Contents

cohomology

# Contents

## Idea

The Euler class is a characteristic class of the special orthogonal group, hence of oriented real vector bundles.

The Euler class of the tangent bundle of a manifold is its Euler characteristic.

## Properties

### Whitney sum formula

###### Proposition

(Euler class takes Whitney sum to cup product)

The Euler class of the Whitney sum of two oriented real vector bundles to the cup product of the separate Euler classes:

$\chi( E \oplus F ) \;=\; \chi(E) \smile \chi(F) \,.$

### Fiber integration

###### Proposition

Let

$\array{ S^4 &\longrightarrow& B Spin(4) \\ && \big\downarrow^{\mathrlap{\pi}} \\ && B Spin(5) }$

be the spherical fibration of classifying spaces induced from the canonical inclusion of Spin(4) into Spin(5) and using that the 4-sphere is equivalently the coset space $S^4 \simeq Spin(5)/Spin(4)$ (this Prop.).

Then the fiber integration of the odd cup powers $\chi^{2k+1}$ of the Euler class $\chi \in H^4\big( B Spin(4), \mathbb{Z}\big)$ (see this Prop) are proportional to cup powers of the second Pontryagin class

$\pi_\ast \left( \chi^{2k+1} \right) \;=\; 2 \big( p_2 \big)^k \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,,$

for instance

\begin{aligned} \pi_\ast \big( \chi \big) & = 2 \\ \pi_\ast \left( \chi^3 \right) & = 2 p_2 \\ \pi_\ast \left( \chi^5 \right) & = 2 (p_2)^2 \end{aligned} \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,;

while the fiber integration of the even cup powers $\chi^{2k}$ vanishes

$\pi_\ast \left( \chi^{2k} \right) \;=\; 0 \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,.$

## References

Review includes

Discussion for projective modules

• Satya Manda, An overview of Euler class theory (pdf)

Discussion of fiber integration:

Discussion of Euler forms (differential form-representatives of Euler classes in real cohomology?):