Bockstein homomorphism
In homological algebra, the Bockstein homomorphism, introduced by Bockstein (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: Hi(C, R) → Hi − 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) → Hi + 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, 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