group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A family of characteristic classes that obstruct orientation, spin structure, spin^c structure, orientation of EO(2)-theory etc.
Stiefel-Whitney classes have coefficients in $\mathbb{Z}_2$, but via the Bockstein homomorphism they are lifted to integral Stiefel-Whitney classes.
(axiomatic definition)
The Stiefel-Whitney classes are characteristic classes $w_i \in H^{i}(B O(n), \mathbb{Z}_2)$ on the classifying space of the orthogonal group in dimension $n$, defined by
$w_0 = 1$ and if $i \gt n$ then $w_i = 0$;
for $n = 1$, $w_1 \neq 0$;
for the inclusion $\iota : B O(n) \hookrightarrow B O(n-1)$ we have $\iota^* w_i^{(n+1)} = w_i^{(n)}$;
sum rule: for all $k,l \in \mathbb{N}$ with the canonical inclusion
we have for all $i \in \mathbb{N}$ that
(on the right the cup product).
For $E \to X$ a real vector bundle/orthogonal group-principal bundle, the total universal Stiefel-Whitney class $w(E)$ is
as an element in the cohomology ring.
For the total SW class of def. 2, the sum rule of def. 1 says equivalently that for $E_1, E_2$ two real vector bundles, then the total SW class of their direct sum of vector bundles is the cup product of the separate classes:
(…)
Every class in $H^\bullet(B O(n), \mathbb{Z}_2)$ can be written uniquely as a polynomial in the Stiefel-Whitney classes. In fact the cohomology ring is the polynomial algebra over $\mathbb{Z}_2$ in the SW classes:
For $X \hookrightarrow \mathbb{R}^q$ an embedding of a compact manifold, write $\tau := T X$ for the tangent bundle and $\nu$ 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
If $E_{\mathbb{C}}$ is a complex vector bundle/ $U(n)$-principal bundle and $E_{\mathbb{R}}$ is the underlying real vector bundle / $O(2n)$-principal bundle then the second Stiefel-Whitney class is given by the first Chern class mod 2:
An almost complex structure on the tangent bundle of a manifold induces a spin^c structure.
This is discussed at Spin^c-striucture – From almost complex structures.
More generally, the SW classes are then given by the Chern character. See for instance Milnor-Stasheff, p. 171.
A classic work on the subject is
A concise introduction is in chapter 23, section 3 of
See also