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 the rational number is called the Pontryagin number of corresponding to ; here is the tangent bundle and is the fundamental class of . The Pontryagin number depends only on the homogeneous component of degree of the class . Let be a partition of , i.e. a set of non-negative integers such that and let . The rational numbers are defined for a closed manifold of dimension and all partitions of the number .

The Pontryagin numbers of two bordant (in the oriented sense, cf. Bordism) manifolds are equal: (Pontryagin's theorem).

According to this theorem each characteristic class induces a homomorphism , and each element induces a homomorphism , . In other words, there is a mapping

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 . The solution of this problem is based on the consideration of Pontryagin numbers in -theory corresponding to the Pontryagin classes (cf. Pontryagin class) in -theory. Let be a set of non-negative integers, let and be the characteristic classes defined by the symmetric series

respectively; here is the minimal symmetric polynomial containing the monomials , . Let be a set of homomorphisms for which , for all tuples . Then the image of the homomorphism

coincides with (the Stong–Hattori theorem).

The characteristic numbers and corresponding to the classes are called the -genus and the -genus of , respectively.

For a closed manifold of dimension divisible by the equality holds, where is the signature of the manifold, i.e. the signature of the quadratic intersection form defined on , (Hirzebruch's theorem). For a closed spin manifold of even dimension the spinor index of , i.e. the index of the Dirac operator on , coincides with .

For references see Pontryagin class.