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First account for the unlabelled coefficient-sequence construction. The two short exact sequences of abelian groups areThe singular chain groups of $X$ are free abelian. Applying $\mathrm{Hom}(C_q(X),-)$ therefore preserves these exact sequences, degree by degree, giving short exact sequences of cochain complexes. The associated long exact sequence from a coefficient sequence gives the displayed maps in cohomology; the connecting maps are the integral Bockstein homomorphism $\hat{\beta }$ and the modulo-$n$ Bockstein homomorphism $\beta$. The first omitted map is multiplication by $n$, and the second is induced by $j$.

For the requested example, attach an $(i+1)$-cell to $S^i$ using a map of degree $n$. The resulting Moore space $X=M(\mathbb{Z}/n,i)$ has positive-degree cellular chain complexin degrees $i+1,i$. This construction also works for $i=1$, using the degree-$n$ map of the circle. In cellular cohomology with coefficients $\mathbb{Z}/n$, the differential is zero, so both $H^i$ and $H^{i+1}$ are $\mathbb{Z}/n$.

Lift the cochain taking value $1$ on the $i$-cell to a cochain with coefficients $\mathbb{Z}/n^2$. Its coboundary takes value $n$ on the $(i+1)$-cell, which is $j(1)$. The definition of the connecting homomorphism therefore sends the degree-$i$ generator to the degree-$(i+1)$ generator. HenceIt is nonzero for every $i\ge 1$ and $n\ge 2$, including composite $n$. This is the Bockstein on a cyclic Moore space.

A singular $n$-cochain with coefficients in an abelian group $A$ is a homomorphism from the singular chain group $C_n(X;\mathbb{Z})$ to $A$. It is specified by its values on all singular simplices, without a finite-support restriction. Its coboundary evaluates on the alternating sum of the faces. The resulting cochain complex computes singular cohomology.

For a topological space $X$ and abelian group $A$, singular cohomology is the cohomology of the cochain complex $C^n(X;A)=\mathrm{Hom}(C_n(X;\mathbb{Z}),A)$, where $C_n$ is the singular chain group. Its differential is dual to the singular boundary. This theory has relative cohomology, long exact sequences and the Excision theorem. On spaces that are not locally contractible it can differ from Čech cohomology.