In Dubrovin/Fomenko/Novikov *Modern geometry--Methods and applications*, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on fixed points of self-maps is treated in the adjacent section 15.3 (see theorem 5.3.2 on page 131).

Even though the sections are adjacent, no mention is made of the relation between the two theorems, perhaps because this is "obvious" (take the flow of the vector field for small time, then apply Lefschetz, etc).

Is there a framework that would allow one to deduce both results as a consequence of a more general theorem?

globalinvariant. While all this is elementary to someone who is already familiar with these techniques, one might wonder if a more conceptual connection exists between these results via a more general one that woulddirectlyspecialize to both. – katz Jan 1 at 14:44