Graph-diagonal formula for the Lefschetz number ($⟨(1,f{)}^{\ast }{\epsilon }_{\Delta },[M]⟩=L(f)$)

For the graph map $(1,f):M\to M\times M$, the cohomology class of the diagonal satisfiesIf $f$ is fixed-point-free, its graph misses the diagonal, so the pullback vanishes. This proves the Lefschetz fixed-point theorem for closed oriented manifolds.