Let $M$ be a smoothly triangulated compact $d$-dimensional manifold.  Consider the subcomplex $C_*^{\pitchfork T}(M)$ of smooth singular chains which are transverse to the triangulation. An inductive chain homotopy construction establishes that these are quasi-isomorphic to all smooth, and thus all singular, chains.  

Define the intersection map $I : C_n^{\pitchfork T}(M; R) \to C^{d-n}_\Delta(M; R)$ (the latter being simplicial cochains arising from the triangulation) by sending $\sigma : \Delta^d \to M$ to the cochain whose value on the an element of the triangulation whose characteristic  map is $\iota : \Delta^{d-n} \to M$ is the count of the zero manifold given by the pullback of $\sigma$ and $\iota$.  Here either $R$ is $\mathbb{Z}/2$ or $M$ must be oriented and the count is with the usual signs, and one uses some version (such as [this][1]) of transversality for manifolds with corners. 

Fun exercise: with appropriate signs, I is a map of chain complexes.  (Hint: as in the proof that degree as defined by counting preimages is homotopy invariant, this relies on the classification of one-manifolds.) Poincar\'e duality implies that the domain and range of $I$ are quasi-isomorphic.  

Question: why is $I$ a quasi-isomorphism?  

I think I can prove this, but only in the mod-two setting, by using Thom's seminal work on bordism and Quillen's elementary approach to cobordism (just the definitions of his "elementary" paper - not the main results, which to me are quite deep despite the title of the paper).  But there must be a more direct argument, which covers the oriented case as well, and it seems like this should be in the literature somewhere - from the 1940's maybe? 

(Motivation: Greg Friedman, Anibal Medina and I have what we think is a new approach to questions such as https://mathoverflow.net/questions/9457/do-chains-and-cochains-know-the-same-thing-about-the-manifold through vector field flows, and would like to build on existing knowledge of the interplay between intersection and duality.)


  [1]: https://arxiv.org/abs/0910.3518