Poincare dual ($\mathrm{PD}[Y]$)

For an oriented codimension-$r$ closed submanifold $Y\subset M$, its Poincare dual is the class in $H^r(M)$ whose cup-product evaluation against any complementary class equals intersection with $Y$.

For a compact oriented submanifold without boundary in an oriented manifold, the normal Thom class gives a compactly supported cohomology class dual to its pushed-forward fundamental class. It is independent of the tubular neighborhood and its ordinary-cohomology image vanishes on the complement of the submanifold. A noncompact submanifold closed only as a subset generally has its dual in ordinary cohomology, using locally finite homology instead.