Bockstein operation

In homological algebra, the Bockstein homomorphism, introduced by Bockstein (1942, 1943, 1958), 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 (some flat module condition should enter). The construction of β is by the usual argument (snake lemma).

A similar construction applies to cohomology groups, this time increasing degree by one. Thus we have

The Bockstein homomorphism β of the coefficient sequence

is used as one of the generators of the Steenrod algebra. This Bockstein homomorphism has the two properties:

in other words it is a superderivation acting on the cohomology mod p of a space.

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