1.1. Simplicial sets 5 1.2. Symmetric spectra 6

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.