Differential character

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Differential characters are a term from the mathematical field of differential topology that generalizes the cohomology groups.

Secondary characteristic classes , for example the Cheeger-Chern-Simons classes of vector bundles, are differential characters. In the case of flat bundles , these are then even cohomology classes.

ℤ-valued differential characters

Let be a smooth manifold and an integer. The group of -valent differential characters of degree is

.

Here the group of - denotes cycles and the notation means that there is a differential form such that

holds for every smooth chain .

ℝ / ℤ-valued differential characters

Let be a smooth manifold and an integer. The group of -valent differential characters of degree is

.

Here the group of -cycles denotes and the notation means that there is a differential form such that

holds for every smooth chain .

Short exact sequences

Korand illustration

You have a short exact sequence

.

Herein refers to the group of closed differential forms integral period and the image

assigns the unique differential form with too.

In particular, one can understand as a subgroup of .

Secondary characteristic classes of vector bundles give invariants in , which in the case of vanishing curvature even lie in.

Bockstein homomorphism

There is a homomorphism

,

whose restriction is precisely the Bockstein homomorphism . It fits into an exact sequence

.

literature

  • Jeff Cheeger, James Simons: Differential characters and geometric invariants. Geometry and topology. In: Lecture Notes in Math. 1167, Springer, Berlin 1985, pp. 50–80.
  • Christian Bär, Christian Becker: Differential characters. In: Lecture Notes in Mathematics. 2112. Springer, Cham 2014, ISBN 978-3-319-07033-9 .