For $[c] \in H_n(C_*)$, choose a cycle $c \in C_n$ representing the class, choose $b \in B_n$ with $p_n(b)=c$, and define $a \in A_{n-1}$ as the unique element satisfying
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\begin{align*}
i_{n-1}(a)=d^B_n b.
\end{align*}
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Then $a$ is a cycle in $A_{n-1}$, and the assignment