Stiefel number

A characteristic number of a closed manifold taking values in $ \mathbf Z _ {2} $, the integers modulo 2. Let $ x \in H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ be an arbitrary stable characteristic class, and let $ M $ be a closed manifold. The residue modulo 2 defined by

$$ x[ M] = \langle x( \tau M), [ M]\rangle $$

is called the Stiefel number (or Stiefel–Whitney number) of $ M $ corresponding to the class $ x $. Here $ \tau M $ is the tangent bundle of $ M $, and $ [ M] \in H _ \star ( M; \mathbf Z _ {2} ) $ is the fundamental class. For $ n $- dimensional manifolds, the Stiefel number depends only on the $ n $- th homogeneous component of the class $ x $. The group $ H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ is isomorphic to a vector space over the field $ \mathbf Z _ {2} $ whose basis is in one-to-one correspondence with the set of all partitions $ w = \{ i _ {1} \dots i _ {k} \} $ of the number $ n $, i.e. tuples $ \{ i _ {1} \dots i _ {k} \} $ of non-negative integers such that $ i _ {1} + \dots + i _ {k} = n $. The classes $ w _ \omega = w _ {i _ {1} } \dots w _ {i _ {k} } $ would be a natural choice for a basis of $ H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $. Thus, to characterize a manifold by its Stiefel numbers it is sufficient to consider the classes $ w _ \omega $, where $ \omega $ is a partition of the dimension of the manifold.

Bordant manifolds have the same Stiefel numbers, since each characteristic class $ x $ determines a homomorphism $ x[ {} ] : \mathfrak N ^ {n} \rightarrow \mathbf Z _ {2} $, where $ \mathfrak N ^ {n} $ is the group of classes of bordant non-oriented $ n $- dimensional manifolds. If for two closed manifolds $ M $, $ N $ the equality $ w _ \omega [ M] = w _ \omega [ N] $ holds for all partitions $ \omega $ of $ n = \mathop{\rm dim} M = \mathop{\rm dim} N $, then the manifolds $ M $ and $ N $ are bordant (Thom's theorem).

Let $ A $ be the vector space $ \mathop{\rm Hom} ( H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ), \mathbf Z _ {2} ) $ over the field $ \mathbf Z _ {2} $. Let $ \{ e _ \omega \} $ be the basis in $ A $ dual to the basis $ \{ w _ \omega \} $ in $ H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $, $ e _ \omega ( w _ {\omega ^ \prime } ) = \delta _ \omega ^ {\omega ^ \prime } $, here $ \omega , \omega ^ \prime $ are partitions of $ n $; and let a mapping $ \phi : \mathfrak N \rightarrow A $ be defined by $ \phi ([ M]) = \sum _ \omega w _ \omega [ M] e _ \omega $. The mapping $ \phi $ is monomorphic, and for a complete description of the group $ \mathfrak N ^ {n} $ 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 $ M $, let $ v \in H ^ \star ( M; \mathbf Z _ {2} ) $ be the so-called Wu class, uniquely defined by $ \langle \alpha \cup v, [ M]\rangle = \langle Sq \alpha [ M]\rangle $, which should hold for all $ \alpha \in H ^ \star ( M; \mathbf Z _ {2} ) $. Then $ w( \tau M) = Sqv $, where $ \tau M $ is the tangent bundle to $ M $( Wu's theorem).

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

$$ v = Sq ^ {-} 1 w \in H ^ \star ( \mathop{\rm BO} ; \mathbf Z _ {2} ), $$

where $ w \in H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ is the complete Stiefel–Whitney class and $ Sq ^ {-} 1 = 1 + Sq ^ {1} + Sq ^ {2} + Sq ^ {2} Sq ^ {1} + \dots $ is the cohomology operation inverse to the complete Steenrod square $ Sq $. Let $ \alpha \in H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ be an arbitrary characteristic class. Then for any closed manifold the numbers $ ( \alpha \cup v)[ M] $ and $ ( Sq \alpha )[ M] $ coincide. Thus, an element $ a \in A $, $ a = \sum a _ \omega e _ \omega $ can be in the image of the mapping $ \phi $ only if $ a( \alpha \cup v) = a( Sq \alpha ) $ holds for all $ \alpha \in H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $. For a homomorphism $ a: H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) \rightarrow \mathbf Z _ {2} $ there exists a manifold $ M ^ {n} $ such that $ x[ M ^ {n} ] = a( x) $ for all $ x \in H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ if and only if $ a( \alpha \cup v) = a( Sq \alpha ) $ for all $ \alpha \in H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $( Dold's theorem).

For references, see Stiefel–Whitney class.

Comments

As is customary $ H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ denotes the direct product of the cohomology groups $ H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ of the classifying space $ \mathop{\rm BO} $, while $ H ^ \star ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ is the direct sum.