I know of two different definitions of the Lefschetz number of a map. I'd like to see a proof of the fact that they coincide (when they are both defined).
The first one would be:
Given a finite simplicial complex $X$, $L(f) = \sum_k (-1)^k tr(f_{*, k}) $, where $f_{*, k} : H_k(X, \mathbb{Q} ) \to H_k(X, \mathbb{Q} ) $ is the map induced by $f$.
And the second one:
Let X be a closed, oriented manifold and $f:X \to X$ a smooth map. Then $L(f) = I( \Delta, graph(f)) $, where $\Delta \subset X \times X$ is the diagonal in $X \times X$, $graph(f) \subset X\times X$ is the graph of $f$, and $I( \Delta, graph(f))$ is their (oriented) intersection number.
When $f$ is the identity, I can prove they coincide by constructing a vector field with index $\chi(X)$ and using the expression for $L(f) $ in terms of the local Lefschetz numbers, but I don't see any way this could be modified to work in the general case.
A sketch/idea of the proof or a reference to where I could find one would be greatly appreciated. Thanks!
