The cap product with the fundamental cycle of an oriented, triangulated $n$-manifold $X$ defines a chain map $-\cap X: C^*(X,A) \to C_{n-*}(X,A)$, where $A$ is any abelian coefficient group.
It's easy to see by computing examples that this map is neither surjective nor injective in general. However, it induces isomorphisms $H^*(X,A) \simeq H_{*-n}(X,A)$ so it is a quasi-isomorphism.
Hence my question: is there any well-known or simple construction of an "inverse map" $$f:C_*(X,A) \to C^{*-n}(X,A)$$ such that $f(-) \cap X$ and $f(- \cap X)$ are homotopic to the identity maps on $C_*$ and $C^*$, respectively?