Bockstein homomorphism

In homological algebra, the Bockstein homomorphism, introduced by Bockstein (1942, 1943, 1958), is a connecting homomorphism associated with a short exact sequence

0 → P → Q → R → 0

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,

β: H_i(C, R) → H_{i − 1}(C, P).

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

β: Hⁱ(C, R) → H^{i + 1}(C, P).

The Bockstein homomorphism β of the coefficient sequence

0 → Z/pZ → Z/p²Z → Z/pZ → 0

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

ββ = 0 if p>2 β(a∪b) = β(a)∪b + (-1)^{dim a} a∪β(b)

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