Bockstein homomorphism

In homological algebra, the Bockstein homomorphism, introduced by Bockstein (1942, 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).

This is important as a source of cohomology operations (see Steenrod algebra). For coefficients in a finite cyclic group of order n as R, the mapping β can be combined with reduction modulo n; and then iterated.

References

• Bockstein, M. (1942), "Universal systems of ∇-homology rings", C. R. (Doklady) Acad. Sci. URSS (N.S.), 37: 243–245, MR0008701
• Bockstein, M. (1943), "A complete system of fields of coefficients for the &nable;-homological dimension", C. R. (Doklady) Acad. Sci. URSS (N.S.), 38: 187–189, MR0009115
• 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, MR0103918
• Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79540-1, MR1867354.
• Spanier, Edwin H. (1981), Algebraic topology. Corrected reprint, New York-Berlin: Springer-Verlag, pp. xvi+528, ISBN 0-387-90646-0, MR0666554