Eliashberg's h-principle for maps with Thom-Boardman singularities – II

In the previous talk the following theorem was formulated: A continuous map $f:M\to N$ of $n$-manifolds is homotopic to a smooth map with prescribed Thom-Boardman singularities if and only if the vector bundle $f^*(TN)$ is isomorphic to a certain vector bundle $T^\phi M$ (which we have constructed from a germ of the prescribed singularities). The plan of a proof was given.
In this talk we will see how the proof works in the case $n=2$, when $M$ and $N$ are closed surfaces. Namely, we will prove a necessary and sufficient condition for a collection of curves with marked points in $M$ to be the set of folds and cusps of some smooth map homotopic to $f$. I will remind all definitions, so this talk is supposed to be independent of the previous one.