Pontryagin-Thom Construction and Poincaré Duality How is the Pontryagin-Thom Construction related to Poincaré Duality? These are two important ideas in topology which I understand separately and I've heard there is a link but I haven't found a reference for it. 
 A: The Pontryagin-Thom construction can be used to prove that 
if $M$ is a (smooth closed) n-manifold embedded in $\mathbf{R}^q$,
then $M_+$ is Spanier-Whitehead $q$-dual to the Thom space $T\nu$ of 
the normal bundle $\nu$ of the embedding.  If the manifold is oriented, 
then, with coefficients in $\mathbf{Z}$ for example,  $H^i(M)$ is isomorphic 
to $\tilde{H}^{i+q-n}(T\nu)$. In turn, by Spanier-Whitehead duality, that
is isomorphic to $H_{n-i}(M)$.  That is the modern proof of Poincare 
duality, and it applies equally well to generalized homology and cohomology
theories.
A: Here is a connection in a rather special, but important case. Suppose $M$ is an oriented compact manifold, $K\subset M$ an oriented compact submanifold. The Poincare dual of (the fundamental class of) $K$ is a cohomology class $\alpha\in H^{\dim M - \dim K}(M)$. Let $U\supset K$ be a tubular neighbourhood ($U$ is diffeomorphic to a tubular neighbourhood of the normal bundle), and let $\beta\in H^{\dim M - \dim K}(U,\partial U)=H^{\dim M - \dim K}(U/\partial U)$ be the Thom class ($U/\partial U$ is the Pontryagin-Thom construction). Then  $\alpha$ is the image of $\beta$ under $H^*(U,\partial U)=H^*(M,M-U)\to H^*(M)$.
Basically, "the Poincare dual of $K$ lives in an arbitrarily small neighbourhood of $K$".
Any explicit construction of Thom class will therefore give you an explicit construction of Poincare duals - see Mathai-Quillen formula in the case of de Rham cohomology.
