# Stiefel-Whitney class

A characteristic class with values in $H ^ \star ( ; \mathbf Z _ {2} )$, defined for real vector bundles. The Stiefel–Whitney classes are denoted by $w _ {i}$, $i > 0$, and for a real vector bundle $\xi$ over a topological space $B$, the class $w _ {i} ( \xi )$ lies in $H ^ {i} ( B; \mathbf Z _ {2} )$. These classes were introduced by E. Stiefel [1] and H. Whitney [2] and have the following properties. 1) For two real vector bundles $\xi , \eta$ over a common base,

$$w _ {k} ( \xi \oplus \eta ) = \sum _ { i } w _ {i} ( \xi ) w _ {k-} i ( \eta ),\ \ w _ {0} = 1;$$

in other words, $w( \xi \oplus \eta ) = w( \xi ) w( \eta )$, where $w = 1+ w _ {1} + w _ {2} + \dots$ is the complete Stiefel–Whitney class. 2) For the one-dimensional universal bundle $\zeta _ {1}$ over $\mathbf R P ^ \infty$ the equality $w( \zeta _ {1} ) = 1 + y$ holds, where $y$ is the non-zero element of the group $H ^ {1} ( \mathbf R P ^ \infty ; \mathbf Z _ {2} ) = \mathbf Z _ {2}$. These two properties together with naturality for induced bundles define the Stiefel–Whitney classes uniquely. The Stiefel–Whitney classes are stable, i.e. $w( \xi \oplus \theta ) = w ( \xi )$, where $\theta$ is the trivial bundle, and $w _ {i} ( \xi ) = 0$ for $i > \mathop{\rm dim} \xi$. For an oriented $n$- dimensional vector bundle $\xi$ over a base $B$, $w _ {n} ( \xi ) \in H ^ {n} ( B; \mathbf Z _ {2} )$ coincides with the reduction modulo 2 of the Euler class.

For a vector bundle $\xi$ over $B$, let $B ^ \xi$ be the Thom space of this bundle. Further, let $\Phi : H ^ \star ( B; \mathbf Z _ {2} ) \rightarrow \widetilde{H} {} ^ {\star+ n } ( B ^ \xi ; \mathbf Z _ {2} )$ be the Thom isomorphism. Then the complete Stiefel–Whitney class $w( \xi )$ coincides with

$$\Phi ^ {-} 1 Sq \Phi ( 1) \in H ^ \star ( B; \mathbf Z _ {2} ),$$

where $Sq = 1 + Sq ^ {1} + Sq ^ {2} + \dots$ is the complete Steenrod square. This property of Stiefel–Whitney classes can be used as their definition. Stiefel–Whitney classes are homotopy invariants in the sense that they coincide for fibre-wise homotopically-equivalent bundles over a common base.

Any characteristic class with values in $H ^ \star ( ; \mathbf Z _ {2} )$, defined for real vector bundles, can be expressed by Stiefel–Whitney classes: The rings $H ^ {\star\star} ( \mathop{\rm BO} _ {n} ; \mathbf Z _ {2} )$ and $H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} )$ are rings of formal power series in the Stiefel–Whitney classes:

$$H ^ {\star\star} ( \mathop{\rm BO} _ {n} ; \mathbf Z _ {2} ) = \mathbf Z _ {2} [[ w _ {1} \dots w _ {n} ]],$$

$$H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) = \mathbf Z _ {2} [[ w _ {1} ,\dots ]].$$

#### References

 [1] E. Stiefel, "Richtungsfelden und Fernparallelismus in -dimensionalen Mannigfaltigkeiten" Comm. Math. Helv. , 8 : 4 (1935–1936) pp. 305–353 [2] H. Whitney, "Topological properties of differentiable manifolds" Bull. Amer. Math. Soc. , 43 (1937) pp. 785–805 [3] J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) [4] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) [5] N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)

The notation $H ^ {\star\star} ( X; G)$ denotes the product of the Abelian groups $H ^ {n} ( X; G)$, while $H ^ \star ( X; G)$ is the direct sum; the notation $H ^ \star ( B; \mathbf Z _ {2} ) \rightarrow \widetilde{H} {} ^ {\star+} n ( B ^ \xi ; \mathbf Z _ {2} )$ means that there is a graded homomorphism of degree $n$: $H ^ {m} ( B; \mathbf Z _ {2} ) \rightarrow \widetilde{H} {} ^ {m+} n ( B ^ \xi ; \mathbf Z _ {2} )$. For the classifying spaces $\mathop{\rm BO} _ {n}$ and $\mathop{\rm BO}$ see Classifying space.