Parametrized homotopy theory

Abstract

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.