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

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.

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 -dimensional complex" Dokl. Akad. Nauk SSSR , 35 (1942) pp. 35–39 (In Russian)
[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