nLab
top Chern class

Contents

Context

Algebraic topology

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

For 𝒱 X\mathcal{V}_X a complex vector bundle of complex rank nn, the highest degree Chern class that may generally be non-vanishing is c nc_n. This is hence often called the top Chern class of the vector bundle.

Properties

Relation to Euler- and Thom-class

Proposition

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

𝒱 Xhas complex ranknc n(𝒱 X)=e(𝒱 X )H 2n(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))

Proposition

The top Chern class of a complex vector bundle 𝒱 X\mathcal{V}_X equals the pullback of any Thom class thH 2n(𝒱 X;)th \;\in\; H^{2n}\big( \mathcal{V}_X; \mathbb{Z} \big) on 𝒱 X\mathcal{V}_X along the zero-section:

𝒱 Xhas complex ranknc n(𝒱 X)=(0 X) *(th)H 2n(X;) \mathcal{V}_X \; \text{has complex rank}\;n \;\;\;\;\; \Rightarrow \;\;\;\;\; c_n \big( \mathcal{V}_X \big) \;\;\; = \;\;\; (0_X)^\ast (th) \;\; \in \; H^{2n} \big( X ; \, \mathbb{Z} \big)

(e.g. Bott-Tu 82, Prop. 12.4)

References

For more references see at Chern class and at characteristic class.

Created on January 26, 2021 at 08:52:54. See the history of this page for a list of all contributions to it.