Bockstein homomorphism
In homological algebra, the Bockstein homomorphism, introduced by, is a connecting homomorphism associated with a short exact sequence
of abelian groups, when they are introduced as coefficients into a chain complex C, and which appears in the homology groups as a homomorphism reducing degree by one,
To be more precise, C should be a complex of free, or at least torsion-free, abelian groups, and the homology is of the complexes formed by tensor product with C. The construction of β is by the usual argument.
A similar construction applies to cohomology groups, this time increasing degree by one. Thus we have
The Bockstein homomorphism associated to the coefficient sequence
is used as one of the generators of the Steenrod algebra. This Bockstein homomorphism has the following two properties:
in other words, it is a superderivation acting on the cohomology mod p of a space.
