Bockstein homomorphism explained

In homological algebra, the Bockstein homomorphism, introduced by, is a connecting homomorphism associated with a short exact sequence

0 \to P \to Q \to R \to 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,

\beta\colon H _{i(C,} R) \to 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

\beta\colon H ^{i(C,} R) \to H ^{i+1} (C,P).

The Bockstein homomorphism

\beta associated to the coefficient sequence0 \to \Z/p\Z\to \Z/p ^{2\Z\to} \Z/p\Z\to 0 is used as one of the generators of the Steenrod algebra. This Bockstein homomorphism has the following two properties:\beta\beta = 0 ,\beta(a\cup b) = \beta(a)\cup b + ( -1 ) ^{\dim} a\cup \beta(b)

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

See also

• Bockstein spectral sequence

References

• • • • .