Pontryagin number
A characteristic number defined for closed oriented manifolds and assuming rational values. Let
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.
Pontryagin number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pontryagin_number&oldid=48242