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.