Is Poincare-Hopf index theorem connected with Leftschetz fixed point theorem?

Is Poincare-Hopf index theorem connected with Leftschetz fixed point theorem?

Lefschetz Fixed Point Theorem:
For a compact triangulable space $X$, and a continuous map $f:X\rightarrow
X$, we have
$$\sum_i(-1)^i\mathrm{Tr}(f_*|H_k(X,\mathbb{Q})=\sum_{x\in\mathrm{Fix}(f)}\mathrm{index}
_x f$$
Poincare-Hopf Index Theorem:
For a compact orientable differentiable manifold $M$, and a vector field
$v$ on $M$ with isolated singularities, we have $$\sum_i(-1)^i\dim
H_k(X,\mathbb{Q})=\sum_{x\in\mathrm{Sing}(v)}\mathrm{index}_xv$$
Observing the formal similarity between these formulae and the
compatibility of the conditions, I came up the idea that they are
connected.
For one possible connection, I imagined that we can construct a map
$f_v:M\rightarrow M$, by letting the points flow along $v$ (in a
sufficiently short time?). Then this map is very similar (homotopic?) to
$\mathrm{id}_M$, so trace is dimension. And by construction the two index
should agree. So we can view the latter as a corollary.
But I was not able to realize this. Can anyone tell me how to do it? Or it
can be found in some texts?