nLab Pontrjagin number

Idea

Pontrjagin numbers are integer topological invariants for real vector bundles over orientable smooth manifolds, which are computed from their Pontrjagin classes. In particular, they exist for orientable smooth manifolds when considering their tangent bundle. According to the Pontrjagin-Thom theorem, two orientable smooth manifolds are orientable bordant if and only if all their Stiefel-Whitney numbers and Pontrjagin numbers are identical.

Definition

Let $E\twoheadrightarrow M$ be a real vector bundle over a $4n$-dimensional orientable smooth manifold $M$ with a fundamental class $[M]\in H_{4n}(M,\mathbb{Z})\cong\mathbb{Z}$ and $i_1+\ldots+i_r=n$ be a partition. Using the Kronecker pairing and the cup product, the Pontrjagin number of $E$ corresponding to this partition is given by:

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

Pontrjagin numbers are usually considered for its tangent bundle with the short notation $p_{i_1}\ldots p_{i_r}[M]\coloneqq p_{i_1}\ldots p_{i_r}[TM]$.

(Milnor & Stasheff 74, p. 185)

Examples

The Pontrjagin numbers of the complex projective space $\mathbb{C}P^{2n}$ (whose underlying real manifold is $4n$-dimensional) are given by:

$$p_{i_1}\ldots p_{i_r}[\mathbb{C}P^{2n}] =\binom{2n+1}{i_1}\ldots\binom{2n+1}{i_r}.$$

(Milnor & Stasheff 74, p. 185)

Related concepts

• Stiefel-Whitney number
• Chern number

References

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