Pontryagin number - Imported Knowlpedia Content

Definition

Let $M$ be a closed oriented $4k$ -manifold and let $E\to M$ be a real vector bundle. For a partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)$ of $k$ , the associated Pontryagin number is

p_\lambda(E)[M] := \left\langle p_{\lambda_1}(E)\smile\cdots\smile p_{\lambda_\ell}(E), [M]\right\rangle\in\mathbb Z.

The monomial has degree $4k$ , since $p_j(E)\in H^{4j}(M;\mathbb Z)$ . A Pontryagin number of an oriented smooth manifold conventionally means the number obtained from its tangent bundle $E=TM$ . The evaluation uses the ordinary integral cup product and the chosen orientation of $M$ .

Bordism and the signature

Tangent-bundle Pontryagin numbers are characteristic numbers and are invariant under oriented bordism. Together with Stiefel–Whitney numbers they detect oriented bordism classes. Rationally, the Pontryagin numbers alone detect the oriented bordism class modulo torsion Milnor–Stasheff, §§16–17.

The Hirzebruch signature theorem expresses the signature of a closed oriented $4k$ -manifold as evaluation of the $L$ -polynomial in its Pontryagin classes. In dimension four this reads

\operatorname{sign}(M)=\frac{1}{3}\left\langle p_1(TM),[M]\right\rangle.

Example: the complex projective plane

For $\mathbb{CP}^2$ , regarded as an oriented real four-manifold, the relation between the underlying real tangent bundle and its complex tangent bundle gives

p_1(T\mathbb{CP}^2) =c_1(T\mathbb{CP}^2)^2-2c_2(T\mathbb{CP}^2) =3h^2.

Therefore $\langle p_1,[\mathbb{CP}^2]\rangle=3$ , in agreement with $\operatorname{sign}(\mathbb{CP}^2)=1$ .

Conventions and scope

Only top-degree monomials produce numbers. On a manifold whose dimension is not divisible by four there are no tangent-bundle Pontryagin numbers, although lower-degree Pontryagin classes may be nonzero.

References

John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. DOI record. Relevant: §§15–17, Pontryagin classes, characteristic numbers, and cobordism.
Friedrich Hirzebruch, Topological Methods in Algebraic Geometry, 3rd ed., Springer, 1966. DOI record. Relevant: chapter 1, the signature theorem and Pontryagin numbers.