Characteristicnumber

Characteristicnumber is a generalized term used to denote a numerical invariant associated with a geometric object, typically a closed oriented manifold or a vector bundle, obtained by evaluating a characteristic class on the fundamental class. In this sense, a characteristicnumber encapsulates topological information by turning a cohomological class into a single integer (or rational number) through pairing with the manifold’s fundamental class.

Construction generally proceeds by selecting a homogeneous polynomial in one or more characteristic classes, such as Chern classes for complex vector bundles, Pontryagin and Stiefel-Whitney classes for real bundles, or the Euler class for oriented tangent bundles. When the total degree of the chosen polynomial equals the dimension of the manifold, evaluating the resulting class on the fundamental class yields the characteristicnumber. Classical examples include Euler characteristics, Chern numbers for complex manifolds, Pontryagin numbers for oriented manifolds, and numbers arising from the Todd class in the Hirzebruch–Riemann–Roch framework.

Examples and interpretations vary: the Euler characteristic χ(M) is the archetypal characteristicnumber given by ⟨e(TM), [M]⟩; Chern numbers such as ⟨c1^n, [M]⟩ or ⟨c1^{n-1}c2, [M]⟩ encode complex-geometric information; Pontryagin numbers arise from real oriented bundles; Todd-genus related numbers connect to index theory. These invariants are stable under oriented diffeomorphisms and, in cobordism theory, remain constant on cobordism classes, providing obstructions and classification data for manifolds and bundles.