Multiplicative properties of Atiyah duality

Abstract

Let $M^n$ be a closed, connected $n$-manifold. Let $M^{-\tau}$ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that $M^{-\tau}$ 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 $M^{-\tau}$ 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, $\alpha :$M^{-\tau}$ \to 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 $M$.