Multiplicative properties of Atiyah duality

Let Mn 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 . Introduction Throughout this paper M will denote a fixed, closed, connected n-manifold. Let e : M →֒ R be an embedding into Euclidean space, and let ηe be the normal bundle. A well known theorem of Atiyah [1] states that the Thom space, Me , is a kSpanier Whitehead dual of M . One can normalize with respect to k in the following way. Let M denote the spectrum, M = ΣMe . The homotopy type of this spectrum is well defined, in that it is independent of the embedding e. Atiyah’s theorem can be restated as saying that there is a homotopy equivalence of spectra, α : M ≃ −−−−→ F (M,S) where S is the sphere spectrum, and F (M,S) is the spectrum whose k space is the unbased mapping space, F (M,S). Recently, symmetric monoidal categories of spectra have been developed ([5], [7], [9]). In these categories, the dual F (M,S) has the structure of a commutative ring spectrum. The goal of this The author was partially supported by a grant from the NSF