= Graph-diagonal formula for the Lefschetz number
{title2=$\langle(1,f)^*\varepsilon_\Delta,[M]\rangle=L(f)$}
For the graph map $(1,f):M\to M\times M$, the <cohomology class of the diagonal> satisfies
$$
\left\langle(1,f)^*\varepsilon_\Delta,[M]\right\rangle
=\sum_p(-1)^p\operatorname{tr}\left(f^*:H^p(M;\mathbb Q)\to H^p(M;\mathbb Q)\right).
$$
If $f$ is fixed-point-free, its graph misses the diagonal, so the pullback vanishes. This proves the <Lefschetz fixed-point theorem> for closed oriented manifolds.
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