Characteristic class
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
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The mapping
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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
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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
:
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and is, by definition, the cohomology class of this cocycle.
The spinor structure (or spin structure). There is an exact sequence of groups
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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
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the class
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The Stiefel–Whitney classes of a bundle are defined as
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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
,
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in the complex case;
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in the real orientable case; and
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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
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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
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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
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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.
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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
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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
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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
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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,
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Here ,
, the summation being over all partitions of
, that is,
,
,
.
The multiplicative sequence defined by the series
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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:
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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
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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
,
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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.
References
[1] | A. Borel, "Collected papers" , 1 , Springer (1973) |
[2a] | M.F. Atiyah, I.M. Singer, "The index of elliptic operators, I" Ann. of Math. (2) , 87 (1968) pp. 484–530 |
[2b] | M.F. Atiyah, I.M. Singer, "The index of elliptic operators, II" Ann. of Math (2) , 87 (1968) pp. 531–545 |
[2c] | M.F. Atiyah, I.M. Singer, "The index of elliptic operators, III" Ann. of Math. (2) , 87 (1968) pp. 546–604 |
[2d] | M.F. Atiyah, I.M. Singer, "The index of elliptic operators, IV" Ann. of Math. (2) , 93 (1971) pp. 119–138 |
[2e] | M.F. Atiyah, I.M. Singer, "The index of elliptic operators, V" Ann. of Math. (2) , 93 (1971) pp. 139–149 |
[3] | R. Bott, "Lectures on characteristic classes" , Lectures on algebraic and differential topology , Lect. notes in math. , 279 , Springer (1972) pp. 1–94 |
[4] | J. Milnor, "Lectures on characteristic classes" , Princeton Univ. Press (1957) (Notes by J. Stasheff) |
[5] | L.S. Pontryagin, "Mappings of the three-dimensional sphere into an ![]() |
[6] | E. Stiefel, Comm. Math. Helv. , 8 : 4 (1935) pp. 305–353 |
[7] | H. Whitney, Bull. Amer. Math. Soc. , 43 (1937) pp. 785–805 |
[8] | S.-S. Chern, "Characteristic classes of Hermitian manifolds" Ann. of Math. (2) , 47 : 1 (1946) pp. 85–121 |
[9] | S.P. Novikov, "Topological invariance of Pontryagin classes" Soviet Math. Doklady , 6 : 4 (1965) pp. 921–923 Dokl. Akad. Nauk SSSR , 163 (1965) pp. 298–300 |
[10] | J. Milnor, "On characteristic classes for spherical fibre spaces" Comm. Math. Helv. , 43 : 1 (1968) pp. 51–77 |
[11] | J. Stasheff, "More characteristic classes for spherical fibre spaces" Comm. Math. Helv. , 43 : 1 (1968) pp. 78–86 |
[12a] | A. Borel, F. Hirzebruch, "Characteristic classes and homogeneous spaces, I" Amer. J. Math. , 80 (1958) pp. 458–538 |
[12b] | A. Borel, F. Hirzebruch, "Characteristic classes and homogeneous spaces, II" Amer. J. Math. , 81 (1959) pp. 315–382 |
[12c] | A. Borel, F. Hirzebruch, "Characteristic classes and homogeneous spaces, III" Amer. J. Math. , 82 (1960) pp. 491–504 |
[13] | R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) |
[14] | J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) |
Comments
Still another way to define (characterize) the Euler class is as follows. Let over
be an oriented
-dimensional real vector bundle. Giving an orientation on a vector space
is the same thing as giving a preferred generator of
, where
(cf. Orientation). An orientation on
therefore defines a generator
for each fibre
of
. It is now a theorem (cf. [14], p. 97) that for an oriented
-dimensional vector bundle
with total space
,
for
and that
contains a unique cohomology class
which restricts to the given
for each fibre
(under the inclusion
). This
is called the fundamental class or Thom class of
. Moreover, taking cup products with
induces isomorphisms
, cf. also Thom isomorphism. The inclusion
defines homomorphisms
and
(induced by
). The image of
under the composite of these two homomorphisms is the Euler class
of
. Some elementary properties of the Euler class are:
if the fibre dimension is odd,
and
, where of course
is the vector bundle with total space
over the base space
and where
has fibre
over
.
The term fundamental class is used in the following sense. Let be an
-dimensional manifold with boundary. Then
for a (generalized) cohomology theory
is a fundamental class if and only if for every
one has that
(
is a generator of
as a module over
. Then, if
is defined by a connected ring spectrum
, a compact, triangulable manifold
is orientable if and only if it has a fundamental class.
References
[a1] | D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) |
[a2] | R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. 360ff |
Characteristic class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Characteristic_class&oldid=13274