nLab Chern number

Idea

Chern numbers are integer topological invariants for complex vector bundles over orientable smooth manifolds, which are computed from their Chern classes. In particular, they exist for complex manifolds (which are always orientable) when considering their tangent bundle. An important Chern number is the Seiberg-Witten invariant.

Definition

Let $E\twoheadrightarrow M$ be a complex vector bundle over a $2n$-dimensional orientable smooth manifold $M$ with a fundamental class $[M]\in H_n(M,\mathbb{Z})\cong\mathbb{Z}$ and $i_1+\ldots+i_r=n$ be a partition. Using the Kronecker pairing and the cup product, the Chern number of $E$ corresponding to this partition is given by:

$$c_{i_1}\ldots c_{i_r}[E] \coloneqq\langle c_{i_1}(E)\smile\ldots\smile c_{i_r}(E),[M]\rangle \in\mathbb{Z}.$$

Chern numbers are usually considered for tangent bundles of complex manifolds with the short notation $c_{i_1}\ldots c_{i_r}[M]\coloneqq c_{i_1}\ldots c_{i_r}[TM]$.

(Milnor & Stasheff 74, p. 184)

Examples

The Chern numbers of the complex projective space $\mathbb{C}P^n$ are given by:

$$c_{i_1}\ldots c_{i_r}[\mathbb{C}P^n] =\binom{n+1}{i_1}\ldots\binom{n+1}{i_r}.$$

(Milnor & Stasheff 74, p. 184)

Related concepts

• Stiefel-Whitney number
• Pontrjagin number

References

• John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974) (ISBN:9780691081229, doi:10.1515/9781400881826, pdf)