Bockstein homomorphism

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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: 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.

[edit] History

The name is for the Soviet topologist from Moscow, Meer Feliksovich Bokshtein (Bokstein), with Bockstein being a French transliteration. Little known in the West, he was born October 4, 1913, and died May 2, 1990.

[edit] 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, MR1867354, ISBN 978-0-521-79540-1, <http://www.math.cornell.edu/%7Ehatcher/AT/ATpage.html> .
• Spanier, Edwin H. (1981), Algebraic topology. Corrected reprint, New York-Berlin: Springer-Verlag, pp. xvi+528, MR0666554, ISBN 0-387-90646-0