cohomology

# Contents

## Idea

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.

## Definition

###### Definition

(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

1. $w_0 = 1$ and if $i \gt n$ then $w_i = 0$;

2. for $n = 1$, $w_1 \neq 0$;

3. for the inclusion $\iota : B O(n) \hookrightarrow B O(n-1)$ we have $\iota^* w_i^{(n+1)} = w_i^{(n)}$;

4. sum rule: for all $k,l \in \mathbb{N}$ with the canonical inclusion

$\iota : B O(k) \times B O(l) \to B O(k+l)$

we have for all $i \in \mathbb{N}$ that

$\iota^* w_i = \sum_{j = 0}^i w_j \cup w_{i-j}$

(on the right the cup product).

###### Definition

For $E \to X$ a real vector bundle/orthogonal group-principal bundle, the total universal Stiefel-Whitney class $w(E)$ is

$w(E) \coloneqq \sum_i w_i(E) \in H^\bullet(X, \mathbb{Z}_2)$

as an element in the cohomology ring.

###### Remark

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:

$w(E_1 \oplus E_2) = w(E_1) \cup w(E_2) \,.$

(…)

## Properties

### Spanning the cohomology ring

###### Proposition

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:

$H^\bullet(B O(n), \mathbb{Z}_2) \simeq \mathbb{Z}_2[w_1, \cdots, w_n] \,.$

### Whitney duality formula

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

$\tau \oplus \nu \simeq T \mathbb{R}^q$

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

$w(\tau) \cup w(\nu) = 1 \,.$

### Relation to Chern-classes

###### Proposition

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:

$w_2(E_{\mathbb{R}}) = c_1(E_{\mathbb{C}}) \; mod 2 \,.$
###### Corollary

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.

###### Remark

More generally, the SW classes are then given by the Chern character. See for instance Milnor-Stasheff, p. 171.

## References

A classic work on the subject is

A concise introduction is in chapter 23, section 3 of