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2006 Continuous functors as a model for the equivariant stable homotopy category
Andrew Blumberg
Algebr. Geom. Topol. 6(5): 2257-2295 (2006). DOI: 10.2140/agt.2006.6.2257

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 {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 AWG the collection {Z(ASW)} 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.

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Andrew Blumberg. "Continuous functors as a model for the equivariant stable homotopy category." Algebr. Geom. Topol. 6 (5) 2257 - 2295, 2006. https://doi.org/10.2140/agt.2006.6.2257

Information

Received: 15 June 2005; Revised: 8 November 2006; Accepted: 9 November 2006; Published: 2006
First available in Project Euclid: 20 December 2017

zbMATH: 1186.55002
MathSciNet: MR2286026
Digital Object Identifier: 10.2140/agt.2006.6.2257

Subjects:
Primary: 55P42
Secondary: 55P47, 55P91

Rights: Copyright © 2006 Mathematical Sciences Publishers

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