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Multiplicative properties of Atiyah duality
 Homology Homotopy Appl
"... Let M n be a closed, connected nmanifold. 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 SpanierWhitehead dual of M with a disjoint basepoint, M+. This dual can be viewed as the function spectrum, F(M, S) ..."
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Let M n be a closed, connected nmanifold. 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 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 [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, SpanierWhitehead, 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 ChasSullivan loop product in the homology of the free loop space of M.
Homology, Homotopy and Applications, vol.6(1), 2004, pp.269–281 MULTIPLICATIVE PROPERTIES OF ATIYAH DUALITY
"... Let Mn be a closed, connected nmanifold. 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 SpanierWhitehead dual ofM with a disjoint basepoint, M+. This dual can be viewed as the function spectrum, F (M,S), ..."
Abstract
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Let Mn be a closed, connected nmanifold. 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 SpanierWhitehead dual ofM 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, SpanierWhitehead, 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 ChasSullivan loop product in the homology of the free loop space of M.