# Characteristic class

A natural association between every bundle $\xi = ( E, p, B)$ of a certain type (as a rule, a vector bundle) and some cohomology class of the base space $B$ (the so-called characteristic class of the given bundle). Natural here means that the characteristic class of the bundle induced by a mapping $f: B ^ { \prime } \rightarrow B$ coincides with the image under $f ^ {*} : H ^ {*} ( B) \rightarrow H ^ {*} ( B ^ { \prime } )$ of the characteristic class of the bundle $\xi$ over $B$. 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.

## Contents

### Examples.

Orientability of a bundle. There is an exact sequence of groups

$$1 \rightarrow \mathop{\rm SO} _ {n} ( \mathbf R ) \rightarrow \ O _ {n} ( \mathbf R ) \mathop \rightarrow \limits ^ {\rm det} \ \mathbf Z _ {2} \rightarrow 1.$$

The mapping

$$w _ {1} = \ ( \mathop{\rm det} ) _ {*} : \ H ^ {1} ( B; O _ {n} ( \mathbf R )) \rightarrow \ H ^ {1} ( B; \mathbf Z _ {2} )$$

associates with every real vector bundle $\xi$ the class $w _ {1} ( \xi )$, which is called the first Stiefel–Whitney class of $\xi$; here $H ^ {1} ( B; O _ {n} ( \mathbf R ))$ is the cohomology group with coefficients in the sheaf of germs of continuous functions with values in $O _ {n} ( \mathbf R )$ (see $G$-fibration). The exact cohomology sequence shows that the group of the bundle $\xi$ reduces to $\mathop{\rm SO} _ {n} ( \mathbf R )$, that is, the bundle is orientable (cf. Orientation), if and only if $w _ {1} ( \xi ) = 0$.

The first Chern class. Consider the short exact sequence

$$0 \rightarrow \mathbf Z \rightarrow \mathbf C \mathop \rightarrow \limits ^ {\rm exp} \mathbf C ^ {0} \rightarrow 0,$$

where $\mathbf C ^ {0} = \mathbf C \setminus \{ 0 \}$. The connecting homomorphism $\delta : H ^ {1} ( B; \mathbf C ^ {0} ) \rightarrow H ^ {2} ( B; \mathbf Z )$ of the corresponding cohomology sequence associates with every one-dimensional complex bundle $\xi$ over $B$ a two-dimensional cohomology class of the base $B$, the so-called first Chern class of $\xi$, which is denoted by $c _ {1} ( \xi )$. In other words, if the $g _ {\alpha \beta } : u _ \alpha \cap u _ \beta \rightarrow \mathbf C ^ {0}$ are the transition functions of $\xi$, then choosing any values for the logarithms $\mathop{\rm ln} g _ {\alpha \beta }$ one obtains a two-dimensional integral cocycle $\{ k _ {\alpha \beta \gamma } \}$:

$$k _ {\alpha \beta \gamma } = \ \frac{1}{2 \pi i } ( \mathop{\rm ln} g _ {\alpha \beta } + \mathop{\rm ln} g _ {\beta \gamma } + \mathop{\rm ln} g _ {\gamma \alpha } )$$

and $c _ {1} ( \xi )$ is, by definition, the cohomology class of this cocycle.

The spinor structure (or spin structure). There is an exact sequence of groups

$$1 \rightarrow \mathbf Z _ {2} \rightarrow \ \mathop{\rm Spin} _ {n} ( \mathbf R ) \rightarrow \ \mathop{\rm SO} _ {n} ( \mathbf R ) \rightarrow 1,$$

where $\mathop{\rm Spin} _ {n} ( \mathbf R )$ is a group defined in the theory of Clifford algebras (cf. Clifford algebra). The connecting mapping $w _ {2} : H ^ {1} ( B; \mathop{\rm SO} _ {n} ( \mathbf R )) \rightarrow H ^ {2} ( B; \mathbf Z _ {2} )$ of the corresponding cohomology sequence is called the second Stiefel–Whitney class. The structure group of an orientable vector bundle $\xi$ can be reduced to $\mathop{\rm Spin} _ {n} ( \mathbf R )$ if and only if $w _ {2} ( \xi ) = 0$.

The Euler class. Suppose that the base $B$ of a real vector bundle $\xi = ( E, p, B)$ is a smooth compact $N$-dimensional manifold with (possibly empty) boundary $\partial B$ and that the null section $i: B \rightarrow E$ is in "general position with itself" . Suppose that an imbedding $i ^ \prime : B \rightarrow E$ close to and isotopic to $i$ is transversally regular with respect to $i ( B) \subset E$. Then $X = i ^ {-1} ( i ^ \prime ( B) \cap i ( B))$ is a submanifold of $B$ and $\partial X \subset \partial B$, $\mathop{\rm codim} X = n = \mathop{\rm dim} \xi$. Consequently, $[ X] \in H _ {N - n } ( B, \partial B)$. The cohomology class dual to $[ X]$ is called the Euler class of $\xi$ and is denoted by $e ( \xi ) \in H ^ {n} ( B)$. The bundle $\xi$ has a nowhere-vanishing section if and only if $e ( \xi ) = 0$. If $B$ is connected, if $\partial B = \emptyset$ and if $\xi$ is the tangent bundle, then $\mathop{\rm dim} X = 0$; consequently, $X$ consists of finitely many points. In this case the class $[ X] \subset H _ {0} ( B)$ is determined by an integer, which is denoted by $\chi ( B)$ and coincides with the Euler characteristic of $B$.

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 $\eta : E \rightarrow ^ {p} B$ be a Serre fibration and let $B$ be a connected complex. Then the homotopy type of the fibre $F = p ^ {-1} ( x)$ does not depend on $x \in B$. If $\pi _ {q} ( F )$ is the first non-trivial homotopy group of $F$ and if $B$ is simply connected, then the first obstruction to the construction of sections $s: B \rightarrow E$ lies in the group $H ^ {q + 1 } ( B; \pi _ {q} ( F ))$. This obstruction $\kappa ( \eta )$ is invariantly associated with $\eta$. Sometimes the invariant $\kappa ( \eta )$ is called the characteristic class of the fibration $\eta$. Let $\xi$ be a complex vector bundle over $B$, $\mathop{\rm dim} \xi = n$. For every $1 \leq q \leq n$ another bundle $\eta ^ {q}$ with fibre $U _ {n} / U _ {q - 1 }$ is associated with $\xi$ (the complex Stiefel manifold). From the exact sequences of bundles it follows that $\pi _ {i} ( U _ {n} / U _ {q - 1 } ) = 0$ for $i < 2q - 1$, $\pi _ {2q - 1 } ( U _ {n} / U _ {q - 1 } ) = \mathbf Z$, so that $\kappa ( \eta ^ {q} ) \in H ^ {2q} ( B)$. This is called the $q$-th Chern class of $\xi$, $c _ {q} ( \xi ) = \kappa ( \eta ^ {q} )$.

If $\xi$ is a real bundle, $F = \mathbf R ^ {n}$, then the fibre of $\eta ^ {q}$ is $O _ {n} /O _ {q - 1 }$. Since

$$\pi _ {q - 1 } ( O _ {n} /O _ {q - 1 } ) = \ \left \{ \begin{array}{l} \mathbf Z _ {2} \\ \mathbf Z \end{array} \ \ \begin{array}{l} \textrm{ if } q \textrm{ is even and } q < n, \\ \textrm{ if } q \textrm{ is odd or } q = n; \end{array} \right .$$

$$\pi _ {i} ( O _ {n} /O _ {q - 1 } ) = 0 \ \textrm{ for } i < q - 1,$$

the class

$$\kappa ( \eta ^ {q} ) \in \left \{ \begin{array}{l} H ^ {q} ( B) \\ H ^ {q} ( B; \mathbf Z _ {2} ) \end{array} \ \ \begin{array}{l} \textrm{ if } q \textrm{ is } \textrm{ odd } \textrm{ or } q = n, \\ \textrm{ if } q \textrm{ is } \textrm{ even } and q < n. \end{array} \right .$$

The Stiefel–Whitney classes of a bundle $\xi$ are defined as

$$w _ {q} ( \xi ) = \ \kappa ( \eta ^ {q} ) \ \mathop{\rm mod} 2 \in H ^ {q} ( B; \mathbf Z _ {2} ).$$

However, if $\xi$ is non-orientable, then the classes $\kappa ( \eta ^ {q} )$ are well defined only with coefficients in $\mathbf Z _ {2}$.

For $q = n$ the Stiefel manifold is the sphere $S ^ {n - 1 }$ in the real and $S ^ {2q - 1 }$ in the complex case. The problem of constructing sections of the bundle $\eta ^ {n}$ is the same as that of constructing non-vanishing sections of the bundle $\xi$. In this case the first obstruction is called the Euler class $e ( \xi )$,

$$e ( \xi ) = \ c _ {n} ( \xi ) \in H ^ {2n} ( B)$$

in the complex case;

$$e ( \xi ) = \ \kappa ( \eta ^ {n} ) \in H ^ {n} ( B)$$

in the real orientable case; and

$$e ( \xi ) = \ w _ {n} ( \xi ) \in H ^ {n} ( B; \mathbf Z _ {2} )$$

in the real non-orientable case.

Let $E _ {D}$ and $E _ {S}$ be the fibre spaces associated with $\xi$ whose fibres are the disc $D ^ {n}$ and the sphere $S ^ {n - 1 }$. If $i: B \rightarrow E _ {D}$ is the null section, then $e ( \xi ) = i ^ {*} ( u)$, where $u \in H ^ {n} ( E _ {D} , E _ {S} )$ is the Thom class. Let $F$ be $\mathbf R$, the field of real numbers, or $\mathbf C$, the field of complex numbers, or $\mathbf H$, the field of quaternions. Let $h ^ {*}$ be a multiplicative cohomology theory having the following property: For every finite-dimensional vector space $V$ over $F$ one can choose in a natural, i.e. functorial, way (relative to imbedding) an element $\alpha _ {V} \in h ^ {d} ( P ( V ))$, where $P ( V )$ is the manifold of all one-dimensional subspaces of $V$, $P ( V ) = F P ^ { \mathop{\rm dim} V - 1 }$, and $d = \mathop{\rm dim} _ {\mathbf R } F$, such that $h ^ {*} ( P ( V )) = h ^ {*} ( \mathop{\rm pt} ) [ \alpha _ {V} ]/( \alpha _ {V} ^ {n} )$, where $n = \mathop{\rm dim} V$. For $\mathop{\rm dim} V = 2$, suppose that $\alpha _ {V}$ coincides with the fundamental class of the (oriented) manifold $P ( V)$.

Let $\xi = ( E, p, B)$ be a vector bundle (in the sense of $F$) over $B$ with fibre $V$, $\mathop{\rm dim} \xi = n$, let $P ( \xi )$ be the projectivization of this bundle, that is, the locally trivial bundle over $B$ with fibre $P ( V )$ whose space $P ( E)$ consists of all one-dimensional subspaces in the fibres of $\xi$. Over the space $P ( E)$ there is a one-dimensional bundle whose space consists of all pairs $( l, x)$, where $l$ is a one-dimensional subspace of a fibre of $\xi$, $l \in P ( E)$, and $x$ is a point in $l$. To this bundle corresponds a classifying mapping (cf. Classifying space) $j: P ( E) \rightarrow P ( V )$. Let $a = j ^ {*} ( \alpha _ {V} )$, $a \in h ^ {d} ( P ( E))$. If the group $h ^ {*} ( P ( E))$ is endowed with the structure of an $h ^ {*} ( B)$-module by means of the homomorphism $\pi ^ {*}$, where $\pi : P ( E) \rightarrow B$ is the projection of the bundle $P ( \xi )$, then this module is free and has basis $1, a \dots a ^ {n - 1 }$. There are uniquely determined homology classes $\sigma _ {1} ( \xi ) \dots \sigma _ {n} ( \xi )$, $\sigma _ {i} ( \xi ) \in h ^ {di} ( B)$, for which

$$a ^ {n} - \pi ^ {*} ( \sigma _ {1} ( \xi )) a ^ {n - 1 } + \dots + (- 1) ^ {n} \pi ^ {*} ( \sigma _ {n} ( \xi )) = 0.$$

For $F = \mathbf R$ the conditions imposed on the theory $h ^ {*}$ are satisfied, for example, by the theory $H ^ {*} ( \cdot ; \mathbf Z _ {2} )$. In this case the characteristic classes defined above are denoted by $w _ {1} \dots w _ {n}$ and are called Stiefel–Whitney classes. For $F = \mathbf C$ one can take as $h ^ {*}$ the ordinary cohomology theory $H ^ {*}$. For $H ^ {*}$ the classes defined above are denoted by $c _ {1} \dots c _ {n}$ and are called the Chern classes. Moreover, for $F = \mathbf C$ any orientable cohomology theory (cf. Cohomology; Generalized cohomology theories) satisfies the conditions required. For $F = \mathbf H$ one may also consider the ordinary theory $H ^ {*}$. In this case the classes defined above are denoted by $p _ {1} ^ {s} \dots p _ {n} ^ {s}$ and are called the symplectic Pontryagin classes.

As before, let $F$ be one of the fields $\mathbf R , \mathbf C$ or $\mathbf H$, and let $h ^ {*}$ be a cohomology theory satisfying the conditions required above. The splitting principle: For an arbitrary vector bundle $\xi$ (in the sense of $F$) over $B$ there exists a space $B ^ { \prime }$ and a mapping $f: B ^ { \prime } \rightarrow B$ for which the bundle $f ^ {*} \xi$ over $B ^ { \prime }$ splits into a direct sum of one-dimensional bundles, and the homomorphism $f ^ {*} : h ^ {*} ( B) \rightarrow h ^ {*} ( B ^ { \prime } )$ is a monomorphism.

In particular, if $\xi$ is the universal complex bundle over $BU _ {n}$ (cf. Classifying space), then for $B ^ { \prime }$ one may take the space $BT _ {n} = \mathbf C P ^ \infty \times \dots \times \mathbf C P ^ \infty$ ($n$ factors), where $T _ {n}$ is a maximal torus in $U _ {n}$, and for $f: BT _ {n} \rightarrow BU _ {n}$ one may take the mapping induced by the inclusion $T _ {n} \subset U _ {n}$. The mapping

$$f ^ {*} : \ H ^ {**} ( BU _ {n} ) \rightarrow \ H ^ {**} ( BT _ {n} ) = \ \mathbf Z [[ x _ {1} \dots x _ {n} ]]$$

is a monomorphism, and the image of $f ^ {*}$ coincides with the ring of all symmetric formal power series in the variables $x _ {1} \dots x _ {n}$, $\mathop{\rm dim} x _ {i} = 2$.

For any topological group $G$ the set of all characteristic classes defined for principal $G$-fibrations and taking values in a cohomology theory $h ^ {*}$ are in one-to-one correspondence with $h ^ {*} ( BG)$, where $BG$ is the classifying space of $G$. In particular, for vector bundles and for the theory $H ^ {*}$, the problem of describing all characteristic classes reduces to a computation of the cohomology rings $H ^ {*} ( BO _ {n} )$, $H ^ {*} ( B \mathop{\rm SO} _ {n} )$, $H ^ {*} ( BU _ {n} )$, etc.

Let $G$ be a compact Lie group and let $T$ be a maximal torus in $G$. The inclusion $T \subset G$ induces a mapping $BT \rightarrow BG$ of classifying spaces. The space $BT$ is homotopically equivalent to the product $\mathbf C P ^ \infty \times \dots \times \mathbf C P ^ \infty$, in which the number of factors equals the dimension of $T$. Therefore, $H ^ {**} ( BT) = \mathbf Z [[ x _ {1} \dots x _ {n} ]]$, where $n = \mathop{\rm dim} T$, $\mathop{\rm dim} x _ {i} = 2$. On the torus $T$ the Weyl group $\Phi ( G) = N ( T)/T$ acts, where $N ( T)$ is the normalizer of $T$, consequently, the Weyl group also acts on $BT$. If $G$ is a connected group and the spaces $G$ and $G/T$ are torsion free in homology, then the homomorphism $\rho ^ {*} : H ^ {**} ( BG ) \rightarrow H ^ {**} ( BT)$ is a monomorphism, and the image of $\rho ^ {*}$ coincides with the subring of all elements of $H ^ {**} ( BT) = Z [[ x _ {1} \dots x _ {n} ]]$ that are invariant under the Weyl group (Borel's theorem).

The group $U _ {n}$ satisfies the conditions of the theorem. The diagonal unitary matrices form a maximal torus $T _ {n}$ in $U _ {n}$. If the elements of a diagonal matrix are denoted by $t _ {1} \dots t _ {n}$, then the Weyl group consists of all permutations of the variables $t _ {1} \dots t _ {n}$. Consequently, $H ^ {**} ( BU _ {n} ) = \mathbf Z [[ c _ {1} \dots c _ {n} ]]$, where $c _ {1} \dots c _ {n}$ are the elementary symmetric functions in the variables $x _ {1} \dots x _ {n} \in H ^ {2} ( BT _ {n} )$ and coincide with the Chern classes. The group $\mathop{\rm Sp} _ {n}$ also satisfies the conditions of Borel's theorem. The Weyl group is generated by all permutations of $x _ {1} \dots x _ {n}$ and arbitrary changes of sign. Consequently, $H ^ {**} ( B \mathop{\rm Sp} _ {n} ) = \mathbf Z [[ \sigma _ {1} \dots \sigma _ {n} ]]$, where $\sigma _ {1} \dots \sigma _ {n}$ are the elementary symmetric functions in $x _ {1} ^ {2} \dots x _ {n} ^ {2}$. The group $\mathop{\rm SO} _ {n}$ does not satisfy the conditions of Borel's theorem; however, if one considers as coefficient ring an arbitrary ring $\Lambda$ containing the element $1/2$, for example, $\mathbf Z /p \mathbf Z$ for odd $p$ or $\mathbf Q$, then the theorem modified in this way is valid. A maximal torus of the group $\mathop{\rm SO} _ {n}$ is formed by the matrices of the form

$$\left \| \begin{array}{lr} \cos \alpha _ {1} &- \sin \alpha _ {1} \\ \sin \alpha _ {1} &\cos \alpha _ {1} \\ \end{array} \ \right \| \oplus \left \| \begin{array}{lr} \cos \alpha _ {2} &- \sin \alpha _ {2} \\ \sin \alpha _ {2} &\cos \alpha _ {2} \\ \end{array} \ \right \| \oplus \dots$$

and has dimension $[ n/2]$. The Weyl group is generated by all permutations of $\alpha _ {1} \dots \alpha _ {[} n/2]$ and changes sign for an even number of the symbols when $n$ is even and for an arbitrary number of symbols when $n$ is odd. Therefore, $H ^ {**} ( \mathop{\rm SO} _ {2k} ; \Lambda ) = \Lambda [[ p _ {1} \dots p _ {k - 1 } , e]]$, where $p _ {1} \dots p _ {k - 1 }$ are the elementary symmetric functions in the variables $x _ {1} ^ {2} \dots x _ {k} ^ {2}$, except the last, and $e = x _ {1} \dots x _ {k}$. The classes $p _ {1} \dots p _ {k - 1 } , p _ {k} = e ^ {2}$ coincide with the Pontryagin classes (see below), $e$ is the Euler class; $H ^ {**} ( \mathop{\rm SO} _ {2k + 1 } ; \Lambda ) = \Lambda [[ p _ {1} \dots p _ {k} ]]$.

The classes $x _ {i} \in H ^ {2} ( BT _ {n} )$, $1 \leq i \leq n$, are called Wu generators. They are not characteristic classes (since they do not lie in $H ^ {**} ( BU _ {n} )$), but any characteristic class can be expressed in terms of them as a symmetric formal power series, and any symmetric formal power series in $\{ x _ {i} \}$ specifies a characteristic class. For example, to the Euler class $c _ {n}$ there corresponds the product $\prod _ {i = 1 } ^ {n} x _ {i}$.

The element (formal power series) $e ^ {x _ {1} } + \dots + e ^ {x _ {n} } \in \mathbf Z ( x _ {1} \dots x _ {n} ) = H ^ {**} ( BT _ {n} )$ is symmetric and gives as characteristic class an inhomogeneous element of the ring $H ^ {**} ( BU _ {n} )$, which is denoted by $\mathop{\rm ch}$ and is called the Chern character. The Chern character is "additive-additive" and "multiplicative-multiplicative" , i.e.

$$\mathop{\rm ch} ( \xi \oplus \eta ) = \ \mathop{\rm ch} ( \xi ) + \mathop{\rm ch} ( \eta ),$$

$$\mathop{\rm ch} ( \xi \otimes \eta ) = \mathop{\rm ch} ( \xi ) \cup \mathop{\rm ch} ( \eta ).$$

Chern classes and curvature. Suppose that the base $B$ of an $n$-dimensional vector bundle $\xi = ( E, p, B)$ is a smooth manifold and that in $\xi$ an arbitrary affine connection is given. If a local trivialization of $\xi$ is fixed in a neighbourhood of some point of the base, then in this neighbourhood the curvature of the given connection is a $2$-form $\Omega$ with values in the vector space $\mathfrak g \mathfrak l ( \mathbf C , n)$ of complex $( n \times n)$-matrices. Under a change of the local trivialization of the bundle, the values of the form $\Omega$ are transformed by the rule $m \rightarrow gmg ^ {-1}$, where $g \in \mathop{\rm GL} ( \mathbf C , n)$ is the transition matrix from one trivialization to the other. If $\phi : \mathfrak g \mathfrak l ( \mathbf C , n) \rightarrow \mathbf C$ is a homogeneous polynomial of degree $j$, then $\phi \circ \Omega$ is a $\mathbf C$-valued exterior form of degree $2j$. If, in addition, the polynomial $\phi$ is invariant under the action

$$\mathop{\rm GL} ( \mathbf C , n) \times \mathfrak g \mathfrak l ( \mathbf C , n) \rightarrow \ \mathfrak g \mathfrak l ( \mathbf C , n),\ ( g, m) \rightarrow \ g mg ^ {-1} ,$$

then the form $\phi \circ \Omega$ does not depend on the local trivializations and is a $\mathbf C$-valued exterior form on the whole manifold $B$. It can be shown that $d ( \phi \circ \Omega ) = 0$ and that a change of the connection changes $\phi \circ \Omega$ only by an exact form. Since the coefficients of the trace $\mathop{\rm tr} ( m ^ {i} )$ of the characteristic polynomial of the matrix $m$ are invariant, by setting $\phi _ {i} ( m) = \mathop{\rm tr} ( m ^ {i} )$, one obtains the cohomology class $[ \phi _ {i} \circ \Omega ] \in H ^ {2i} ( B; \mathbf C )$. Here $[ \phi _ {i} \circ \Omega ] = ( c _ {i} ( \xi )) ^ {\mathbf C }$, where $( c _ {i} ( \xi )) ^ {\mathbf C }$ are the Chern classes with complex coefficients.

The Pontryagin classes of a real vector bundle $\xi$ are defined as the classes $p _ {i} ( \xi ) = (- 1) ^ {i} c _ {2i} ( \xi \otimes \mathbf C ) \in H ^ {4i} ( B)$, where $\xi \otimes \mathbf C$ is the complexification of the bundle $\xi$. (For another definition, see [5].) Suppose that the base $B$ of an $n$-dimensional bundle $\xi = ( E, p, B)$ is an $N$-dimensional manifold with boundary and that $\sigma$ is an integer-valued non-decreasing function of the argument $h = 0 \dots n - 1$. A system of vectors $u _ {1} \dots u _ {m} \in \mathbf R ^ {h}$ is called a lifting of $\sigma$ if $\mathop{\rm dim} \{ u _ {1} \dots u _ {h + \sigma ( h) } \} \leq h$ for all $h = 0 \dots n - 1$. Suppose that in the bundle sections $v _ {1} ( x) \dots v _ {m} ( x)$ in general position are chosen. The subset $\{ {x \in B } : {v _ {1} ( x) \dots v _ {m} ( x) \textrm{ is a lifting of } \sigma } \}$ of the base is a pseudo-manifold of codimension $\sigma ( 0) + \dots + \sigma ( n - 1)$. It realizes a relative homology class in $H _ {N - \sum _ {0} ^ {n - 1 } \sigma ( i) } ( B, \partial B)$, and the homology class dual to it in $H ^ {\sum _ {0} ^ {n - 1 } \sigma ( i) } ( B)$ is a characteristic class of the bundle $\xi$. The class $p _ {r} ( \xi )$ is obtained if for $\sigma$ one takes the function

$$\sigma ( 0) = \dots = \sigma ( n - 2r - 1) = 0,\ \$$

$$\sigma ( n - 2r) = \dots = \sigma ( n - 1) = 2.$$

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 $\mathbf Q$-algebra $A$, let $\Gamma _ {1} A$ be the group (under multiplication) of series of the form $1 + a _ {1} + a _ {2} + \dots$, $\mathop{\rm deg} a _ {i} = i$. A multiplicative sequence is a sequence of polynomials $\{ K _ {i} ( x _ {1} \dots x _ {i} ) \}$, $K _ {i} \in \mathbf Q [ x _ {1} \dots x _ {i} ]$, such that the correspondence

$$a = ( 1 + a _ {1} + \dots ) \rightarrow \ ( 1 + K _ {1} ( a _ {1} ) + K _ {2} ( a _ {1} , a _ {2} ) + \dots ) = K ( a)$$

is a group homomorphism $\Gamma _ {1} A \rightarrow \Gamma _ {1} A$ for any graded $\mathbf Q$-algebra $A$. In particular, $K _ {i} ( x _ {1} \dots x _ {i} )$ is homogeneous of degree $i$ if $\mathop{\rm deg} x _ {j} = j$. If $A = \mathbf Q [ t]$, then $\Gamma _ {1} A$ is the group of formal power series starting from 1. For any $f ( t) \in \Gamma _ {1} ( \mathbf Q [ t])$ there exists a unique multiplicative sequence $K = \{ K _ {i} \}$ with $K ( 1 + t) = f ( t)$. Moreover,

$$K _ {n} ( x _ {1} \dots x _ {n} ) = \ \sum _ {\omega \in \Pi ( n) } \lambda _ {i _ {1} } \dots \lambda _ {i _ {r} } S _ {i _ {1} \dots i _ {r} } ( x _ {1} \dots x _ {n} ).$$

Here $f ( t) = \sum _ {0} ^ \infty \lambda _ {i} t ^ {i}$, $\lambda _ {0} = 1$, the summation being over all partitions of $n$, that is, $\omega = \{ i _ {1} \dots i _ {r} \}$, $i _ {1} \dots i _ {r} \geq 0$, $i _ {1} + \dots + i _ {r} = n$.

The multiplicative sequence defined by the series

$$\frac{\sqrt t }{ \mathop{\rm tanh} \sqrt t } = \ 1 + { \frac{1}{3} } t - \dots + (- 1) ^ {k - 1 } \frac{2 ^ {2k} }{( 2k)! } B _ {k} t ^ {k} + \dots ,$$

where $B _ {k}$ are the Bernoulli numbers, is usually denoted by $L = \{ L _ {i} \}$. Let $M ^ {n}$ be a manifold, let $A = H ^ {4*} ( M, \mathbf Q )$, and let $p ( M) = 1 + p _ {1} ( M) + \dots + p _ {[} n/2] ( M) \in \Gamma _ {1} A$ be the complete Pontryagin class. The rational number $\langle L ( p ( M)), [ M]\rangle$ is called the $L$-genus of $M ^ {n}$. The $L$-genera of bordant manifolds (cf. Bordism) are equal. If $n$ is not divisible by 4, then $\langle L ( p ( M)), [ M]\rangle = 0$. If $M ^ {n}$ is a closed manifold of dimension $n = 4k$, then $\langle L ( p ( M)), [ M]\rangle = I [ M]$, where $I [ M]$ is the signature of the quadratic intersection form on $H _ {2k} ( M, \mathbf Q )$ (Hirzebruch's signature theorem).

Many special multiplicative sequences are important for applications, for example, the series of $f ( t) = t/( 1 - e ^ {-t} )$ gives a multiplicative sequence $T$. For a complex bundle $\xi$ the class defined by $\tau ( \xi ) = T ( c ( \xi )) \in H ^ {*} ( B ( \xi ))$, is called the Todd class of $\xi$. The Todd class is connected with the Chern character $\mathop{\rm ch}$ in the following way:

$$1 = (- 1) ^ {n} \tau ( \xi ) \cdot \phi _ \xi ^ {-1} \ \mathop{\rm ch} \psi _ \xi ( 1) \in H ^ {*} ( B, \mathbf Q ),\ \ n = \mathop{\rm dim} \xi ,$$

where $\psi _ \xi ( 1)$ is the Thom class in $K _ {\mathbf C }$-theory and $\phi _ \xi$ is the Thom isomorphism in $H ^ {*}$. For a real bundle $\xi$ the class defined by $J ( \xi ) = T ( \xi _ {\mathbf C } ) \in H ^ {*} ( B ( \xi ), \mathbf Q )$ is called the index class. The following index theorem holds (the Atiyah–Singer index theorem): The index of an elliptic operator $D$ on a compact manifold $M$ of dimension $n$ is equal to

$$(- 1) ^ {n} \{ \mathop{\rm ch} u \cdot J ( M) \} [ M ^ \tau ],$$

where $M ^ \tau$ is the Thom space of the tangent bundle and $u \in K ( M ^ \tau )$ is the class of the symbol of the operator $D$.

The characteristic classes of a spherical bundle are in one-to-one correspondence with the cohomology spaces of the classifying space $BG _ {n}$.

For an odd prime number $p$, in dimensions less than $2p ( p - 1) - 1$,

$$H ^ {*} ( BG; \mathbf Z _ {p} ) = \ \mathbf Z _ {p} [ q _ {1} , q _ {2} , . . . ] \otimes \Lambda [ \beta q _ {1} , \beta q _ {2} , . . . ],$$

where for all $n$ the classes $q _ {i} \in H ^ {2i ( p - 1) } ( BG _ {n} ; \mathbf Z _ {p} )$ can be expressed by the formula $q _ {i} = \phi ^ {- 1 } P ^ {i} \phi ( 1)$; here the $P ^ {i}$ are the Steenrod cyclic reduced powers (cf. Steenrod reduced power), $\phi$ is the Thom isomorphism, and $\Lambda [ \beta q _ {1} , \beta q _ {2} , . . . ]$ is an exterior $\mathbf Z _ {p}$-algebra (Milnor's theorem).

The classes $q _ {i}$ 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 $BG$. Finally, $H ^ {*} ( BSG _ {2k} ; \mathbf Q ) = \mathbf Q [ e]$, where $e$ is the Euler class and $H ^ {*} ( BG _ {2k} ; \mathbf Q ) = \mathbf Q [ e ^ {2} ]$.

It can be shown that the formula quoted above concerning $H ^ {*} ( BG; \mathbf Z _ {p} )$ is not true even in dimension $2p ( p - 1) - 1$: $H ^ {2p ( p - 1) - 1 } ( BG; \mathbf Z _ {p} ) \cong \mathbf Z _ {p}$, and a generator of this group cannot be expressed in terms of $q _ {i}$ and $\beta q _ {i}$, 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)

Still another way to define (characterize) the Euler class is as follows. Let $\xi$ over $B$ be an oriented $n$-dimensional real vector bundle. Giving an orientation on a vector space $V$ is the same thing as giving a preferred generator of $H ^ {n} ( V, V _ {0} ; \mathbf Z ) \simeq \mathbf Z$, where $V _ {0} = V \setminus \{ 0 \}$ (cf. Orientation). An orientation on $\xi$ therefore defines a generator $u _ {F} \in H ^ {n} ( F, F _ {0} ; \mathbf Z )$ for each fibre $F$ of $\xi$. It is now a theorem (cf. [14], p. 97) that for an oriented $n$-dimensional vector bundle $\xi$ with total space $E$, $H ^ {i} ( E, E _ {0} ; \mathbf Z ) = 0$ for $i < n$ and that $H ^ {n} ( E, E _ {0} ; \mathbf Z )$ contains a unique cohomology class $u$ which restricts to the given $u _ {F}$ for each fibre $F$ (under the inclusion $( F, F _ {0} ) \subset ( E, E _ {0} )$). This $u$ is called the fundamental class or Thom class of $\xi$. Moreover, taking cup products with $u$ induces isomorphisms $H ^ {k} ( E; \mathbf Z ) \rightarrow H ^ {k + n } ( E, E _ {0} ; \mathbf Z )$, cf. also Thom isomorphism. The inclusion $( E, \phi ) \subset ( E, E _ {0} )$ defines homomorphisms $H ^ {n} ( E, E _ {0} ; \mathbf Z ) \rightarrow H ^ {n} ( E; \mathbf Z )$ and $H ^ {n} ( E; \mathbf Z ) \rightarrow H ^ {n} ( B; \mathbf Z )$ (induced by $B \subset E$). The image of $u$ under the composite of these two homomorphisms is the Euler class $e ( \xi )$ of $\xi$. Some elementary properties of the Euler class are: $2e ( \xi ) = 0$ if the fibre dimension is odd, $e ( \xi \oplus \xi ^ \prime ) = e ( \xi ) \cup e ( \xi ^ \prime )$ and $e ( \xi \times \xi ^ \prime ) = e ( \xi ) \times e ( \xi ^ \prime )$, where of course $\xi \times \xi ^ \prime$ is the vector bundle with total space $E \times E ^ \prime$ over the base space $B \times B ^ { \prime }$ and where $\xi \oplus \xi ^ \prime$ has fibre $E _ {x} \oplus E _ {x} ^ { \prime }$ over $x \in B$.
The term fundamental class is used in the following sense. Let $( M, \partial M )$ be an $n$-dimensional manifold with boundary. Then $z \in h _ {n} ( M, \partial M)$ for a (generalized) cohomology theory $h _ {*}$ is a fundamental class if and only if for every $x \in M \setminus \partial M$ one has that $j _ {*} ( z) \in h _ {n} ( M, M \setminus \{ x \} )$ ($\simeq h _ {n} ( \mathbf R ^ {n} , \mathbf R ^ {n} \setminus \{ 0 \} )$ is a generator of $h _ {*} ( M, M \setminus \{ x \} )$ as a module over $h _ {*} ( \mathop{\rm pt} )$. Then, if $h$ is defined by a connected ring spectrum $E$, a compact, triangulable manifold $M$ is orientable if and only if it has a fundamental class.