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.

References

Bockstein, M. (1942), "Universal systems of ∇-homology rings", C. R. (Doklady) Acad. Sci. URSS (N.S.) 37: 243–245, MR 0008701
Bockstein, M. (1943), "A complete system of fields of coefficients for the ∇-homological dimension", C. R. (Doklady) Acad. Sci. URSS (N.S.) 38: 187–189, MR 0009115
Bockstein, Meyer (1958), "Sur la formule des coefficients universels pour les groupes d'homologie", Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 247: 396–398, MR 0103918
Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79540-1, MR 1867354.
Spanier, Edwin H. (1981), Algebraic topology. Corrected reprint, New York-Berlin: Springer-Verlag, pp. xvi+528, ISBN 0-387-90646-0, MR 0666554

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