  We give rigorous foundations for parametrized homotopy theory in this
monograph. After preliminaries on point-set topology, base change functors, and
proper actions of non-compact Lie groups, we develop the homotopy theory of
equivariant ex-spaces (spaces over and under a given space) and of equivariant
parametrized spectra. We emphasize several issues of independent interest and
include much new material on the general theory of topologically enriched model
categories. The essential point is to resolve problems in parametrized homotopy
theory that have no nonparametrized counterparts. In contrast to previously
encountered situations, model theoretic techniques are intrinsically
insufficient for this. Instead, a rather intricate blend of model theory and
classical homotopy theory is required. Stably, we work with equivariant
orthogonal spectra, which are simpler for the purpose than alternative kinds of
spectra and give highly structured smash products. We then give a fiberwise
duality theorem that allows fiberwise recognition of dualizable and invertible
parametrized spectra. This allows application of formal duality theory to the
construction and analysis of transfer maps. A construction of fiberwise bundles
of spectra plays a central role and leads to a simple conceptual proof of a
generalized Wirthmuller isomorphism theorem that calculates the right adjoint
to base change along an equivariant bundle with manifold fibers in terms of a
shift of the left adjoint. Due to the generality of our bundle theoretic
context, the Adams isomorphism theorem relating orbit and fixed point spectra
is a direct consequence.
