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](https://arxiv.org/abs/0910.3518)) 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.) Poincaré 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 [Do chains and cochains know the same thing about the manifold?](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.)



Intersection map giving rise to Poincaré duality
In a related and highly relevant comment thread, Mike Miller pointed me to this [preprint](https://arxiv.org/abs/1409.1121) of Lipyanskiy. I'm sure there are arguments which work, such as what Joshua and Dmitri and I discuss in the comments. But before I forget I'd like to point to Lipyanskiy's work, especially Section 12, and mark the question as answered.


