Cohomology class of a cooriented submanifold (${\epsilon }_Y\in H^{\mathrm{codim}Y}(X;\mathbb{Z})$)

An oriented normal bundle of a compact embedded submanifold has a Thom class. A tubular neighborhood and excision identify it with a relative class in $H^d(X,X\setminus Y)$. Its image in absolute cohomology is ${\epsilon }_Y$. It restricts to zero off Y and, in an oriented closed ambient manifold, represents the Poincare dual of Y with the normal-first induced orientation.