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• W2079238422 abstract "We construct a (mod-I) Spanier-Whitehead dual for the etale homotopy type of any geometrically unibranched and projective variety over an algebraically closed field of arbitrary characteristic. The Thom space of the normal bundle to imbedding any compact complex manifold in a large sphere as a real submanifold provides a Spanier-Whitehead dual for the disjoint union of the manifold and a base point. Our construction generalises this to any characteristic. We also observe various consequences of the existence of a (mod-I) Spanier-Whitehead dual. Introduction. In this paper we establish the existence of a (mod-i) SpanierWhitehead dual for the etale homotopy type of any geometrically unibranched and projective variety over an algebraically closed field of arbitrary characteristic. This generalises the familiar construction of the Spanier-Whitehead dual for a compact complex manifold. In the forthcoming papers [J-2 and J-3] we make use of this to establish a Becker-Gottlieb type transfer for proper and smooth maps of smooth quasi-projective varieties. Recall that associated to every finite spectrum X there exists another spectrum, denoted DX, and called the Spanier-Whitehead dual of X, which is characterised by the following property. Let EO denote the sphere spectrum, while E denotes any arbitrary spectrum. Then there exists a map ,u: E* X A DX of spectra which induces isomorphisms [u]: h-q(X, E) -* hq(DX, E) and [T,p]: h-q(DX, E) -* hq(X, E) for all q. Here r is the map interchanging the two factors X and DX while h* ( , E) (h* ( , E)) is the generalised homology (generalised cohomology, respectively) with respect to the spectrum E. (See [Sw, pp. 321-335] for a general reference on the familiar notion of Spanier-Whitehead duality in topology.) If X happens to be the suspension spectrum associated to a compact closed real manifold M, there exists an explicit geometric construction of its SpanierWhitehead dual. If a is the normal bundle to imbedding M in a large sphere as a smooth closed submanifold, then a suitable desuspension of the Thom space of this bundle forms a Spanier-Whitehead dual for M+. We observe that this construction therefore provides a Spanier-Whitehead dual for any compact complex manifold, by merely forgetting its complex structure. This construction is generalised here for projective and geometrically unibranched varieties over any algebraically closed field. Received by the editors November 14, 1984 and, in revised form, June 23, 1985. 1980 Mathematics Subject Classification. Primary 14F35; Secondary 14F20, 55P25, 55P42, 55N20. (@)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page" @default.
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• W2079238422 date "1986-01-01" @default.
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• W2079238422 title "Spanier-Whitehead duality in étale homotopy" @default.
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• W2079238422 doi "https://doi.org/10.1090/s0002-9947-1986-0837804-4" @default.
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