 Cohomology class of the diagonal

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