nLab
top Chern class

Definition

For $\mathcal{V}_X$ a complex vector bundle of complex rank $n$, the highest degree Chern class that may generally be non-vanishing is $c_n$. 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 $\mathcal{V}_X$ 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))

Proposition

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

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

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

Remark

This relation to Thom classes generalizes to Conner-Floyd-Chern classes in complex oriented Whitehead-generalized cohomology. See at universal complex orientation on MU.

Related concepts

• first Chern class

• Euler class, Thom class

References

• Raoul Bott, Loring Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/BFb0063500)

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