A natural association between every bundle
of a certain type (as a rule, a
vector bundle)
and some cohomology class of the base space
(the so-called characteristic class of the given bundle). Natural
here means that the characteristic class of the bundle induced by a mapping
coincides with the image under
of the characteristic class of the bundle
over
.
The
characteristic class of a manifold
is the cohomology class of the manifold that is
the characteristic class of its tangent bundle. The
characteristic classes of manifolds are connected with important
topological characteristics of manifolds such as orientability, the
Euler characteristic,
the
signature,
etc.
Examples.
Orientability of a bundle.
There is an exact sequence of groups
The mapping
associates with every real vector bundle
the class
,
which is called the
first Stiefel–Whitney class
of
;
here
is the cohomology group with coefficients in the
sheaf of germs of continuous functions with values in
(see
-fibration).
The exact cohomology sequence shows that the group of the bundle
reduces to
,
that is, the bundle is orientable (cf.
Orientation),
if and only if
.
The
first Chern class.
Consider the short exact sequence
where
.
The connecting homomorphism
of the corresponding cohomology sequence associates
with every one-dimensional complex bundle
over
a two-dimensional cohomology class of the base
,
the so-called
first Chern class
of
,
which is denoted by
.
In other words, if the
are the transition functions of
,
then choosing any values for the logarithms
one obtains a two-dimensional integral cocycle
:
and
is, by definition, the cohomology class of this cocycle.
The
spinor structure
(or
spin structure).
There is an exact sequence of groups
where
is a group defined in the theory of Clifford algebras (cf.
Clifford algebra).
The connecting mapping
of the corresponding cohomology sequence is called the
second Stiefel–Whitney class.
The structure group of an orientable vector bundle
can be reduced to
if and only if
.
The
Euler class.
Suppose that the base
of a real vector bundle
is a smooth compact
-dimensional
manifold with (possibly empty) boundary
and that the null section
is in
"general position with itself" .
Suppose that an imbedding
close to and isotopic to
is transversally regular with respect to
.
Then
is a submanifold of
and
,
.
Consequently,
.
The cohomology class dual to
is called the
Euler class
of
and is denoted by
.
The bundle
has a nowhere-vanishing section if and only if
.
If
is connected, if
and if
is the tangent bundle, then
;
consequently,
consists of finitely many points. In this case the class
is determined by an integer, which is denoted by
and coincides with the Euler characteristic of
.
The construction of the Stiefel–Whitney and Chern classes in
the language of the theory of obstructions (see
[6]–[8]
and
Obstruction)
proceeds as follows. Let
be a
Serre fibration
and let
be a connected complex. Then the homotopy type of the fibre
does not depend on
.
If
is the first non-trivial homotopy group of
and if
is simply connected, then the first obstruction to the construction of sections
lies in the group
.
This obstruction
is invariantly associated with
.
Sometimes the invariant
is called the
characteristic class of the fibration
.
Let
be a complex vector bundle over
,
.
For every
another bundle
with fibre
is associated with
(the
complex Stiefel manifold).
From the exact sequences of bundles it follows that
for
,
,
so that
.
This is called the
-th Chern class
of
,
.
If
is a real bundle,
,
then the fibre of
is
.
Since
the class
The
Stiefel–Whitney classes
of a bundle
are defined as
However, if
is non-orientable, then the classes
are well defined only with coefficients in
.
For
the Stiefel manifold is the sphere
in the real and
in the complex case. The problem of constructing sections of the bundle
is the same as that of constructing non-vanishing sections of the bundle
.
In this case the first obstruction is called the
Euler class
,
in the complex case;
in the real orientable case; and
in the real non-orientable case.
Let
and
be the fibre spaces associated with
whose fibres are the disc
and the sphere
.
If
is the null section, then
,
where
is the
Thom class.
Let
be
,
the field of real numbers, or
,
the field of complex numbers, or
,
the field of quaternions. Let
be a multiplicative cohomology theory having the
following property: For every finite-dimensional vector space
over
one can choose in a natural, i.e. functorial, way (relative to imbedding) an element
,
where
is the manifold of all one-dimensional subspaces of
,
,
and
,
such that
,
where
.
For
,
suppose that
coincides with the fundamental class of the (oriented) manifold
.
Let
be a vector bundle (in the sense of
)
over
with fibre
,
,
let
be the
projectivization
of this bundle, that is, the locally trivial bundle over
with fibre
whose space
consists of all one-dimensional subspaces in the fibres of
.
Over the space
there is a one-dimensional bundle whose space consists of all pairs
,
where
is a one-dimensional subspace of a fibre of
,
,
and
is a point in
.
To this bundle corresponds a classifying mapping (cf.
Classifying space)
.
Let
,
.
If the group
is endowed with the structure of an
-module
by means of the homomorphism
,
where
is the projection of the bundle
,
then this module is free and has basis
.
There are uniquely determined homology classes
,
,
for which
For
the conditions imposed on the theory
are satisfied, for example, by the theory
.
In this case the characteristic classes defined above are denoted by
and are called
Stiefel–Whitney classes.
For
one can take as
the ordinary cohomology theory
.
For
the classes defined above are denoted by
and are called the
Chern classes.
Moreover, for
any orientable cohomology theory (cf.
Cohomology;
Generalized cohomology theories)
satisfies the conditions required. For
one may also consider the ordinary theory
.
In this case the classes defined above are denoted by
and are called the
symplectic Pontryagin classes.
As before, let
be one of the fields
or
,
and let
be a cohomology theory satisfying the conditions required above. The
splitting principle:
For an arbitrary vector bundle
(in the sense of
)
over
there exists a space
and a mapping
for which the bundle
over
splits into a direct sum of one-dimensional bundles, and the homomorphism
is a monomorphism.
In particular, if
is the universal complex bundle over
(cf.
Classifying space),
then for
one may take the space
(
factors), where
is a maximal torus in
,
and for
one may take the mapping induced by the inclusion
.
The mapping
is a monomorphism, and the image of
coincides with the ring of all symmetric formal power series in the variables
,
.
For any topological group
the set of all characteristic classes defined for principal
-fibrations
and taking values in a cohomology theory
are in one-to-one correspondence with
,
where
is the
classifying space
of
.
In particular, for vector bundles and for the theory
,
the problem of describing all characteristic classes reduces
to a computation of the cohomology rings
,
,
,
etc.
Let
be a compact Lie group and let
be a maximal torus in
.
The inclusion
induces a mapping
of classifying spaces. The space
is homotopically equivalent to the product
,
in which the number of factors equals the dimension of
.
Therefore,
,
where
,
.
On the torus
the Weyl group
acts, where
is the normalizer of
,
consequently, the Weyl group also acts on
.
If
is a connected group and the spaces
and
are torsion free in homology, then the homomorphism
is a monomorphism, and the image of
coincides with the subring of all elements of
that are invariant under the Weyl group
(Borel's theorem).
The group
satisfies the conditions of the theorem. The
diagonal unitary matrices form a maximal torus
in
.
If the elements of a diagonal matrix are denoted by
,
then the Weyl group consists of all permutations of the variables
.
Consequently,
,
where
are the elementary symmetric functions in the variables
and coincide with the Chern classes. The group
also satisfies the conditions of Borel's theorem. The
Weyl group is generated by all permutations of
and arbitrary changes of sign. Consequently,
,
where
are the elementary symmetric functions in
.
The group
does not satisfy the conditions of Borel's theorem; however,
if one considers as coefficient ring an arbitrary ring
containing the element
,
for example,
for odd
or
,
then the theorem modified in this way is valid. A maximal torus of the group
is formed by the matrices of the form
and has dimension
.
The Weyl group is generated by all permutations of
and changes sign for an even number of the symbols when
is even and for an arbitrary number of symbols when
is odd. Therefore,
,
where
are the elementary symmetric functions in the variables
,
except the last, and
.
The classes
coincide with the Pontryagin classes (see below),
is the Euler class;
.
The classes
,
,
are called
Wu generators.
They are not characteristic classes (since they do not lie in
),
but any characteristic class can be expressed in terms of them as
a symmetric formal power series, and any symmetric formal power series in
specifies a characteristic class. For example, to the Euler class
there corresponds the product
.
The element (formal power series)
is symmetric and gives as characteristic class an inhomogeneous element of the ring
,
which is denoted by
and is called the
Chern character.
The Chern character is
"additive-additive"
and
"multiplicative-multiplicative" ,
i.e.
Chern classes and curvature.
Suppose that the base
of an
-dimensional
vector bundle
is a smooth manifold and that in
an arbitrary
affine connection
is given. If a local trivialization of
is fixed in a neighbourhood of some point of
the base, then in this neighbourhood the curvature of the given connection is a
-form
with values in the vector space
of complex
-matrices.
Under a change of the local trivialization of the bundle, the values of the form
are transformed by the rule
,
where
is the transition matrix from one trivialization to the other. If
is a homogeneous polynomial of degree
,
then
is a
-valued
exterior form of degree
.
If, in addition, the polynomial
is invariant under the action
then the form
does not depend on the local trivializations and is a
-valued
exterior form on the whole manifold
.
It can be shown that
and that a change of the connection changes
only by an exact form. Since the coefficients of the trace
of the characteristic polynomial of the matrix
are invariant, by setting
,
one obtains the cohomology class
.
Here
,
where
are the Chern classes with complex coefficients.
The
Pontryagin classes
of a real vector bundle
are defined as the classes
,
where
is the complexification of the bundle
.
(For another definition, see
[5].)
Suppose that the base
of an
-dimensional
bundle
is an
-dimensional
manifold with boundary and that
is an integer-valued non-decreasing function of the argument
.
A system of vectors
is called a
lifting
of
if
for all
.
Suppose that in the bundle sections
in general position are chosen. The subset
of the base is a pseudo-manifold of codimension
.
It realizes a relative homology class in
,
and the homology class dual to it in
is a characteristic class of the bundle
.
The class
is obtained if for
one takes the function
The Pontryagin classes can be expressed in terms of the curvature of the
connection of a real bundle, just as this was done for the Chern classes.
For an arbitrary graded
-algebra
,
let
be the group (under multiplication) of series of the form
,
.
A
multiplicative sequence
is a sequence of polynomials
,
,
such that the correspondence
is a group homomorphism
for any graded
-algebra
.
In particular,
is homogeneous of degree
if
.
If
,
then
is the group of formal power series starting from 1. For any
there exists a unique multiplicative sequence
with
.
Moreover,
Here
,
,
the summation being over all partitions of
,
that is,
,
,
.
The multiplicative sequence defined by the series
where
are the Bernoulli numbers, is usually denoted by
.
Let
be a manifold, let
,
and let
be the complete Pontryagin class. The rational number
is called the
-genus
of
.
The
-genera
of bordant manifolds (cf.
Bordism)
are equal. If
is not divisible by 4, then
.
If
is a closed manifold of dimension
,
then
,
where
is the signature of the quadratic intersection form on
(
Hirzebruch's signature theorem).
Many special multiplicative sequences are important for
applications, for example, the series of
gives a multiplicative sequence
.
For a complex bundle
the class defined by
,
is called the
Todd class
of
.
The Todd class is connected with the Chern character
in the following way:
where
is the Thom class in
-theory
and
is the
Thom isomorphism
in
.
For a real bundle
the class defined by
is called the
index class.
The following
index theorem
holds (the
Atiyah–Singer index theorem):
The index of an elliptic operator
on a compact manifold
of dimension
is equal to
where
is the
Thom space
of the tangent bundle and
is the class of the symbol of the operator
.
The characteristic classes of a spherical bundle are in one-to-one
correspondence with the cohomology spaces of the classifying space
.
For an odd prime number
,
in dimensions less than
,
where for all
the classes
can be expressed by the formula
;
here the
are the Steenrod cyclic reduced powers (cf.
Steenrod reduced power),
is the Thom isomorphism, and
is an exterior
-algebra
(Milnor's theorem).
The classes
are precise analogues of the Stiefel–Whitney classes, and,
just as the latter, can be regarded as characteristic
classes of spherical bundles or as cohomology classes of the space
.
Finally,
,
where
is the Euler class and
.
It can be shown that the formula quoted above concerning
is not true even in dimension
:
,
and a generator of this group cannot be expressed in terms of
and
,
that is, it is the first
exotic characteristic class.