Bockstein homomorphism
In mathematics, the Bockstein homomorphism in homological algebra 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 abelian groups, or at least torsion free, 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.
The name is for the Soviet topologist from Moscow Meer Feliksovich Bokshtein (Bokstein), with Bockstein being a francophone transliteration. Little known in the West, he was born October 4, 1913 and died May 2, 1990.
References
- Meyer Bockstein, Sur la formule des coefficients universels pour les groupes d'homologie C. R. Acad. Sci.Paris 247 1958 396--398.
- Edwin Spanier, Algebraic Topology