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Bockstein homomorphism

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homological algebra

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nonabelian homological algebra

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Idea

A Bockstein homomorphism is a connecting homomorphism induced from a short exact sequence whose injective map is given by multiplication with an integer.

The archetypical examples are the Bockstein homomorphisms induced this way from the short exact sequence

2/2. \mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \stackrel{}{\to} \mathbb{Z}/2\mathbb{Z} \,.

These relate notably degree-nn cohomology with coefficients in 2\mathbb{Z}_2 (such as Stiefel-Whitney classes) to cohomology with integral coefficients in degree n+1n+1 (such as integral Stiefel-Whitney classes).

Definition

Let AA be an abelian group and mm be an integer. Then multiplication by mm

AmA A \stackrel{m\cdot}{\to} A

induces a short exact sequence of abelian groups

0A/A mtorsmAA/mA0, 0\to A/A_{m-tors} \stackrel{m\cdot}{\to} A \to A/m A\to 0,

where A mtorsA_{m-tors} is the subgroup of mm-torsion elements of AA, and so a long fiber sequence

B n(A/A mtors)B nAB n(A/mA)B n+1(A/A mtors) \cdots \mathbf{B}^n (A/A_{m-tors}) \to \mathbf{B}^n A \to \mathbf B^n(A/ m A) \to \mathbf{B}^{n+1} (A/A_{m-tors}) \to \cdots

of ∞-groupoids, where B n()\mathbf{B}^n(-) denotes the nn-fold delooping (hence B nA\mathbf{B}^n A is the Eilenberg-MacLane object on AA in degree nn).

This induces, in turn, for any object XHX \in \mathbf{H} (for H\mathbf{H} the ambient (∞,1)-topos, such as Top \simeq ∞Grpd) , a long fiber sequence

H(X,B n(A/A mtors))H(X,B nA)H(X,B n(A/mA))β mH(X,B n+1(A/A mtors)) \cdots \mathbf{H}(X,\mathbf{B}^n (A/A_{m-tors})) \to \mathbf{H}(X,\mathbf{B}^n A) \to \mathbf{H}(X,\mathbf B^n(A/ m A)) \stackrel{\beta_m}{\to} \mathbf{H}(X,\mathbf{B}^{n+1} (A/A_{m-tors})) \to \cdots

of cocycle ∞-groupoids.

Here the connecting homomorphisms β m\beta_m are called the Bockstein homomorphisms.

Notice that often this term is used to refer only to the image of the above in cohomology, hence to the image of β m\beta_m under 0-truncation/0th homotopy group π 0\pi_0:

β m:H n(X,A/mA)H n+1(X,(A/A mtors)). \beta_m : H^n(X,A/ m A) \to H^{n+1}(X,(A/A_{m-tors})) \,.

Examples

  • When A=A=\mathbb{Z}, the equivalence B n+1B nU(1)\vert \mathbf{B}^{n+1}\mathbb{Z} \vert \cong \vert \mathbf{B}^n U(1)\vert (which holds in ambient contexts such as H=\mathbf{H} = ETop∞Grpd or Smooth∞Grpd under geometric realization :ETopGrpdΠGrpdTop\vert - \vert : ETop \infty Grpd \stackrel{\Pi}{\to} \infty Grpd \stackrel{\simeq}{\to} Top) identifies the morphisms B n( m)B n+1\mathbf{B}^n(\mathbb{Z}_m)\to \mathbf{B}^{n+1}\mathbb{Z} with the morphisms B n( m)B nU(1)\mathbf{B}^n(\mathbb{Z}_m)\to \mathbf{B}^{n} U(1) induced by the inclusion of the subgroup of mm-th roots of unity into U(1)U(1). This identifies the Bockstein homomorphism β m:H n(X; m)H n+1(X;)\beta_m: H^n(X;\mathbb{Z}_m)\to H^{n+1}(X;\mathbb{Z}) with the natural homomorphism H n(X; m)H n(X;U(1))H^n(X;\mathbb{Z}_m)\to H^{n}(X;U(1)).

  • The Bockstein homomorphism β\beta for the sequence 2 2\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \stackrel{}{\to} \mathbb{Z}_2 serves to defined integral Stiefel-Whitney classes W n+1:=βw nW_{n+1} := \beta w_n in degree n+1n+1 from 2\mathbb{Z}_2-valued Stiefel-Whitney classes in degree nn.

  • For pp any prime number the multiplication by pp on p 2\mathbb{Z}_{p^2} induces the short exact sequence p p 2 p\mathbb{Z}_p \to \mathbb{Z}_{p^2} \to \mathbb{Z}_p. The corresponding Bockstein homomorphism β p\beta_p appears as one of the generators of the Steenrod algebra.

References

Original references include

  • M. Bockstein,

    Universal systems of nablka\nablka-homology rings, C. R. (Doklady) Acad. Sci. URSS (N.S.) 37 (1942), “: 243–245, MR0008701

    A complete system of fields of coefficients for the \nabla-homological dimension , C. R. (Doklady) Acad. Sci. URSS (N.S.) (1943), 38: 187–189, MR0009115

  • Bockstein, Meyer Sur la formule des coefficients universels pour les groupes d’homologie , Comptes Rendus de l’académie des Sciences. Série I. Mathématique (1958), 247: 396–398, MR0103918

Revised on September 12, 2012 18:00:57 by Urs Schreiber (131.174.188.61)