Bockstein homomorphism

In algebraic topology, a branch of mathematics, the Bockstein sequence is a tool for comparison of cohomology groups with different coefficients, it is named after Meir Bockstein.

Construction

Homology

Be

A short exact sequence of abelian groups and a topological space. From the short exact sequence of chain complexes

Obtained by means of the Schlangenlemmas a long exact sequence of homology groups

The so-called Bockstein sequence or Bockstein sequence. The connecting homomorphism is called Bockstein homomorphism.

Cohomology

Also provides a short exact sequence of Kokettenkomplexen

And again with the Schlangenlemma a long exact sequence of cohomology groups

Which is also called the Bockstein sequence or Bockstein sequence and the connecting homomorphism as Bockstein homomorphism.

Examples

• The short exact sequence is the Bockstein homomorphism

• The code associated to the short exact sequence Bockstein homomorphism

Is of importance for the construction of the Steenrod algebra.

• The to the short exact sequences and associated Bockstein homomorphisms

Are of importance in the construction of secondary characteristic classes and in the Deligne cohomology.