nLab Segre class

Idea

Segre classes (or dual Chern classes) are $\mathbb{Z}$-valued characteristic classes for complex vector bundles. The total Chern class is invertible in the cohomology ring (using $c_0=1$ and the geometric series) and the Segre classes are then the components of its inverse, meaning in particular that they can be expressed by Chern classes again and carry the exact same information. Nonetheless, Segre classes can be used to simplfy calculations.

Definition

Let $E\twoheadrightarrow B$ be a complex vector bundle of rank $n$ with total Chern class $c(E)\in H^*(B,\mathbb{Z})$, then its total Segre class $s(E)\in H^*(B,\mathbb{Z})$ is defined by:

$$c(E)s(E) =1.$$

If the Chern classes are known, then the equation can be solved inductively due to their linear and triangular form. Solving the equation in order $n$ yields:

$$s_n(E) =-\sum_{k=1}^n c_k(E)s_{n-k}(E).$$

Another direct computation uses the geometric series:

$$s(E) =c(E)^{-1} =\left( 1+\sum_{k=1}^n c_k(E) \right)^{-1} =\sum_{l\in 0}^\infty\left( \sum_{k=1}^n c_k(E) \right)^l.$$

Examples

Low Segre classes are:

$$s_0 =1;$$

$$s_1 =-c_1;$$

$$s_2 =-c_2 +c_1^2.$$

Properties

For complex vector bundles $E,F\twoheadrightarrow B$, one has $c(E\oplus F)=c(E)c(F)$, meaning that $c(F)=s(E)c(E\oplus F)$. In particular, if $B$ is a compact Hausdorff space, then for every $E$ there exists a $F$ with $E\oplus F\cong\mathbb{R}^n$, (Hatcher 17, Prop. 1.4) which with $w(E\oplus F)=w(\mathbb{R}^n)=1$ makes the formula simplify to:

$$s(E) =c(F).$$

Hence in this case the Segre classes can be seen as the Chern classes of a complement to a trivial bundle.

Related concepts

• dual Stiefel-Whitney class

References

• Allen Hatcher: Vector bundles and K-Theory, book draft (2017) [webpage, pdf]