nLab Bockstein homomorphism





Special and general types

Special notions


Extra structure



Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories




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).


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})) \,.



(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.


(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.


(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:


(mod 2 Bockstein homomorphism and the exponential exact sequence)


  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.


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


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} }

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


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


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


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 .


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).


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 January 31, 2021 at 05:56:41. See the history of this page for a list of all contributions to it.