Characteristic number - Imported Knowlpedia Content

Definition

Let $M$ be a closed $R$ -oriented $n$ -manifold, let $E\to M$ be a vector bundle, and let $P$ be a homogeneous polynomial of total cohomological degree $n$ in characteristic classes $\alpha_i(E)$ . The characteristic number determined by $P$ is

\bigl\langle P(\alpha_1(E),\alpha_2(E),\ldots),[M]\bigr\rangle\in R,

where products are taken in the cohomology ring and $[M]$ is the fundamental class. A characteristic number of the manifold usually means one formed from characteristic classes of its tangent bundle $TM$ .

Standard families

Chern numbers evaluate degree- $2n$ monomials in Chern classes on a closed almost-complex $2n$ -manifold. Pontryagin numbers use degree- $4k$ monomials on an oriented manifold, while Stiefel–Whitney numbers use mod- $2$ classes and the mod- $2$ fundamental class. The Euler number is

\langle e(E),[M]\rangle

when $E$ is oriented of rank $n$ .

Geometric significance

Characteristic numbers are unchanged by bundle isomorphisms over the identity. For tangent bundles they are invariants under orientation-preserving diffeomorphisms. Stiefel–Whitney numbers determine unoriented cobordism classes, while Pontryagin and Stiefel–Whitney numbers together determine oriented cobordism classes Milnor–Stasheff, chapters 16–17.

Conventions and scope

The coefficient ring, orientation, bundle, and polynomial are part of the data. Only the component of total degree $n$ can pair with $[M]$ . Mod- $2$ numbers do not require an orientation, whereas integral Chern, Pontryagin, and Euler numbers use the appropriate integral orientation.

References

John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. DOI record. Relevant: chapters 16–17, characteristic numbers and cobordism.
Friedrich Hirzebruch, Topological Methods in Algebraic Geometry, 3rd ed., Springer, 1966. DOI record. Relevant: characteristic numbers and genera of manifolds.