Contents

cohomology

# 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

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

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

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

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).

## References

Original references include

• 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 10:56:41. See the history of this page for a list of all contributions to it.