nLab Bockstein homomorphism

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

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

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

Definition

Let $A$ be an abelian group and $m$ be an integer. Then multiplication by $m$

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

induces a short exact sequence of abelian groups

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

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

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

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

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

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

of cocycle ∞-groupoids.

Here the connecting homomorphisms $\beta_m$ 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 $\beta_m$ under 0-truncation/0th homotopy group $\pi_0$:

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

Examples

Example

(mod 2 Bockstein homomorphism into integral cohomology)

The Bockstein homomorphism $\beta$ for the sequence

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

serves to define integral Stiefel-Whitney classes

$$W_{n+1} \coloneqq \beta w_n$$

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

Example

(first Steenrod square)

The Bockstein homomorphism for the sequence

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

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

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

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

More generally, for $p$ any prime number the multiplication by $p$ on $\mathbb{Z}_{p^2}$ induces the short exact sequence $\mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/{p^2}\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}$. The corresponding Bockstein homomorphism $\beta_p$ appears as one of the generators of the mod $p$ Steenrod algebra.

Example

(integral Steenrod squares)

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

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

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

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

because, by (1) we have

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

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

More in detail:

Example

(mod 2 Bockstein homomorphism and the exponential exact sequence)

Let

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

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

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

Then

$$\beta \;=\; \delta \circ \iota \,.$$

Because

$$\array{ \mathbb{Z} &\overset{\cdot 2}{\longrightarrow}& \mathbb{Z} &\overset{mod\, 2}{\longrightarrow}& \mathbb{Z}/2\mathbb{Z} &\simeq& \mathbb{Z}/2\mathbb{Z} &\overset{\beta}{\longrightarrow}& B \mathbb{Z} \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{\cdot \pi}} && \downarrow^{\mathrlap{\cdot \pi}} && \downarrow^{\iota \coloneqq \mathrlap{\cdot \pi}} && \downarrow^{\mathrlap{id}} \\ \mathbb{Z} &\underset{\cdot 2 \pi}{\longrightarrow}& \mathbb{R} &\underset{mod\, 2\pi}{\longrightarrow}& U(1) &\simeq& \mathbb{R}/2\pi\mathbb{Z} &\underset{\delta}{\longrightarrow}& B \mathbb{Z} }$$

Proposition

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

Let

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

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

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

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

$$\hat H \hat \cup \hat H = - \hat H \hat \cup \hat H$$

hence

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

hence

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

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

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

with $\iota$ from example .

Then

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

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

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

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

Related concepts

• Steenrod squares, cohomology operation

• Bockstein spectral sequence

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

• Kiyonori Gomi, Differential characters and the Steenrod squares, In Groups of diffeomorphisms, volume 52 of Adv. Stud. Pure Math., pages 297?308. Math. Soc. Japan, Tokyo, 2008

• Ulrich Bunke, problem 3.106 in Differential cohomology (arXiv:1208.3961)

• Daniel Grady, Hisham Sati, prop. 22 in: Primary operations in differential cohomology, Adv. Math. 335 (2018), 519-562 (arXiv:1604.05988, doi:10.1016/j.aim.2018.07.019)