group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
and
nonabelian homological algebra
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
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).
Let $A$ be an abelian group and $m$ be an integer. Then multiplication by $m$
induces a short exact sequence of abelian groups
where $A_{m-tors}$ is the subgroup of $m$-torsion elements of $A$, and so a long fiber sequence
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
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 $\pi_0$:
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.
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