Multiplicative properties of Atiyah duality

Abstract

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. © 2004, Ralph L. Cohen.