Spanier-Whitehead duality in étale homotopy

Abstract

We construct a ( mod - l ) (\bmod {\text {-}}l) 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 - l ) (\bmod {\text {-}}l) Spanier-Whitehead dual.