On the notion of multiplicity of a fixed point [closed]

Question

I am trying to understand the notion of multiplicity of a fixed point of a map $f: M \to M$, say, $M$ being a smooth closed manifold, and $f$ being a smooth diffeomorphism.

There is a notion of fixed point index that appears in the Lefschetz number of $f$, but I don't really understand it geometrically.

On the other hand, there seems to be a notion of multiplicity of fixed points. For this notion, which I would like to make precise, I would like to show that the sum of fixed points of $f$, counted with multiplicity, is greater than its Lefschtez number.

Is there a precise way to define / understand the notion of multiplicity so that the above statement holds ?

Answer 1

Note that the fixed points of $f$ can be characterized as the intersection points of the graph of $f$, which is a submanifold of $M\times M$, with the diagonal of $M\times M$

$$\Delta_M=\big\{ (x,y)\in M\times M:\;\;x=y\big\}. $$

The Lefchetz number of $f$ at a fixed point $x_0$ is then the local intersection number of the Graph $\Gamma_f$ with the diagonal $\Delta_m $ at $(x_0,x_0)\in M$. See Sec 7.3. of these notes.