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!

  • 1
    $\begingroup$ You might start with pp. 126-129 of Bott/Tu's Differential Forms in Algebraic Topology. $\endgroup$ Nov 9, 2017 at 23:53
  • $\begingroup$ You might also look at comments for this question. Another good source is Godbillon's little book Topologie Algébrique (in French, of course). $\endgroup$ Nov 10, 2017 at 18:14
  • $\begingroup$ Thank you very much. It's been a couple months, but I finally got there. Using the exercise in page 129 of Bott/Tu's book and the formula for the intersection number $I(\Delta, graph(f))$ in terms of the signs of $d_xf - id$ for $x$ in the intersection (found in Guillemin/Pollack's book, for example), the coincidence of the definitions is proved! $\endgroup$
    – Mauro
    Feb 4, 2018 at 2:36


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