Let $f\colon X\to Y$ be a smooth surjective map of smooth manifolds of dimension $n$ which are not necessarily orientable. A relative orientation of $X$ over $Y$ consists of an isomorphism $\psi\colon L\otimes L\to\textrm{Hom}(f^*\Omega_Y^n,\Omega_X^n)$ where $L$ is a line bundle on $X$. For every $y\in Y$ outside the branch locus of $f$, for a small enough neighbourhood $U\subseteq Y$ of $y$ and an orientation of $U$, such a relative orientation determines an orientation of $f^{-1}(U)$. Can we define this solely in topological terms, not making use of the smooth structure?

Let $f\colon X\to Y$ be a smooth surjective map of smooth manifolds of dimension $n$ which are not necessarily orientable. A relative orientation of $X$ over $Y$ consists of an isomorphism $\psi\colon L\otimes L\to\textrm{Hom}(f^*\Omega_Y^n,\Omega_X^n)$ where $L$ is a line bundle on $X$. For every $y\in Y$ outside the branch locus of $f$, for a small enough neighbourhood $U\subseteq Y$ of $y$ and an orientation of $U$, such a relative orientation determines an orientation of $f^{-1}(U)$. Can we define this solely in topological terms, not making use of the smooth structure?