Let $F: W \to N$ be a smooth map, where $W$ is a compact manifold with boundary, $Z \subset N$ is closed and all manifolds are oriented. Also $\partial F \pitchfork Z$ and $F^{-1}(Z)$ is a compact, oriented, 1-dimensional manifold diffeomorphic to $[0,1]$.
I would like to show that the intersection numbers of of the two endpoints of this line segment have opposite sign, that is $(\partial F \cdot Z)_x = - (\partial F \cdot Z)_y$ for $\{x, y\} = (\partial F)^{-1}(Z)$.
In the notes that I read we show that $(\partial F \cdot Z)_x = (F^{-1}(Z) \cdot \partial W)_x$ and then the result seems to follow quite intuitively. But to prove this equality we need results that I still have some troubles with (about short exact sequences) and I was wondering if there is an easier way to get the same result.