I'm reading the basics of differential topology to try to understand the Poincaré-Hopf theorem, its proof and its applications. My plan is as follows:
1) Study transversality: its homotopy stability + genericity (any application is homotopic (isotopic) to a transverse application to a given submanifold).
2) Consider the intersection number modulo 2 + oriented intersection number.
3) Study Lefschetz maps (and their fixed points).
So far, this is technical but not too complicated...I do not know how to show that:
1) the formula $\sum_i\mathrm{ind}(X,x_i)$ gives the Euler-Poincaré characteristic (that I can define from de Rham cohomology I know and not from the singular homology, I do not know yet).
2) Show that the formula in the Poincaré-Hopf theorem does not depend on the choice of the vector field. In particular how to link this issue to the Lefschetz fixed point?
On the other hand, I have a problem to calculate the index of a vector field in an isolated singular point. If the point is non-degenerate, the calculation is simple. Otherwise, I do not know whether to go through the calculation of topological index of the Gauss map (where calculations are not always easy, because I must start from a regular value $y$, determine the fiber $f^{-1}(y)=\{x_1,...,x_k\}$ and calculate the sum $\sum\mathrm{sign}\det T_{x_i}f$).
I am afraid the following reasoning I made is wrong: the Euler characteristic of any Lie group is zero. For this, I start with a compact Lie group, as it is parallelizable, it admits a vector field without singularities and from the Poincaré-Hopf theroem, its Euler characteristic is zero. By the rigisity theorem of Mostow, any Lie group has the homotopy type of a compact Lie group and therefore also of Euler characteristic zero.
Thank you to enlighten me on these points and suggest me the necessary literature to treat this questions.