Let $K$ be a simplicial complex of dimension $n$, $M$ be a topological manifold, and $f \colon |K| \to M$ be a continuous map. Let $X$ be an embedded manifold in $M$ of codimension $n$, such that $f(|K|) \cap X$ is a finite set of good points (i.e., a neighborhood of each point homeomorphic to a disc where $f(|K|)$ and $X$ intersect as planes in general position), and their preimages lie in $|K \setminus sk_{n-1} K|$.
Denote by $PD([X])$ the Poincaré dual of $X$ cohomology class lying in $H^n(M, \mathbb{Z}_2)$. It seems correct that the pullback $f^*PD([X])$ in $H^n(K, \mathbb{Z}_2)$ is represented, for example, by a cochain that associates to each $n$-simplex $A$ the count of points in $f(A) \cap X$ modulo two.
Is this statement true? Perhaps there is a formal proof of it somewhere, or did I miss something, and it can be proved in just a few lines?
