Does someone have a nice proof for Proposition 11.14 in Farb&Margalits "Primer to Mapping Class Groups", which states the following:
Let $S$ be a closed surface with a singular foliation and $P_s$ be the number of prongs of this foliation at a singular point $s$. A copy of their picture of a 4-pronged singularity is
We then have the Euler-Poincaré-formula
$$ 2 \chi(S)=\sum(2-P_s)$$
i.e. the Euler-characteristic determines the "sum of multiplicities" of the singularities. They refer to a proof in a french paper and the Poincaré-Hopf formula for vector fields. If the foliation is orientable I can believe the reference to the Poincaré-Hopf formula, but for uneven prongs it needs a little bit more work. Since my Google-skills only yielded the traditional Euler-Poincaré-formula, my questions are:
a) Is there a more hands on proof of this proposition (or any proof which is not french)? You can find my approach below, but I would like someone to check it.
b) Is there a similar formula for punctured surfaces (since the puncture might hide singularities)
Otherwise can someone please proofread my approach (and assure if it is understandable for students): Cut along the leaves which go through the singularities and for all the leaves which do not hit another cut, close them up transversally whenever they get close (i.e. cover the mannifold with charts s.t. the leaves are horizontal in each leave and close them whenever they are in the same chart). This cuts the mannifold into a bunch of areas which are topologically cylinders, since we can continuously walk along a leave (with smooth jumps whenever we closed up) until we get "back" to the start. Hence we can cut each of these cylinders by a transversal arc to get a disk. In total we get a decomposition into
$\begin{align*} \#Vertices&=(\#s+2\#jumps+2\#cylinders)\\ \#Edges&=(\sum P_s/2+2\#jumps+\#cylinders)\\ \#Areas&=\#cylinders\\[1 em] \Rightarrow \chi(S)&=(V-E+A)\\ &=(\#s+2\#jumps+2\#cylinders)-(\sum P_s/2+2\#jumps+3\#cylinders)+\#cylinders\\ &=\#s-\sum P_s/2=\sum (1-P_s/2) \end{align*}$
problem: For me it seems there is ad hoc an issue with the „jumps“ as its not clear, if we cut another prong with it. Hopefully someone here can resolve this issue.