nLab
Euler class

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:

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

Poincaré–Hopf theorem

• Poincaré–Hopf theorem

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

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

(Bott-Cattaneo 98, Lemma 2.1)

Related concepts

• Pontryagin class, Stiefel-Whitney class, Wu class

• one-loop anomaly polynomial

References

Review includes

• Allen Hatcher, Euler and Pontryagin classes, section 3.2 in Vector bundles and K-theory (pdf)

• Anant R. Shastri, section 2.1 of Vector bundles and Characteristic Classes (pdf)

• Michael Hutchings, section 5 of Cup product and intersections (pdf)

Discussion for projective modules

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

Discussion of fiber integration:

• Raoul Bott, Alberto Cattaneo, Integral Invariants of 3-Manifolds, J. Diff. Geom., 48 (1998) 91-133 (arXiv:dg-ga/9710001)

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

• Varghese Mathai, Daniel Quillen, below (7.3) of Superconnections, Thom classes, and equivariant differential forms, Topology Volume 25, Issue 1, 1986 (10.1016/0040-9383(86)90007-8)

• Siye Wu, Section 2.2 of Mathai-Quillen Formalism, pages 390-399 in Encyclopedia of Mathematical Physics 2006 (arXiv:hep-th/0505003)

• Liviu Nicolaescu, Section 8.3.2 of Lectures on the Geometry of Manifolds, 2018 (pdf, MO comment)

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)