Bockstein homomorphism

Homological map

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

0 → P → Q → R → 0 {\displaystyle 0\to P\to Q\to R\to 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 ) . {\displaystyle \beta \colon H_{i}(C,R)\to 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 ) . {\displaystyle \beta \colon H^{i}(C,R)\to H^{i+1}(C,P).}

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

0 → Z / p Z → Z / p 2 Z → Z / p Z → 0 {\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 the Steenrod algebra. This Bockstein homomorphism has the following two properties:

β β = 0 {\displaystyle \beta \beta =0} ,
β ( a ∪ b ) = β ( a ) ∪ b + ( − 1 ) dim ⁡ a a ∪ β ( b ) {\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 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, 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