Bockstein homomorphism

Template:Short description In homological algebra, the Bockstein homomorphism, introduced by Template:Harvs, 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,

$\beta :H_i(C,R)\to H_{i-1}(C,P).$

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

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

The Bockstein homomorphism $\beta$ associated to the coefficient sequence

$0\to \mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p^2\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}\to 0$

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

$\beta \beta =0$,

$\beta (a\cup b)=\beta (a)\cup b+(-1{)}^{\dim a}a\cup \beta (b)$;

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

See also

• Bockstein spectral sequence

References

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