I think I have a near solution, but one doubt. Problem 2.4.4 on page 83 of Guillemin and Pollack asks
If $f\colon X\to Y$ is homotopic to a constant map, show that $I_2(f,Z)=0$ for all complementary dimensional closed $Z$ in $Y$, except parhaps if $\dim X=0$.
Assume $\dim X\geq 1$, so $\dim Z<\dim Y$. Suppose $f\sim c_{y_0}$, where $c_{y_0}$ is the constant map with image $\{y_0\}$. If $y_0\notin Z$, the claim is clear. So suppose $y_0\in Z$. Since $Z$ is a proper subset of $Y$, there exists $y\notin Z$. If $\gamma\colon I\to Y$ is a smooth path from $y_0$ to $y$, then the map $F\colon X\times I\to Y$ defined by $F(x,t)=\gamma(t)$ is a homotopy from $c_{y_0}$ to $c_y$. So I see that $f\sim c_y$. Since the mod $2$ intersection number is homotopy invariant, it follows that $$ I_2(f,Z)=I_2(c_y,Z)=0 $$ since $c_y^{-1}(Z)=\emptyset$.
The thing bothering me is, how can I be sure there exists at least one $y\notin Z$ that is path connected to $y_0$?