Duality for smooth families in equivariant stable homotopy theory - Introduction

In this paper, we formulate and prove a duality theorem for the equivariant stable homotopy category, using the language of Verdier duality from sheaf theory. We work with the category of G-equivariant spectra (for a compact Lie group. G) parametrized over a G-space X, and consider a smooth equivariant family f : X --> Y, which is a G-equivariant bundle whose fiber is a smooth compact manifold, and with actions of subgroups of G varying smoothly over Y. Then our main theorem is a natural equivalence between a certain direct image functor f(*) and a "direct image with proper support functor" f(l), in the stable equivariant homotopy category over Y. In particular, the Wirthmuller and Adams isomorphisms in equivariant stable homotopy theory turn out to be special cases of this duality theorem.