Bockstein homomorphism

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: "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").

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.

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.

References

* citation
last= Bockstein
first= Meyer
title= Sur la formule des coefficients universels pour les groupes d'homologie
journal=Comptes Rendus de l'académie des Sciences. Série I. Mathématique
volume= 247
year= 1958
pages= 396–398
url=
doi=
id= MR|0103918
* citation
first= Allen
last= Hatcher
author-link= Allen Hatcher
title= Algebraic Topology
url= http://www.math.cornell.edu/%7Ehatcher/AT/ATpage.html
year= 2002
publisher= Cambridge University Press
isbn= 978-0-521-79540-1
id= MR|1867354 .
*citation|id=MR|0666554|last= Spanier|first= Edwin H.|author-link=Edwin Spanier| title= Algebraic topology. Corrected reprint |publisher=Springer-Verlag|publication-place= New York-Berlin|year= 1981|pages= xvi+528| isbn= 0-387-90646-0


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