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cohomology

# Contents

## Idea

The Euler class $\chi$ (or $e$) is a characteristic class of the special orthogonal group, hence of oriented real vector bundles.

The Euler class of the tangent bundle of a smooth manifold $X$, evaluated on its fundamental class, is its Euler characteristic $\chi[X]$.

## Properties

### Cup square

For $E$ a vector bundle of even rank $rank(E) = 2 k$, the cup product of the Euler class with itself equals the $k$th Pontryagin class

(1)$\chi(E) \smile \chi(E) \;=\; p_k(E) \,.$

When the Euler class is represented by the Euler form of a connection $\nabla$ on $E$, which then is fiber-wise proportional to the Pfaffian of the curvature form $F_\nabla$ of $\nabla$, the relation (1) corresponds to the fact that the product of a Pfaffian with itself is the determinant: $\big( Pf(F_\nabla) \big)^2 = det(F_\nabla)$.

### 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:

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

(e.g. Walschap 04, Section 6.4)

### Relation to top Chern class

###### Proposition

The top Chern class of a complex vector bundle $\mathcal{V}_X$ equals the Euler class $e$ of the underlying real vector bundle $\mathcal{V}^{\mathbb{R}}_X$:

$\mathcal{V}_X \; \text{has complex rank}\;n \;\;\;\;\; \Rightarrow \;\;\;\;\; c_n \big( \mathcal{V}_X \big) \;\; = \;\; e \big( \mathcal{V}^{\mathbb{R}}_X \big) \;\;\;\; \in H^{2n} \big( X; \, \mathbb{Z} \big) \,.$

(e.g. Bott-Tu 82 (20.10.6))

For more see at top Chern class.

### On unit sphere bundles

###### Proposition

Let $X$ be a smooth manifold and $E \overset{\pi}{\longrightarrow} X$ an oriented real vector bundle of even rank, $rank(E) = 2k + 2$.

For any choice of connection $\nabla$ on $E$ ($SO(dim(X))$-connection), let $\chi(\nabla_E) \in \Omega^{2k}(X)$ denote the corresponding Euler form.

Then the pullback of the Euler form $\chi(\nabla_E)$ to the unit sphere bundle $S(E) \overset{S(\pi)}{\longrightarrow} X$ is exact

$\big( S(\pi) \big)^\ast \chi(\nabla_E) \;=\; d \Omega$

such that the trivializing form has (minus) unit integral over any of the (2k+1)-sphere-fibers $S^{2k+1}_x \overset{\iota_x}{\hookrightarrow} S(E)$:

(2)$\int_{S^{2k+1}} \iota_x^\ast \Omega \;=\; -1 \,.$

### 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

### General

Discussion of fiber integration:

Discussion for projective modules

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

See also

• Wikipedia Euler class

• Robert F. Brown, On the Lefschetz number and the Euler class, Transactions of the AMS 118, (1965) (JSTOR)

• Solomon Jekel, A simplicial formula and bound for the Euler class, Israel Journal of Mathematics 66, n. 1-3, 247-259 (1989)

### Euler forms

Discussion of Euler forms (differential form-representatives of Euler classes in de Rham cohomology) as Pfaffians of curvature forms:

Last revised on February 23, 2021 at 13:04:06. See the history of this page for a list of all contributions to it.