 Compactly supported dual of a compact submanifold

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.