= Compactly supported dual of a compact submanifold
{title2=$\varepsilon_Y\in H_c^{\operatorname{codim}Y}(M)$}
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.
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