Parametrized homotopy theory
J. P. May and J. Sigurdsson
may@math.uchicago.edu, jsigurds@nd.edu
University of Chicago, University of Notre Dame
Primary 19D99, 55N20, 55P42; Secondary 19L99, 55N22, 55T25
Abstract: We provide rigorous modern foundations for parametrized
(equivariant, stable) homotopy theory in this four part monograph.
In Part I, we give preliminaries on the necessary point-set topology,
on base change and other relevant functors, and on generalizations
of various standard results to the context of proper actions of
non-compact Lie groups.
In Part II, we give a leisurely development of the homotopy theory
of ex-spaces that emphasizes several issues of independent
interest. It includes much new material on the general theory
of topologically enriched model categories. The essential
point is to resolve problems in the homotopy theory of ex-spaces
that have no nonparametrized counterparts. In contrast to
previously encountered situations, model theoretic techniques
are intrinsically insufficient for this purpose. Instead, a
rather intricate blend of model theory and classical homotopy
theory is required.
In Part III, we develop the homotopy theory of parametrized spectra.
We work 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, there are many difficulties that have no
nonparametrized counterparts and cannot be dealt with model
theoretically.
In Part IV, we give a fiberwise duality theorem that allows fiberwise
recognition of dualizable and invertible parametrized spectra. This
allows application of the formal theory of duality in symmetric
monoidal categories to the construction and analysis of transfer maps.
A construction of fiberwise bundles of spectra, which are like bundles
of tangents along fibers but with spectra replacing spaces as fibers,
plays a central role. Using it, we obtain 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.