Classifying space for U(n) - AbsoluteAstronomy.com

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 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 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...

U(n) is a space B(U(n)) together with a universal bundle E(U(n)) such that any hermitian bundle on a paracompact space

Paracompact space

In mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by ....

X is the pull-back of E by a map X → B unique up to homotopy.

This space with its universal fibration may be constructed as either

the 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...

of n-planes in an infinite-dimensional complex Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

; or,
the direct limit, with the induced topology, of 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...

s of n planes.

Both constructions are detailed here.

Construction as an infinite Grassmannian

The total space

of the universal bundle

Universal bundle

In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group G, is a specific bundle over a classifying space BG, such that every bundle with the given structure group G over M is a pullback by means of a continuous map-In the CW complex...

is given by

Here, H is an infinite-dimensional complex Hilbert space, the

are vectors in H, and

is the Kronecker delta. The symbol

is the inner product on H. Thus, we have that EU(n) is the space of orthonormal n-frames in H.

The group action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

of U(n) on this space is the natural one. The base space is then

and is the set of 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...

n-dimensional subspaces (or n-planes) in H. That is,

so that V is an n-dimensional vector space.

Case of line bundles

In the case of

, one has

known to be a contractible space.

The base space is then

the infinite-dimensional complex projective space

Complex projective space

In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...

. Thus, the set of isomorphism class

Isomorphism class

An isomorphism class is a collection of mathematical objects isomorphic to each other.Isomorphism classes are often defined if the exact identity of the elements of the set is considered irrelevant, and the properties of the structure of the mathematical object are studied. Examples of this are...

es of circle bundles over a manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

are in one-to-one correspondence with the homotopy classes of maps from

.

One also has the relation that

that is,

is the infinite-dimensional projective unitary group

Projective unitary group

In mathematics, the projective unitary group PU is the quotient of the unitary group U by the right multiplication of its center, U, embedded as scalars....

. See that article for additional discussion and properties.

For a torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

T, which is abstractly isomorphic to

, but need not have a chosen identification, one writes

.

The 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...

is given by numerical polynomial

Numerical polynomial

In mathematics, a numerical polynomial is a polynomial with rational coefficients that takes integer values on integers. They are also called integer-valued polynomials....

s; more details below.

Construction as an inductive limit

Let

be the space of orthonormal families of

vectors in

and let

be the Grassmannian of

-dimensional subvector spaces of

. The total space of the universal bundle can be taken to be the direct limit of the

goes to infinity, while the base space is the direct limit of the

goes to infinity.

Validity of the construction

In this section, we will define the topology on EU(n) and prove that EU(n) is indeed contractible.

Let

be the space of orthonormal families of

vectors in

. The group

acts
freely on

and the quotient is the Grassmannian

-dimensional subvector spaces of

. The map

is a fibre bundle of fibre

. Thus because

is trivial and because of the long exact sequence of the fibration

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...

, we have

whenever

. By taking

big enough, precisely for

, we can repeat the process and get

This last group is trivial for k > n + p. Let

be 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 all the

(with the induced topology). Let

be the direct limit

Direct limit

of all the

(with the induced topology).

Lemma

The group

is trivial for all

.

Proof
Let

be a map from the sphere

to EU(n). As

is compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

,
there exists

such that

is included in

. By taking

big enough,
we see that

is homotopic, with respect to the base point, to the constant map.

In addition,

acts freely on

. The spaces

and

are CW-complexes

CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...

. One can
find a decomposition of these spaces into CW-complexes such that the decomposition of

, resp.

, is induced by restriction of the one for

, resp.

. Thus

(and also

) is a CW-complex. By
Whitehead Theorem

Whitehead theorem

In homotopy theory , the Whitehead theorem states that if a continuous mapping f between topological spaces X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence provided X and Y are connected and have the homotopy-type of CW complexes. This result was proved by J....

and the above Lemma,

is contractible.

Cohomology of

Proposition

The 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...

of the classifying space

is a ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

of polynomials in

variables

where

is of degree

.

Proof
Let us first consider the case

. In this case,

is the circle

and the universal bundle
is

. It is well known that the cohomology of

is isomorphic to

, where

is the Euler class

Euler class

In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is...

of
the

-bundle

, and that the injections

,
for

, are compatible with these presentations of the cohomology of the projective spaces.
This proves the Proposition for

.

In the general case, let

be the subgroup of diagonal matrices. It is a maximal torus

Maximal torus

In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.A torus in a Lie group G is a compact, connected, abelian Lie subgroup of G . A maximal torus is one which is maximal among such subgroups...

. Its
classifying space is

and its cohomology is

, where

is the Euler class

Euler class

of the tautological bundle over the i-th

. The
Weyl group

Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...

acts on

by permuting the diagonal entries, hence it acts on

by
permutation of the factors. The induced action on its cohomology is the permutation of the

's. We deduce

where the

's are the symmetric polynomials in the

's.

K-theory of

The topological K-theory

Topological K-theory

is known explicitly in terms of numerical

Numerical polynomial

In mathematics, a numerical polynomial is a polynomial with rational coefficients that takes integer values on integers. They are also called integer-valued polynomials....

symmetric polynomial

Symmetric polynomial

In mathematics, a symmetric polynomial is a polynomial P in n variables, such that if any of the variables are interchanged, one obtains the same polynomial...

s.

The K-theory reduces to computing

, since K-theory is 2-periodic by the Bott periodicity theorem

Bott periodicity theorem

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...

, and

is a limit of complex manifolds, so it has a CW-structure with only cells in even dimensions, so odd K-theory vanishes.

Thus

, where

, where t is the Bott generator.

is the ring of numerical polynomial

Numerical polynomial

In mathematics, a numerical polynomial is a polynomial with rational coefficients that takes integer values on integers. They are also called integer-valued polynomials....

s in w, regarded as a subring of

, where w is element dual to tautological bundle.

For the n-torus,

is numerical polynomials in n variables.
The map

is onto, via a splitting principle

Splitting principle

In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles.In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computations are well understood for line bundles and for direct...

, as

is the maximal torus

Maximal torus

. The map is the symmetrization map

and the image can be identified as the symmetric polynomials satisfying the integrality condition that

where

is the multinomial coefficient and

contains r distinct integers, repeated

times, respectively.

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.