Assume $p \colon Q \to P$ is a piecewise linear $n$-dimensional disc bundle. Let's consider triangulations $L$ and $K$ of $Q$ and $P$ in which $p$ is simplicial, and the subpolyhedron of spheres in $Q$ is triangulated by a subcomplex $S$.
Is it possible to provide a representative of the Thom class as a relative simplicial cochain in $C^n(L, S; \mathbb{Z}_2)$? Perhaps there is a way to formalize the intuition that the Thom class is dual to the zero section (without reference to duality, which I realize is not applicable here)?