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Bockstein homomorphism

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

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

Example

(mod 2 Bockstein homomorphism into integral cohomology)

The Bockstein homomorphism β\beta for the sequence

2mod2/2 \mathbb{Z} \stackrel{\cdot 2}{\longrightarrow} \mathbb{Z} \stackrel{mod\, 2}{\longrightarrow} \mathbb{Z}/2\mathbb{Z}

serves to define integral Stiefel-Whitney classes

W n+1βw n W_{n+1} \coloneqq \beta w_n

in degree n+1n+1 from /2\mathbb{Z}/2\mathbb{Z}-valued Stiefel-Whitney classes in degree nn.

Example

(first Steenrod square)

The Bockstein homomorphism for the sequence

/22/4mod2/2 \mathbb{Z}/2\mathbb{Z} \overset{\cdot 2}{\longrightarrow} \mathbb{Z}/4\mathbb{Z} \overset{mod\, 2}{\longrightarrow} \mathbb{Z}/2\mathbb{Z}

is also called the first Steenrod square, denoted Sq 1Sq^1.

This is often equivalently denoted β\beta, as in example . The difference between the two is just the mod-2 reduction in their codomain:

(1) 2 mod2 /2 /2 β B id mod4 id id B(mod2) /2 2 /4 mod2 /2 (/4)/(/2) Sq 1 B(/2) \array{ \mathbb{Z} &\overset{\cdot 2}{\longrightarrow}& \mathbb{Z} &\overset{mod\, 2}{\longrightarrow}& \mathbb{Z}/2\mathbb{Z} &\simeq& \mathbb{Z}/2\mathbb{Z} &\overset{\beta}{\longrightarrow}& B \mathbb{Z} \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{mod\, 4}} && \downarrow^{\mathrlap{id}} && \downarrow^{ id } && \downarrow^{\mathrlap{B(mod\, 2)}} \\ \mathbb{Z}/2\mathbb{Z} &\underset{\cdot 2 }{\longrightarrow}& \mathbb{Z}/4\mathbb{Z} &\underset{mod\, 2}{\longrightarrow}& \mathbb{Z}/2\mathbb{Z} &\simeq& (\mathbb{Z}/4\mathbb{Z})/(\mathbb{Z}/2\mathbb{Z}) &\underset{Sq^1}{\longrightarrow}& B (\mathbb{Z}/2\mathbb{Z}) }

More generally, 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\mathbb{Z} \to \mathbb{Z}/{p^2}\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}. The corresponding Bockstein homomorphism β p\beta_p appears as one of the generators of the mod pp Steenrod algebra.

Example

(integral Steenrod squares)

For odd 2n+12n + 1 \in \mathbb{N} defines the integral Steenrod squares to be

Sq 2n+1βSq 2n. Sq^{2n + 1}_{\mathbb{Z}} \;\coloneqq\; \beta \circ Sq^{2n} \,.

By example and by the first Adem relation Sq 1Sq 2n=Sq 2n+1Sq^1 \circ Sq^{2n} = Sq^{2n+1} (this example) these indeed are lifts of the odd Steenrod squares:

(mod2)Sq 2n+1=Sq 2n+1, (mod\, 2) \circ Sq^{2n + 1}_{\mathbb{Z}} \;=\; Sq^{2n+1} \,,

because, by (1) we have

Sq 2n+1 : B +2n(/2) Sk 2n B +2n(/2) β B +2n+1 id id B k+2n+1(mod2) Sq 2n+1 : B +2n(/2) Sk 2n B +2n(/2) Sq 1 B +2n+1(/2) \array{ Sq^{2n+1}_{\mathbb{Z}} &\colon& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\overset{Sk^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\overset{ \beta }{\longrightarrow}& B^{\bullet + 2n + 1} \mathbb{Z} \\ && \downarrow^{ id } && \downarrow^{ id } && \downarrow^{\mathrlap{B^{k + 2 n + 1}(mod\, 2)}} \\ Sq^{2n+1} &\colon& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\underset{Sk^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\underset{ Sq^1 }{\longrightarrow}& B^{\bullet + 2n + 1} (\mathbb{Z}/2\mathbb{Z}) }

When A=A=\mathbb{Z}, the equivalence |B n+1||B 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)).

More in detail:

Example

(mod 2 Bockstein homomorphism and the exponential exact sequence)

Let

  1. β:/2B\beta \;\colon\; \mathbb{Z}/2\mathbb{Z} \longrightarrow B \mathbb{Z} be the ordinare Bockstein homomorphism

  2. ι(π):/2U(1)\iota\coloneqq (\cdot \pi) \;\colon\; \mathbb{Z}/2\mathbb{Z} \hookrightarrow U(1) the canonical inclusion;

  3. δ:U(1)B\delta \;\colon\; U(1) \longrightarrow B\mathbb{Z} the classifying map.

Then

β=δι. \beta \;=\; \delta \circ \iota \,.

Because

2 mod2 /2 /2 β B id π π ιπ id 2π mod2π U(1) /2π δ B \array{ \mathbb{Z} &\overset{\cdot 2}{\longrightarrow}& \mathbb{Z} &\overset{mod\, 2}{\longrightarrow}& \mathbb{Z}/2\mathbb{Z} &\simeq& \mathbb{Z}/2\mathbb{Z} &\overset{\beta}{\longrightarrow}& B \mathbb{Z} \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{\cdot \pi}} && \downarrow^{\mathrlap{\cdot \pi}} && \downarrow^{\iota \coloneqq \mathrlap{\cdot \pi}} && \downarrow^{\mathrlap{id}} \\ \mathbb{Z} &\underset{\cdot 2 \pi}{\longrightarrow}& \mathbb{R} &\underset{mod\, 2\pi}{\longrightarrow}& U(1) &\simeq& \mathbb{R}/2\pi\mathbb{Z} &\underset{\delta}{\longrightarrow}& B \mathbb{Z} }
Proposition

(Deligne-Beilinson cup product on odd-degree ordinary differential cohomology)

Let

H^:XB 2nU(1) conn) \hat H \;\colon\; X \longrightarrow \mathbf{B}^{2n} U(1)_{conn})

be a class in ordinary differential cohomology with underlying class in odd degree

[H]:XH^B nU(1) connχB 2n+1 [H] \;\colon\; X \overset{\hat H}{\longrightarrow} \mathbf{B}^{n} U(1)_{conn} \overset{\chi}{\longrightarrow} B^{2n+1} \mathbb{Z}

This implies that its Beilinson-Deligne cup product with itself satisfies

H^^H^=H^^H^ \hat H \hat \cup \hat H = - \hat H \hat \cup \hat H

hence

2H^^H^0 2 \hat H \hat \cup \hat H \;\simeq\; 0

hence

2[H][H]0 2 [H] \cup [H] \;\simeq\; 0

hence that the ordinary cup product [H][H][H] \cup [H] is a 2-torsion class. Let then

j:B 4n+1/2B 4n+1(ι)B 4n+1U(1)B 4n+1U(1) conn j \;\colon\; \mathbf{B}^{4n+1} \mathbb{Z}/2\mathbb{Z} \overset{ B^{4n+1} (\iota) }{\hookrightarrow} \flat \mathbf{B}^{4n+1} U(1) \longrightarrow \mathbf{B}^{4n+1} U(1)_{conn}

with ι\iota from example .

Then

H^^H^jSq 2n([H] mod2). \hat H \hat \cup \hat H \;\simeq\; j Sq^{2n}([H]_{mod\,2}) \,.

This is a differential cohomology-refinement of the first Adem relation Sq 1Sq 2n=Sq 2n+1Sq^1 \circ Sq^{2n} = Sq^{2n+1} on the Steenrod squares (this example) in that, by example , its image in ordinary cohomology with coefficients in /2\mathbb{Z}/2\mathbb{Z} is

([H][H]) mod2 Sq 1βSq 2n([H] mod2) = [H] mod2[H] mod,2 = Sq 2n+1([H] mod2). \array{ ([H] \cup [H])_{mod 2} & \simeq & \underset{ \beta }{ \underbrace{ Sq^1 }} \circ Sq^{2n}([H]_{mod\,2}) \\ = \\ [H]_{mod\, 2} \cup [H]_{mod,2} \\ = \\ Sq^{2n+1}([H]_{mod\, 2}) } \,.

This was first observed in (Gomi 08). Streamlined proofs are given in (Bunke 12, propblem 3.106, Grady-Sati 16, prop. 22).

References

Original references include

  • Meyer Bockstein,

    Universal systems of \nabla-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

  • Meyer Bockstein, 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

The relation to the Beilinson-Deligne cup product is discussed in

Last revised on September 7, 2018 at 13:04:37. See the history of this page for a list of all contributions to it.