I am troubled by the following fact:
If $f,g : X\to Y$ are homotopic and both transversal to $Z$ then the mod.$2$ intersection numbers are equal $I_2(f,Z)=I_2(g(Z)$. (the book by Guillemin and Pollack, Differential Topology, page 78).
If $f,g : [-2,2]\to[-2,2]\times\mathbb{R}$, with $f(x)=(x,x^2)$ and $g(x)=(x,x^3)$ then $f$ and $g$ are homotopic (by $H(t,x)=(x,tx^3+(1-t)x^2)$) to $\Delta=\{(x,x),\ x\in[-2,2]\}$ but $I_2(f,\Delta)=2[2]=0[2]$ and $I_2(g,\Delta)=3[2]=1[2]$.
I don't know what I am missing here! Thanks for any help!