Given a manifold $X$ and short exact sequence of abelian groups $$ 1\rightarrow A_1\overset{\iota}{\rightarrow} A_2\overset{\pi}{\rightarrow} A_3\rightarrow 1 $$ we get the Bockstein map in cohomology $\beta : H^p(X,A_3)\rightarrow H^{p+1}(X,A_1)$. If $X$ is orientable, by using Poincarè duality for homology/cohomology with coefficients in $A_1$ and $A_3$ we induce the Bockstein map in homology $b: H_p(X,A_3)\rightarrow H_{p-1}(X,A_1)$. It is not clear to me how to define it explicitely on chains.
I was expacting something very similar to the cohomology case, but I think there is some difference because regarding cohains the abelian groups appear as value of the maps, while about chains they appear as coefficients. Let me explain better. Given $f\in Z^p(X,A_3)$, this is a map valued in $A_3$, then choosing a section $s:A_3\rightarrow A_2$, $\pi \circ s=id_{A_3}$ we can construct $s(\delta f), \delta s(f)\in C^{p+1}(X,A_2)$ but they are not equal. However since $$ \pi\left(\delta s(f)-s(\delta f)\right)=0 $$ there exist a cochain $\beta(f)\in C^{p+1}(X,A_1)$ such that $$ \delta s(f)-s(\delta f)=\iota (\beta(f)) $$ This is the well known algebraic construction of the Bockstein map on cochains, and it really uses the fact that cochains are valued in groups. How one can imitate this on chains?