.gif|thumb|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 Cl<sub>p,q</sub>(<math>\R</math>)=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 Bott periodicity results then refine to a sequence of [[homotopy equivalence]]s:

For complex ''K''-theory:

:<math>\begin{align}
\Omega U           &\simeq \Z\times BU = \Z\times U/(U \times U)\\
\Omega(\Z\times BU) &\simeq U = (U \times U)/U
\end{align}</math>

For real and quaternionic ''KO''- and KSp-theories:

:<math>\begin{align}
\Omega(\Z\times BO) &\simeq O = (O \times O)/O & \Omega(\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 \Z\times \operatorname{BSp} = \Z\times \operatorname{Sp}/(\operatorname{Sp} \times \operatorname{Sp}) & \Omega(U/O) &\simeq \Z\times BO  = \Z \times O/(O \times O)
\end{align}</math>

The resulting spaces are homotopy equivalent to the classical reductive [[symmetric space]]s, 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,<ref group="note">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.''</ref> (for groups, the [[principal homogeneous space]] is also listed):
{| class="wikitable"
! Loop space !! Quotient !! Cartan's label !! Description
|-
| <math>\Omega^0</math> || <math>\Z\times O/(O \times O)</math> || BDI || Real [[Grassmannian]]
|-
| <math>\Omega^1</math> || <math>O = (O \times O)/O</math> || || [[Orthogonal group]] (real [[Stiefel manifold]])
|-
| <math>\Omega^2</math> || <math>O/U</math> || DIII || space of complex structures compatible with a given orthogonal structure
|-
| <math>\Omega^3</math> || <math>U/\mathrm{Sp}</math> || AII || space of quaternionic structures compatible with a given complex structure
|-
| <math>\Omega^4</math> || <math>\Z\times \mathrm{Sp}/(\mathrm{Sp} \times \mathrm{Sp})</math> || CII || Quaternionic  [[Grassmannian]]
|-
| <math>\Omega^5</math> || <math>\mathrm{Sp} = (\mathrm{Sp} \times \mathrm{Sp})/\mathrm{Sp}</math> || || [[Symplectic group]] (quaternionic [[Stiefel manifold]])
|-
| <math>\Omega^6</math> || <math>\mathrm{Sp}/U</math> || CI || complex [[Lagrangian Grassmannian]]
|-
| <math>\Omega^7</math> || <math>U/O</math> || AI || [[Lagrangian Grassmannian]]
|}

==Proofs==
Bott's original proof {{harv|Bott|1959}} used [[Morse theory]], which {{harvtxt|Bott|1956}} had used earlier to study the homology of [[Lie group]]s. Many different proofs have been given.

==Notes==
<references group="note"/>

==References==
{{Reflist}}

*{{Citation | last1=Bott | first1=Raoul | author1-link=Raoul Bott | title=An application of the Morse theory to the topology of Lie-groups | mr=0087035 | year=1956 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=84 | pages=251–281| doi=10.24033/bsmf.1472 | doi-access=free }}
*{{Citation | last1=Bott | first1=Raoul | author1-link=Raoul Bott | title=The stable homotopy of the classical groups | jstor=89403 | mr=0102802 | year=1957 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | volume=43 | issue=10 | pages=933–5 | doi=10.1073/pnas.43.10.933| pmid=16590113 | pmc=528555 | bibcode=1957PNAS...43..933B | doi-access=free }}
*{{Citation | last1=Bott | first1=Raoul | author1-link=Raoul Bott | title=The stable homotopy of the classical groups | jstor=1970106 | mr=0110104 | year=1959 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=70 | issue=2 | pages=313–337 | doi=10.2307/1970106 | pmid=16590113 | pmc=528555 }}
*{{citation |author-link=Raoul Bott |last=Bott|first=Raoul |title=The periodicity theorem for the classical groups and some of its applications |journal=[[Advances in Mathematics]] |volume=4 |issue=3 |year=1970 |pages=353–411 |doi=10.1016/0001-8708(70)90030-7|doi-access=free}}. An expository account of the theorem and the mathematics surrounding it.
*{{citation |last=Giffen |first=C.H. |chapter=Bott periodicity and the Q-construction |volume=199 |pages=107–124 |editor-first=Grzegorz |editor-last=Banaszak |editor2-first=Wojciech |editor2-last=Gajda |editor3-first=Piotr |editor3-last=Krasoń |title=Algebraic K-Theory |chapter-url=https://books.google.com/books?id=Bs0aCAAAQBAJ&pg=PA108 |year=1996 |publisher=American Mathematical Society |series=Contemporary Mathematics |isbn=978-0-8218-0511-4 |mr=1409620 |url=https://www.ams.org/books/conm/199/}}
*{{cite book |first=J. |last=Milnor |title=Morse Theory |publisher=Princeton University Press |year=1969 |isbn=0-691-08008-9 }}
*{{cite web |first=John |last=Baez |title=Week 105 |date=21 June 1997 |work=This Week's Finds in Mathematical Physics |url=http://math.ucr.edu/home/baez/week105.html}}

[[Category:Topology of Lie groups]]
[[Category:Theorems in homotopy theory]]