How to prove that the Lefschetz number is invariante under homotopy?
We define the Lefschetz number as the number of $f : M \to M$ as the number of intersection of the map $g(x) = (x,f(x))$ with the diagonal of $M\times M$. Here, $M$ is a smooth manifold.
I think that is too confusing the definition of Lefschetz number. But I heard that if $f$ is homotopic to identity then $L(f) = \chi(M)$.
So I am trying to prove that Lefschetz number is invariant under homotopy, I mean, if two functions are homotopic, they have the same Lefschetz number.....
If we take the definition of a (global) lefschetz number as in $7.1$ of these notes, then the proof goes as follows: If $f$ is homotopic to $g$, the inclusion maps $id×f$ and $id×g$ of graph $f$ and graph $g$ are homotopic. Thus the global Lefschetz number of a map is homotopy invariant.