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.

  • 1
    $\begingroup$ Your argument that the Euler characteristic of (connected !) compact lie groups is zero works perfectly fine. I don't know the rigidity theorem, so I don't know if your conclusion is ok. I hope you considered having a look at Milnor, Topology from the differentiable viewpoint. He explains (or sketches) two points you're looking for: using the Gauss map to get independence of the vector field, and using Morse theory to build the bridge to the Euler characteristic. I don't exactly know how to avoid Morse theory at that point, maybe you have to have a look at that too. $\endgroup$ – Ben Jul 2 '13 at 21:22
  • $\begingroup$ Many thanks, Ben! Now, I am sure, the Euler Characteristic of any Lie group is zero. The book by Milnor is great, but he only sketches the proof of the Poincaré-Hopf theorem... $\endgroup$ – amine Jul 11 '13 at 17:18

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