Multiplicative properties of Atiyah duality
Abstract
Let $M^n$ be a closed, connected $n$manifold. Let $\mtm$ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that $\mtm$ is homotopy equivalent to the SpanierWhitehead 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 \cite{hss}, \cite{ship}. In this paper we prove that $\mtm$ also has a natural, geometrically defined, structure of a commutative, symmetric ring spectrum, in such a way that the classical duality maps of Alexander, SpanierWhitehead, and Atiyah define an equivalence of symmetric ring spectra, $\alpha : \mtm \to F(M, S)$. We discuss applications of this to Hochshield cohomology representations of the ChasSullivan loop product in the homology of the free loop space of $M$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2004
 arXiv:
 arXiv:math/0403486
 Bibcode:
 2004math......3486C
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Geometric Topology;
 55P25;
 57R40;
 55P43
 EPrint:
 minor revisions. published version https://projecteuclid.org/euclid.hha/1139839554