Olinto De Pretto

In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Template:Harvs, 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.

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

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

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 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:

$\pi _{k}(U)=\pi _{k+2}(U)\,\!$

$\pi _{k}(O)=\pi _{k+4}(\operatorname {Sp} )\,\!$

$\pi _{k}(\operatorname {Sp} )=\pi _{k+4}(O),\ \ k=0,1,\dots .\,\!$

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

$\pi _{k}(O)=\pi _{k+8}(O)\,\!$

$\pi _{k}(\operatorname {Sp} )=\pi _{k+8}(\operatorname {Sp} ),\ \ k=0,1,\dots .\,\!$

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, Ω²BU 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,

$\Omega ^{2}BU\simeq \mathbf {Z} \times BU\,$

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

$\Omega ^{2}U\simeq U.\,$

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,

$\Omega ^{8}BO\simeq \mathbf {Z} \times BO;\,$

or equivalently,

$\Omega ^{8}O\simeq O,\,$

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

$\Omega ^{8}\operatorname {BSp} \simeq \mathbf {Z} \times \operatorname {BSp} ;\,$

or equivalently

$\Omega ^{8}\operatorname {Sp} \simeq \operatorname {Sp} .\,$

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.

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:

$U\times U\subset U\subset U\times U.\,$

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:

$\mathbf {C} \oplus \mathbf {C} \subset \mathbf {C} \subset \mathbf {C} \oplus \mathbf {C} .\,$

Over the real numbers and quaternions:

$\mathbf {R} \oplus \mathbf {R} \subset \mathbf {R} \subset \mathbf {C} \subset \mathbf {H} \subset \mathbf {H} \times \mathbf {H} \subset \mathbf {H} \subset \mathbf {C} \subset \mathbf {R} \subset \mathbf {R} \oplus \mathbf {R} \,$

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:

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

For real and quaternionic KO- and KSp-theories:

${\begin{aligned}\Omega (\mathbf {Z} \times BO)&\simeq O=(O\times O)/O&\Omega (\mathbf {Z} \times \operatorname {BSp} )&\simeq \operatorname {Sp} =(\operatorname {Sp} \times \operatorname {Sp} )/\operatorname {Sp} \\\Omega O&\simeq O/U&\Omega \operatorname {Sp} &\simeq \operatorname {Sp} /U\\\Omega (O/U)&\simeq U/\operatorname {Sp} &\Omega (\operatorname {Sp} /U)&\simeq U/O\\\Omega (U/\operatorname {Sp} )&\simeq \mathbf {Z} \times \operatorname {BSp} =\mathbf {Z} \times \operatorname {Sp} /(\operatorname {Sp} \times \operatorname {Sp} )&\Omega (U/O)&\simeq \mathbf {Z} \times BO=\mathbf {Z} \times O/(O\times O)\end{aligned}}$

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

Proofs

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

References

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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Bott, R. "The periodicity theorem for the classical groups and some of its applications" An expository account of the theorem and the mathematics surrounding it.
• 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