Cohomology class of the diagonal
= Cohomology class of the diagonal
{title2=$\varepsilon_\Delta$}
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 $e_{p,i}^*\in H^{n-p}(M;\mathbb Q)$ by $\langle e_{p,i}e_{p,j}^*,[M]\rangle=\delta_{ij}$. With the product-orientation convention, the <Poincare dual> of the diagonal is
$$
\varepsilon_\Delta=\sum_{p,i}(-1)^{np}e_{p,i}^*\times e_{p,i}.
$$