TITLE: What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?
QUESTION [8 upvotes]: 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?

REPLY [9 votes]: The Poincaré-Hopf theorem is a consequence of the Lefschetz fixed point theorem if you accept the fairly standard fact that the Euler characteristic of a compact manifold $M$ is equal to the self-intersection of the diagonal $\Delta \subset M\times M$. Here is why :
Since $M$ is compact, the vector field $X$ gives rise to a flow $(\varphi _t)_{t\in\mathbb{R}}$; the fixed points of $\varphi _t$ are the zeros of $X$. Suppose these zeros are isolated, and let $p$ be one of them. In local coordinates, we have $\varphi _t(x)= x+tX(x)+o(t)$ around $p$, hence $d\varphi _t(p)=I+t\,dX(p)+o(t)$. This implies that for small $t$, the index of $X$ at $p$ is equal to the index of $p$ as a fixed point of $\varphi _t$. Lefschetz theorem tells you that the sum of these indices is the intersection number $(\Delta .\Delta _t)$, where $\Delta _t$ is the graph of $\varphi _t$. This integer depends continuously on $t$, hence is constant, hence equal to $(\Delta .\Delta) =e(M)$.