Continuous functors as a model for the equivariant stable homotopy category

Abstract

It is a classical observation that a based continuous functor $X$ from the category of finite CW–complexes to the category of based spaces that takes homotopy pushouts to homotopy pullbacks “represents” a homology theory—the collection of spaces $\left\{X\left(S^n\right)\right\}$ obtained by evaluating $X$ on spheres yields an $\Omega$–prespectrum. Such functors are sometimes referred to as linear or excisive. The main theorem of this paper provides an equivariant analogue of this result. We show that a based continuous functor from finite $G$–CW–complexes to based $G$–spaces represents a genuine equivariant homology theory if and only if it takes $G$–homotopy pushouts to $G$–homotopy pullbacks and satisfies an additional condition requiring compatibility with Atiyah duality for orbit spaces $G∕H$.

Our motivation for this work is the development of a recognition principle for equivariant infinite loop spaces. In order to make the connection to infinite loop space theory precise, we reinterpret the main theorem as providing a fibrancy condition in an appropriate model category of spectra. Specifically, we situate this result in the context of the study of equivariant diagram spectra indexed on the category ${\mathcal{W}}_G$ of based $G$–spaces homeomorphic to finite $G$–CW–complexes for a compact Lie group $G$. Using the machinery of Mandell–May–Schwede–Shipley, we show that there is a stable model structure on this category of diagram spectra which admits a monoidal Quillen equivalence to the category of orthogonal $G$–spectra. We construct a second “absolute” stable model structure which is Quillen equivalent to the stable model structure. There is a model-theoretic identification of the fibrant continuous functors in the absolute stable model structure as functors $Z$ such that for $A\in {\mathcal{W}}_G$ the collection $\left\{Z\left(A\wedge S^W\right)\right\}$ forms an $\Omega$–$G$–prespectrum as $W$ varies over the universe $U$. Thus, our main result provides a concrete identification of the fibrant objects in the absolute stable model structure.

This description of fibrant objects in the absolute stable model structure makes it clear that in the equivariant setting we cannot hope for a comparison between the category of equivariant continuous functors and equivariant $\Gamma$–spaces, except when $G$ is finite. We provide an explicit analysis of the failure of the category of equivariant $\Gamma$–spaces to model connective $G$–spectra, even for $G=S^1$.