Bockstein Operation

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,

:$\beta\colon H_i(C, R) \to H_(C,P).$

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

:$\beta\colon H^i(C, R) \to H^(C,P).$

The Bockstein homomorphism $\beta$ associated to the coefficient sequence
:$0 \to \Z/p\Z\to \Z/p^2\Z\to \Z/p\Z\to 0$
is used as one of the generators of the Steenrod algebra. This Bockstein homomorphism has the following two properties:
:$\beta\beta = 0$,
:$\beta(a\cup b) = \beta(a)\cup b + (-1)^ a\cup \beta(b)$;
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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*{{citation, mr=0666554, last= Spanier, first= Edwin H., author-link=Edwin Spanier, title= Algebraic topology. Corrected reprint , publisher=Springer-Verlag, publication-place= New York-Berlin, year= 1981, pages= xvi+528, isbn= 0-387-90646-0

Algebraic topology
Homological algebra