In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the
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'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free acti ...
for the
unitary group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is an ...
U(''n'') is a space BU(''n'') together with a universal bundle EU(''n'') such that any hermitian bundle on a
paracompact space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...
''X'' is the pull-back of EU(''n'') by a map ''X'' → BU(''n'') unique up to homotopy.
This space with its universal fibration may be constructed as either
# the
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
of ''n''-planes in an infinite-dimensional complex
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
; or,
# the direct limit, with the induced topology, of
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
s of ''n'' planes.
Both constructions are detailed here.
Construction as an infinite Grassmannian
The
total space
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
EU(''n'') of the
universal bundle In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group , is a specific bundle over a classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topo ...
is given by
:
Here, ''H'' denotes an infinite-dimensional complex Hilbert space, the ''e''
''i'' are vectors in ''H'', and
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
. The symbol
is the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
on ''H''. Thus, we have that EU(''n'') is the space of
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
''n''-frames in ''H''.
The
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of U(''n'') on this space is the natural one. The
base space is then
:
and is the set of
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
''n''-dimensional subspaces (or ''n''-planes) in ''H''. That is,
:
so that ''V'' is an ''n''-dimensional vector space.
Case of line bundles
For ''n'' = 1, one has EU(1) = S
∞, which is
known to be a contractible space. The base space is then BU(1) = CP
∞, the infinite-dimensional
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 ...
. Thus, the set of
isomorphism class
In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other.
Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the stru ...
es of
circle bundle
In mathematics, a circle bundle is a fiber bundle where the fiber is the circle S^1.
Oriented circle bundles are also known as principal ''U''(1)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circ ...
s over a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
''M'' are in one-to-one correspondence with the
homotopy class
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
es of maps from ''M'' to CP
∞.
One also has the relation that
:
that is, BU(1) is the infinite-dimensional
projective unitary group In mathematics, the projective unitary group is the quotient group, quotient of the unitary group by the right multiplication of its centre of a group, center, , embedded as scalars.
Abstractly, it is the Holomorphic function, holomorphic isometry ...
. See that article for additional discussion and properties.
For a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
''T'', which is abstractly isomorphic to U(1) × ... × U(1), but need not have a chosen identification, one writes B''T''.
The
topological K-theory
In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early ...
''K''
0(B''T'') is given by
numerical polynomials; more details below.
Construction as an inductive limit
Let ''F
n''(C
''k'') be the space of orthonormal families of ''n'' vectors in C
''k'' and let ''G
n''(C
''k'') be the Grassmannian of ''n''-dimensional subvector spaces of C
''k''. The total space of the universal bundle can be taken to be the direct limit of the ''F
n''(C
''k'') as ''k'' → ∞, while the base space is the direct limit of the ''G''
''n''(C
''k'') as ''k'' → ∞.
Validity of the construction
In this section, we will define the topology on EU(''n'') and prove that EU(''n'') is indeed contractible.
The group U(''n'') acts freely on ''F''
''n''(C
''k'') and the quotient is the Grassmannian ''G''
''n''(C
''k''). The map
:
is a fibre bundle of fibre ''F''
''n''−1(C
''k''−1). Thus because
is trivial and because of the
long exact sequence of the fibration, we have
:
whenever
. By taking ''k'' 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
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
of all the ''F''
''n''(C
''k'') (with the induced topology). Let
:
be the
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
of all the ''G''
''n''(C
''k'') (with the induced topology).
Lemma: The group is trivial for all ''p'' ≥ 1.
Proof: Let γ : S
''p'' → EU(''n''), since S
''p'' is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, there exists ''k'' such that γ(S
''p'') is included in ''F''
''n''(C
''k''). By taking ''k'' big enough, we see that γ is homotopic, with respect to the base point, to the constant map.
In addition, U(''n'') acts freely on EU(''n''). The spaces ''F''
''n''(C
''k'') and ''G''
''n''(C
''k'') are
CW-complexes
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
. One can find a decomposition of these spaces into CW-complexes such that the decomposition of ''F''
''n''(C
''k''), resp. ''G''
''n''(C
''k''), is induced by restriction of the one for ''F''
''n''(C
''k''+1), resp. ''G''
''n''(C
''k''+1). Thus EU(''n'') (and also ''G''
''n''(C
∞)) is a CW-complex. By
Whitehead Theorem
In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping ''f'' between CW complexes ''X'' and ''Y'' induces isomorphisms on all homotopy groups, then ''f'' is a homotopy equivalence. This result w ...
and the above Lemma, EU(''n'') is contractible.
Cohomology of BU(''n'')
Proposition: The cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
of the classifying space ''H*''(BU(''n'')) is a ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
of polynomials
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
in ''n'' variables
''c''1, ..., ''cn'' where ''cp'' is of degree 2''p''.
Proof: Let us first consider the case ''n'' = 1. In this case, U(1) is the circle S
1 and the universal bundle is S
∞ → CP
∞. It is well known that the cohomology of CP
''k'' is isomorphic to
, where ''c''
1 is the
Euler class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of ...
of the U(1)-bundle S
2''k''+1 → CP
''k'', and that the injections CP
''k'' → CP
''k''+1, for ''k'' ∈ N*, are compatible with these presentations of the cohomology of the projective spaces. This proves the Proposition for ''n'' = 1.
There are homotopy fiber sequences
:
Concretely, a point of the total space
is given by a point of the base space
classifying a complex vector space
, together with a unit vector
in
; together they classify
while the splitting
, trivialized by
, realizes the map
representing direct sum with
Applying the
Gysin sequence In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. The Gysin sequence is a useful tool f ...
, one has a long exact sequence
:
where
is the
fundamental class
In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundamen ...
of the fiber
. By properties of the Gysin Sequence,
is a multiplicative homomorphism; by induction,
is generated by elements with
, where
must be zero, and hence where
must be surjective. It follows that
must always be surjective: by the
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
of
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
s, a choice of preimage for each generator induces a multiplicative splitting. Hence, by exactness,
must always be injective. We therefore have
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
s split by a ring homomorphism
:
Thus we conclude