nLab
Euler class

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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:

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

Poincaré–Hopf theorem

Fiber integration

Proposition

Let

S 4 BSpin(4) π BSpin(5) \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 4Spin(5)/Spin(4)S^4 \simeq Spin(5)/Spin(4) (this Prop.).

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

π *(χ 2k+1)=2(p 2) kH 4(BSpin(5),), \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

π *(χ) =2 π *(χ 3) =2p 2 π *(χ 5) =2(p 2) 2H 4(BSpin(5),); \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 χ 2k\chi^{2k} vanishes

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

(Bott-Cattaneo 98, Lemma 2.1)

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?):

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)

Last revised on April 18, 2019 at 06:38:58. See the history of this page for a list of all contributions to it.