Special and general types
A family of characteristic classes that obstruct orientation, spin structure, spin^c structure, etc.
Stiefel-Whitney classes have coefficients in , but via the Bockstein homomorphism they are lifted to integral Stiefel-Whitney classes.
The Stiefel-Whitney classes are characteristic classes on the classifying space of the orthogonal group in dimension , defined by
and if then ;
for , ;
for the inclusion we have ;
sum rule: for all with the canonical inclusion
we have for all that
(on the right the cup product).
Spanning the cohomology ring
Every class in can be written uniquely as a polynomial in the Stiefel-Whitney classes. In fact the cohomology ring is the polynomial algebra over in the SW classes:
Whitney duality formula
For an embedding of a compact manifold, write for the tangent bundle and for the corresponding normal bundle. Then since
and the class of the vector bundle on the right is trivial, the sum rule for the SW classes says gives the cup product duality
Relation to Chern-classes
If is a complex vector bundle/ -principal bundle and is the underlying real vector bundle / -principal bundle then the second Stiefel-Whitney class is given by the first Chern class mod 2:
This is discussed at Spin^c-striucture – From almost complex structures.
A classic work on the subject is
A concise introduction is in chapter 23, section 3 of