Inhomological algebra, theBockstein homomorphism, introduced byMeyer Bockstein (1942,1943,1958), is aconnecting homomorphism associated with ashort exact sequence

${\displaystyle 0\to P\to Q\to R\to 0}$

ofabelian groups, when they are introduced as coefficients into achain complexC, and which appears in thehomology groups as a homomorphism reducing degree by one,

${\displaystyle \beta :H_i(C,R)\to H_{i-1}(C,P).}$

To be more precise,C should be a complex offree, or at leasttorsion-free, abelian groups, and the homology is of the complexes formed bytensor product withC (someflat module condition should enter). The construction of β is by the usual argument (snake lemma).

A similar construction applies tocohomology groups, this time increasing degree by one. Thus we have

${\displaystyle \beta :H^i(C,R)\to H^{i+1}(C,P).}$

The Bockstein homomorphism${\displaystyle \beta }$ associated to the coefficient sequence

${\displaystyle 0\to \mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p^2\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}\to 0}$

is used as one of the generators of theSteenrod algebra. This Bockstein homomorphism has the following two properties:

${\displaystyle \beta \beta =0}$,

${\displaystyle \beta (a\cup b)=\beta (a)\cup b+(-1{)}^{\dim a}a\cup \beta (b)}$;

in other words, it is a superderivation acting on the cohomology modp of a space.

• Bockstein spectral sequence

• Bockstein, Meyer (1942), "Universal systems of ∇-homology rings",C. R. (Doklady) Acad. Sci. URSS, New Series,37:243–245,MR 0008701
• Bockstein, Meyer (1943), "A complete system of fields of coefficients for the ∇-homological dimension",C. R. (Doklady) Acad. Sci. URSS, New Series,38:187–189,MR 0009115
• Bockstein, Meyer (1958), "Sur la formule des coefficients universels pour les groupes d'homologie",Comptes Rendus de l'Académie des Sciences, Série I,247:396–398,MR 0103918
• Hatcher, Allen (2002),Algebraic Topology,Cambridge University Press,ISBN 978-0-521-79540-1,MR 1867354.
• Spanier, Edwin H. (1981),Algebraic topology. Corrected reprint, New York-Berlin:Springer-Verlag, pp. xvi+528,ISBN 0-387-90646-0,MR 0666554

• Algebraic topology
• Homological algebra

• Articles with short description
• Short description matches Wikidata