Bockstein homomorphism

In homological algebra, the Bockstein homomorphism, introduced by Meyer Bockstein (1942, 1943, 1958), is a connecting homomorphism associated with a short exact sequence

[math]\displaystyle{ 0 \to P \to Q \to R \to 0 }[/math]

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,

[math]\displaystyle{ \beta\colon H_i(C, R) \to H_{i-1}(C,P). }[/math]

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

[math]\displaystyle{ \beta\colon H^i(C, R) \to H^{i+1}(C,P). }[/math]

The Bockstein homomorphism [math]\displaystyle{ \beta }[/math] associated to the coefficient sequence

[math]\displaystyle{ 0 \to \Z/p\Z\to \Z/p^2\Z\to \Z/p\Z\to 0 }[/math]

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

[math]\displaystyle{ \beta\beta = 0 }[/math],

[math]\displaystyle{ \beta(a\cup b) = \beta(a)\cup b + (-1)^{\dim a} a\cup \beta(b) }[/math];

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

See also

• Bockstein spectral sequence

References

• Bockstein, Meyer (1942), "Universal systems of ∇-homology rings", C. R. (Doklady) Acad. Sci. URSS, New Series 37: 243–245 
• 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 
• 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 
• Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, 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, ISBN 0-387-90646-0