Pontryagin number

A characteristic number defined for closed oriented manifolds and assuming rational values. Let $x \in H ^ {**} ( \mathop{\rm BO} ; \mathbf Q )$ be an arbitrary (not necessarily homogeneous) stable characteristic class. For a closed oriented manifold $M$ the rational number $x [ M ] = \langle x ( \tau M ) , [ M] \rangle$ is called the Pontryagin number of $M$ corresponding to $x$; here $\tau M$ is the tangent bundle and $[ M]$ is the fundamental class of $M$. The Pontryagin number $x [ M]$ depends only on the homogeneous component of degree $\mathop{\rm dim} M$ of the class $x$. Let $\omega = \{ i _ {1} \dots i _ {k} \}$ be a partition of $n$, i.e. a set of non-negative integers $i _ {1} \dots i _ {k}$ such that $i _ {1} + \dots + i _ {k} = n$ and let $p _ \omega = p _ {i _ {1} } \dots p _ {i _ {k} } \in H ^ {4n} ( \mathop{\rm BO} )$. The rational numbers $p _ \omega [ M]$ are defined for a closed manifold $M$ of dimension $4n$ and all partitions $\omega$ of the number $n$.

The Pontryagin numbers $x[ M] , x [ N]$ of two bordant (in the oriented sense, cf. Bordism) manifolds $M , N$ are equal: $x [ M] = x [ N]$( Pontryagin's theorem).

According to this theorem each characteristic class $x \in H ^ {**} ( \mathop{\rm BO} ; \mathbf Q )$ induces a homomorphism $x [ {} ] : \Omega _ {*} ^ { \mathop{\rm SO} } \rightarrow \mathbf Q$, and each element $[ M] \in \Omega _ {*} ^ { \mathop{\rm SO} }$ induces a homomorphism $H ^ {**} ( \mathop{\rm BO} ; \mathbf Q ) \rightarrow \mathbf Q$, $x \rightarrow x [ M]$. In other words, there is a mapping

$$\phi : \Omega _ {*} ^ { \mathop{\rm SO} } \rightarrow \mathop{\rm Hom} ( H ^ {**} ( \mathop{\rm BO} ; \mathbf Q ) ; \mathbf Q ) .$$

If all Pontryagin numbers and Stiefel numbers (cf. Stiefel number) of two oriented closed manifolds coincide, then these manifolds are bordant (in the oriented sense).

A problem similar to the Milnor–Hirzebruch problem for quasi-complex manifolds consists in describing the image of the mapping $\phi$. The solution of this problem is based on the consideration of Pontryagin numbers in $K$- theory corresponding to the Pontryagin classes (cf. Pontryagin class) $\pi _ {i}$ in $K$- theory. Let $\omega = \{ i _ {1} \dots i _ {n} \}$ be a set of non-negative integers, let $S _ \omega ( p)$ and $S _ \omega ( e _ {p} )$ be the characteristic classes defined by the symmetric series

$$S ^ \omega ( x _ {1} ^ {2} \dots x _ {n} ^ {2} ) \textrm{ and } \ S ^ \omega ( e ^ {x _ {1} } + e ^ {- x _ {1} } - 2 \dots e ^ {x _ {n} } + e ^ {- x _ {n} } - 2 ) ,$$

respectively; here $S ^ \omega ( t _ {1} \dots t _ {n} )$ is the minimal symmetric polynomial containing the monomials $t _ {1} ^ {i _ {1} } \dots t _ {k} ^ {i _ {k} }$, $n \geq i _ {1} + \dots + i _ {k}$. Let $B _ {*} \subset \mathop{\rm Hom} ( H ^ {**} ( \mathop{\rm BO} ; \mathbf Q ); \mathbf Q )$ be a set of homomorphisms $b : H ^ {**} ( \mathop{\rm BO} ; \mathbf Q ) \rightarrow \mathbf Q$ for which $b ( S _ \omega ( p) ) \in \mathbf Z$, $b ( S _ \omega ( e _ {p} ) L ) \in \mathbf Z [ 1/2]$ for all tuples $\omega$. Then the image of the homomorphism

$$\phi : \Omega _ {*} ^ { \mathop{\rm SO} } \rightarrow \mathop{\rm Hom} ( H ^ {**} ( \mathop{\rm BO} ; \mathbf Q ) ; \mathbf Q )$$

coincides with $B _ {*}$( the Stong–Hattori theorem).

The characteristic numbers $L [ M]$ and $\widehat{A} [ M]$ corresponding to the classes $L , \widehat{A} \in H ^ {**} ( \mathop{\rm BO} ; \mathbf Q )$ are called the $L$- genus and the $\widehat{A}$- genus of $M$, respectively.

For a closed manifold $M$ of dimension divisible by $4$ the equality $L [ M] = I ( M)$ holds, where $I ( M)$ is the signature of the manifold, i.e. the signature of the quadratic intersection form defined on $H _ {n/2} ( M)$, $n = \mathop{\rm dim} M$( Hirzebruch's theorem). For a closed spin manifold $M$ of even dimension the spinor index of $M$, i.e. the index of the Dirac operator on $M$, coincides with $\widehat{A} [ M]$.

For references see Pontryagin class.