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
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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
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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.
Stiefel number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stiefel_number&oldid=11762