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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.

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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.

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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.