Let us consider a closed oriented smooth manifold M. It is well known that a smooth combinatorial triangulation can be constructed for it. That is to say, a homeomorphism from the geometric realization of a simplicial complex such that the characteristic map of each simplex extends to a smooth map from an open neighborhood of the standard simplex into M, and such that the link of every simplex is a topological sphere. We can construct a dual CW structure to this triangulation by considering the barycentric subdivision of the triangulation, and amalgamating for each original simplex $\sigma$ the simplices of the form $(\sigma, \ldots)$ in the subdivision. A nice proof of Poincare duality can now be given by defining a map to the integers from the simplicial chains tensor the cellular chains with values determined by the signed intersection of the cells; from which it seems natural to ask: how can the dual cell structure be smoothed so that the characteristic maps are transverse to the triangulation?
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$\begingroup$ Are you asking for an explicit construction or an argument that it can be done? To my mind, this would just be repeated applications of isotopy extension. $\endgroup$– Connor MalinJan 25, 2021 at 16:34
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$\begingroup$ A historical note about Anibal's (pertinent and uneasy) question: this proof was Poincare's original proof of the Poincare duality. $\endgroup$– Gael MeigniezFeb 16, 2021 at 7:17
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