Multiplicative properties of Atiyah duality

Let $M^n$ be a closed, connected $n$-manifold. Let $\text{mtm}$ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that $\text{mtm}$ is homotopy equivalent to the Spanier-Whitehead dual of $M$ with a disjoint basepoint, $M_{+}$. This dual can be viewed as the function spectrum, $F(M,S)$, where $S$ is the sphere spectrum. $F(M,S)$ has the structure of a commutative, symmetric ring spectrum in the sense of \cite{hss}, \cite{ship}. In this paper we prove that $\text{mtm}$ also has a natural, geometrically defined, structure of a commutative, symmetric ring spectrum, in such a way that the classical duality maps of Alexander, Spanier-Whitehead, and Atiyah define an equivalence of symmetric ring spectra, $\alpha :\text{mtm}\to F(M,S)$. We discuss applications of this to Hochshield cohomology representations of the Chas-Sullivan loop product in the homology of the free loop space of $M$.