Chernzahl

ChernZahl, sometimes rendered as Chernzahl in German-language texts, refers to the Chern number, a fundamental topological invariant associated with complex vector bundles over smooth manifolds. It is a key concept in differential geometry and algebraic topology, linking geometric data to discrete, integral quantities. The ChernZahl is used to distinguish bundles up to certain equivalences and to extract global information from local curvature data.

Definition and calculation

Let E be a complex vector bundle over a closed oriented manifold M of real dimension 2m. The bundle has Chern classes c1(E), c2(E), ..., each lying in even-degree cohomology groups of M with integer coefficients. A ChernZahl is obtained by evaluating a homogeneous polynomial P in these Chern classes of total degree m on the fundamental class [M], written as ⟨P(c1(E), c2(E), ..., cm(E)), [M]⟩. The most familiar case is when m = 1, where the ChernZahl reduces to the integral of the first Chern class over M, ⟨c1(E), [M]⟩, which measures the degree of a line bundle. More generally, ChernZahlen are the integer invariants determined by the isomorphism class of E and the topology of M.

Significance and applications

ChernZahlen are topological invariants: they depend only on the bundle’s isomorphism class and the base manifold, not on the particular geometric representative. They appear in several areas of mathematics and physics, including the classification of complex vector bundles, index theory via the Hirzebruch–Riemann–Roch and Atiyah–Singer theorems, and the study of Berry phases and quantized conductance in condensed matter physics.

Origin

The concept is named after Shiing-Shen Chern, with “Chernzahl” used in German literature to denote these integer invariants. The ChernZahl framework unifies many special cases under a single topological invariant.

See also

Chern class, Chern character, characteristic classes, topological invariants, index theorem.

References

Standard texts on differential geometry and topology discuss Chern classes and Chern numbers, including works by Chern, Milnor–Stasheff, Bott–Tu, and Hatcher.