Bott periodicity theorem
Encyclopedia
In mathematics
, the Bott periodicity theorem describes a periodicity in the homotopy group
s of classical group
s, discovered by , 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) 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.
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 (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 Sp. In this context, stable refers to taking the union U (also known as the direct limit
) of the sequence of inclusions
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
comes via the so called stable J-homomorphism
from the (unstable) homotopy groups of the (stable) classical groups to these stable homotopy groups
. 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:
Note: The second and third of these isomorphisms intertwine to give the 8-fold periodicity results:
, 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, 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,
is essentially (that is, homotopy equivalent to) the union of a countable number of copies of BU. An equivalent formulation is
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,
or equivalently,
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
or equivalently
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.
The loop spaces in Bott periodicity are then homotopy equivalent to the symmetric space
s of successive quotients, with additional discrete factors of
.
Over the complex numbers:
Over the real numbers and quaternions:
These sequences corresponds to sequences in Clifford algebra
s – 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:
For real and quaternionic KO- and KSp-theories:
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,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. (for groups, the principal homogeneous space
is also listed):
, which had used earlier to study the homology of Lie groups. Many different proofs have been given.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Bott periodicity theorem describes a periodicity in the homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s of classical group
Classical group
In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...
s, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
of stable complex vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
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
Unitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
. See for example topological K-theory
Topological K-theory
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general topological spaces, by means of ideas now recognised as K-theory that were introduced by Alexander Grothendieck...
.
There are corresponding period-8 phenomena for the matching theories, (real) KO-theory and (quaternionic) KSp-theory, associated to the real orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
and the quaternionic symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
, respectively. The J-homomorphism
J-homomorphism
In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by , extending a construction of .-Definition:...
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 groupHomotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s of sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
s, which would be expected to play the basic part in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
by analogy with homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
, have proved elusive (and the theory is complicated). The subject of stable homotopy theory
Stable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
was conceived as a simplification, by introducing the suspension (smash product
Smash product
In mathematics, the smash product of two pointed spaces X and Y is the quotient of the product space X × Y under the identifications ∼ for all x ∈ X and y ∈ Y. The smash product is usually denoted X ∧ Y...
with a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
) 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
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
with characteristic class
Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
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 Sp. In this context, stable refers to taking the union U (also known as the direct limit
Direct limit
In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...
) of the sequence of inclusions
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
Stable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
groups.
The important connection of Bott periodicity with the stable homotopy groups of spheres
comes via the so called stable J-homomorphismJ-homomorphism
In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by , extending a construction of .-Definition:...
from the (unstable) homotopy groups of the (stable) classical groups to these stable homotopy groups
. Originally described by George W. WhiteheadGeorge W. Whitehead
George William Whitehead, Jr. was a professor of mathematics at the Massachusetts Institute of Technology, a member of the United States National Academy of Sciences, and a Fellow of the American Academy of Arts and Sciences. He is known for his work on algebraic topology...
, 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:
Note: The second and third of these isomorphisms intertwine to give the 8-fold periodicity results:
Loop spaces and classifying spaces
For the theory associated to the infinite unitary groupUnitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
, U, the space BU is the classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
for stable complex vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s (a Grassmannian
Grassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...
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
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
construction. Bott periodicity states that this double loop space is essentially BU again; more precisely,
is essentially (that is, homotopy equivalent to) the union of a countable number of copies of BU. An equivalent formulation is
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
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
, O, the space BO is the classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
for stable real vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s. In this case, Bott periodicity states that, for the 8-fold loop space,
or equivalently,
which yields the consequence that KO-theory is an 8-fold periodic theory. Also, for the infinite symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
, Sp, the space BSp is the classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
for stable quaternionic vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s, and Bott periodicity states that
or equivalently
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
Symmetric space
A symmetric space is, in differential geometry and representation theory, a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point...
s of successive quotients, with additional discrete factors of
.Over the complex numbers:
Over the real numbers and quaternions:
These sequences corresponds to sequences in Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
s – see classification of Clifford algebras
Classification of Clifford algebras
In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras have been completely classified...
; 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:
For real and quaternionic KO- and KSp-theories:
The resulting spaces are homotopy equivalent to the classical reductive symmetric space
Symmetric space
A symmetric space is, in differential geometry and representation theory, a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point...
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,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. (for groups, the principal homogeneous space
Principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial...
is also listed):
| Loop space | Quotient | Cartan's label | Description |
|---|---|---|---|
 |
 |
BDI | Real Grassmannian Grassmannian In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological... |
 |
 |
Orthogonal group Orthogonal group In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication... (real Stiefel manifold Stiefel manifold In mathematics, the Stiefel manifold Vk is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. It is named after Swiss mathematician Eduard Stiefel... ) |
|
 |
 |
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 Grassmannian In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological... |
 |
 |
Symplectic group Symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually... (quaternionic Stiefel manifold Stiefel manifold In mathematics, the Stiefel manifold Vk is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. It is named after Swiss mathematician Eduard Stiefel... ) |
|
 |
 |
CI | complex Lagrangian Grassmannian Lagrangian Grassmannian In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is n/2 . It may be identified with the homogeneous space... |
 |
 |
AI | Lagrangian Grassmannian Lagrangian Grassmannian In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is n/2 . It may be identified with the homogeneous space... |
Proofs
Bott's original proof used Morse theoryMorse theory
In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
, which had used earlier to study the homology of Lie groups. Many different proofs have been given.
Applications
- Bott periodicity expresses a considerable amount regarding the topology of the orthogonal group, in particular the homotopy groups