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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Continuous functors as a model for the equivariant stable homotopy category

Andrew J Blumberg

Algebraic & Geometric Topology 6 (2006) 2257–2295

arXiv: math.AT/0505512


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 {X(Sn)} obtained by evaluating X on spheres yields an Ω–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 GH.

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 WG 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 WG the collection {Z(A SW)} forms an ΩG–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 Γ–spaces, except when G is finite. We provide an explicit analysis of the failure of the category of equivariant Γ–spaces to model connective G–spectra, even for G = S1.

equivariant infinite loop space theory, excisive functors, Atiyah duality
Mathematical Subject Classification 2000
Primary: 55P42
Secondary: 55P47, 55P91
Forward citations
Received: 15 June 2005
Revised: 8 November 2006
Accepted: 9 November 2006
Published: 8 December 2006
Andrew J Blumberg
Department of Mathematics
Stanford University
450 Serra Mall
Stanford, California 94305