How to convince myself of the following apparently paradoxical statement?
Poincaré-Hopf Index formula: The Euler characteristic is equal to the sum of winding numbers of a smooth vector field on the compact, oriented manifold $M$.
On one hand, winding numbers on an $n$-manifold $M$, depend only on the top homology group; i.e. $H_n(M)$. On the other hand $$\chi(M)=\sum_i (-1)^i b_i,\qquad b_i=\mathrm{rank}(H_i(M)).$$
In first case, $\chi(M)$ depends only $H_n(M)$ and in second one it depends on all homology groups. What is wrong?
The statement that the winding number only depends on the top homology group of $M$ is misleading; it depends on the choice of vector field as well. Which vector fields will be available is very much dependent on the topology of $M$ and not just on its top homology group; e.g. there is one with no zeros on a torus but not the sphere.