# 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.

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.