ar X iv : m at h / 04 11 65 6 v 1 [ m at h . A T ] 3 0 N ov 2 00 4 Parametrized Homotopy Theory

Part I: We set the stage for our homotopical work with preliminary chapters on the point-set topology necessary to parametrized homotopy theory, the base change and other functors that appear in over and under categories, and generalizations of several classical results about equivariant bundles and fibrations to the context of proper actions of non-compact Lie groups. Part II: Despite its long history, the homotopy theory of ex-spaces requires further development before it can serve as the starting point for a rigorous modern treatment of parametrized stable homotopy theory. We give a leisurely account that emphasizes several issues that are of independent interest in the theory and applications of topological model categories. The essential point is to resolve problems about the homotopy theory of ex-spaces that are absent from the homotopy theory of spaces. In contrast to previously encountered situations, model theoretic techniques are intrinsically insufficient to a full development of the basic foundational properties of the homotopy category of ex-spaces. Instead, a rather intricate blend of model theory and classical homotopy theory is required. However, considerable new material on the general theory of topologically enriched model categories is also required. Part III: We give a systematic treatment of the foundations of parametrized stable homotopy theory, working equivariantly and with highly structured smash products and function spectra. The treatment is based on equivariant orthogonal spectra, which are simpler for the purpose than alternative kinds of spectra. Again, the parametrized context introduces many difficulties that have no nonparametrized counterparts and cannot be dealt with using standard model theoretic techniques. The space level techniques of Part II only partially extend to the spectrum level, and many new twists are encountered. Most of the difficulties are already present in the nonequivariant special case. Equivariantly, we show how change of universe, passage to fixed points, and passage to orbits behave in the parametrized setting. Part IV: We give a fiberwise duality theorem that allows fiberwise recognition of dualizable and invertible parametrized spectra. This allows direct application of the formal theory of duality in symmetric monoidal categories to the construction and analysis of transfer maps. The relationship between transfer for general Hurewicz fibrations and for fiber bundles is illuminated by the construction of fiberwise bundles of spectra, which are like bundles of tangents along fibers, but with spectra replacing spaces as fibers. Using this construction, we obtain a simple conceptual proof of a generalized Wirthmüller 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.