# .

# 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:
*H*^{i}(*C*,*R*) →*H*^{i + 1}(*C*,*P*).

The Bockstein homomorphism β of the coefficient sequence

- 0 →
**Z**/*p***Z**→**Z**/*p*^{2}**Z**→**Z**/*p***Z**→ 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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