Stiefel number

A characteristic number of a closed manifold taking values in , the integers modulo 2. Let be an arbitrary stable characteristic class, and let be a closed manifold. The residue modulo 2 defined by

is called the Stiefel number (or Stiefel–Whitney number) of corresponding to the class . Here is the tangent bundle of , and is the fundamental class. For -dimensional manifolds, the Stiefel number depends only on the -th homogeneous component of the class . The group is isomorphic to a vector space over the field whose basis is in one-to-one correspondence with the set of all partitions of the number , i.e. tuples of non-negative integers such that . The classes would be a natural choice for a basis of . Thus, to characterize a manifold by its Stiefel numbers it is sufficient to consider the classes , where is a partition of the dimension of the manifold.

Bordant manifolds have the same Stiefel numbers, since each characteristic class determines a homomorphism , where is the group of classes of bordant non-oriented -dimensional manifolds. If for two closed manifolds , the equality holds for all partitions of , then the manifolds and are bordant (Thom's theorem).

Let be the vector space over the field . Let be the basis in dual to the basis in , , here are partitions of ; and let a mapping be defined by . The mapping is monomorphic, and for a complete description of the group by the Stiefel numbers it is necessary to find its image. This problem is analogous to the Milnor–Hirzebruch problem for Chern classes (cf. Chern class). For a closed manifold , let be the so-called Wu class, uniquely defined by , which should hold for all . Then , where is the tangent bundle to (Wu's theorem).

This theorem implies that the Wu class can be defined as a characteristic class: Let

where is the complete Stiefel–Whitney class and is the cohomology operation inverse to the complete Steenrod square . Let be an arbitrary characteristic class. Then for any closed manifold the numbers and coincide. Thus, an element , can be in the image of the mapping only if holds for all . For a homomorphism there exists a manifold such that for all if and only if for all (Dold's theorem).

For references, see Stiefel–Whitney class.

Comments

As is customary denotes the direct product of the cohomology groups of the classifying space , while is the direct sum.