Multiplicative properties of Atiyah duality

Let M n be a closed, connected n-manifold.Let M -τ denote the Thom spectrum of its stable normal bundle.A well known theorem of Atiyah states that M -τ 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 [7], [12] [9].In this paper we prove that M -τ 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, α : M -τ → F (M, S).We discuss applications of this to Hochschild cohomology representations of the Chas-Sullivan loop product in the homology of the free loop space of M .