I'm studying compact surfaces and in particular the relationship between zeros of vector fields defined on them and Euler characteristic of the surface herself.
Let be $S$ a compact (smooth) surface and let be $V$ a vector field over $S$ (that is, a continuous function $V:S\longrightarrow \mathbf{R}^3$ such that $V(p)\in T_p S$ for every $p\in S$). I've understood that:
1) If $\xi_1,\ldots,\xi_k$ are the critical points of $V$ (say, his zeros in $S$), we can associate to every $\xi_j$ an integer $I(V,\xi_j)$ called index of the critical point defined as the degree of the map $V\circ f$, where $f:\mathbf{S}^1\longrightarrow S$ is a closed simple curve in $S$ (homeomorphic to the plane circle) around $\xi_j$.
2) The sum of the integers $I(V,\xi_j) $ for $j=1,\ldots,k$ does not depend by the vector field and equals in fact Euler characteristic $\chi(S)$ of the surface (this is called Poincaré-Hopf theorem).
3) An immediate consequence should be that, if $S$ admits a non-vanishing vector field, then $\chi (S)=0$.
4) I would be pleased to prove also the converse: if $\chi (S)=0$ then $S$ admits a non-vanishing vector field. My approach uses the Classification theorem of surfaces, observing that the only surface (up to homeomorphism) with zero Euler characteristic are the torus (in case of orientable surfaces) and the Klein bottle (say $\mathbf{RP}^2\odot \mathbf{RP}^2$ for non orientable surfaces). So if I show two non-vanishing vector fields respectively in torus and Klein bottle, the fact should be proved.
I got a little struck regarding this whole path in algebraic topology, so I ask if my considerations are correct. I would be very grateful for references, especially if they contains topological proofs of Poincaré-Hopf theorem (without use of analytic instruments such that Morse functions or Brouwer analytic degree).
Thank you in advance!
Best Answer
Basically, this is correct, but $V$ should map to $\mathbb R^3$ (but this makes sense for surfaces not sitting in $\mathbb R^3$ as well) and to make sense of degree you need to look at a local-coordinate formulation of $(V\circ f)/\|V\circ f\|\colon S^1\to S^1$.
The converse result is quite deep and holds in all dimensions. I recommend you look at Milnor's Topology from a Differentiable Viewpoint and Guillemin and Pollack's Differential Topology. Morris Hirsch's book by the same title is quite a bit more advanced.
Of course, it's quite easy to give a non-vanishing vector field on the torus. Just comb the hairs on each circle. If you pick the right set of circles, this vector field should descend to a well-defined vector field on the Klein bottle (remember the torus is a two-fold covering space of the Klein bottle).