Cohomology class of the diagonal (${\epsilon }_{\Delta }$)

Let $M$ be a closed oriented $n$-manifold, let $e_{p,i}$ be a homogeneous basis of $H^p(M;\mathbb{Q})$, and choose its Poincare-dual basis $e_{p,i}^{\ast }\in H^{n-p}(M;\mathbb{Q})$ by $⟨e_{p,i}{e}_{p,j}^{\ast },[M]⟩={\delta }_{ij}$. With the product-orientation convention, the Poincare dual of the diagonal is

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.