First recall that for any spectrum X one can define the Spanier-Whitehead dual DX which is characterized by the property that there exists \(\mu: \Sigma^ 0\to X\wedge DX\) \((\Sigma^ 0\) is the sphere spectrum) such that for any spectrum E one has isomorphisms \([\mu]: h^{-q}(X,E)\to h_ q(DX,E)\) and \([\tau\mu]: h^{-q}(DX,E)\to h_ q(X,E)\) (\(\tau\) interchanges the factors). If M is a real manifold then the Spanier- Whitehead dual to \(M_+\) is the Thom space of the stable normal bundle of M in \(S^ N.\)
The author proposes a construction of Spanier-Whitehead dual for the spectrum \(\hat Z{}_{ht+}\) which is the Bousfield-Kan completion at the prime \(\ell\) of the etale homotopy type of Z where Z is a connected and unibranched projective variety. The author first defines the Thom space of a bundle in etale topology and the Thom-Pontryagin collapsing map \[ TP: S^{2N} \to [(t_{E/Z})_{et}/(t_{E/Z}-Z)_{et}]^{\wedge} \] where E is a total space of the \({\mathbb{P}}^ n\)-bundle \(\alpha =\epsilon | (n+1)O_{{\mathbb{P}}^ n}(-1)\) (playing the role of the stable normal bundle), with \(\epsilon\) a trivial bundle on \({\mathbb{P}}^ n\) of sufficiently large rank, \(t_{E/Z}\) is the tubular neighborhood defined by D. Cox [Math. Scand. 42, 229-242 (1978; Zbl 0418.14012)] and \(t_{E/Z}-Z\) is the deleted tubular neighborhood. Then the author considers the diagonal map \[ diag: (t_{E/Z})_{et}/(t_{E/Z}- Z)_{et} \to (t_{E/Z})_{et}/(t_{E/Z}-Z)_{et}\wedge (t_{E/Z})_{et+} \] and the composition of the Thom-Pontryagin collapse and diag gives the duality map \[ \mu: S^{2N} \to [(t_{E/Z})_{et}/(t_{E/Z}-Z)_{et}\wedge (Z_{et+})]^{\wedge}. \] Standard constructions produce the maps between homology and cohomology mentioned above as in the classical case. Finally the author shows that some standard applications of Spanier-Whitehead duality can be carried through in the etale case and constructs the Becker-Gottlieb transfer in etale topology and mod \(\ell^{\nu}\) Poincaré duality between etale K-homology and K-cohomology.