# nLab Bockstein homomorphism

cohomology

### Theorems

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# 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

• 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 $H^n(X;\mathbb{Z}_m)\to H^{n}(X;U(1))$.

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

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

## References

Original references include

• M. Bockstein,

Universal systems of $\nablka$-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)