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)

Comments

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.

References

[a1]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)