## Geometry & Topology

### Equivariant homotopy theory for pro–spectra

Halvard Fausk

#### Abstract

We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The $G$–homotopy theory is “pieced together” from the $G∕U$–homotopy theories for suitable quotient groups $G∕U$ of $G$; a motivation is the way continuous group cohomology of a profinite group is built out of the cohomology of its finite quotient groups. In the model category of equivariant spectra Postnikov towers are studied from a general perspective. We introduce pro–$G$–spectra and construct various model structures on them. A key property of the model structures is that pro–spectra are weakly equivalent to their Postnikov towers. We discuss two versions of a model structure with “underlying weak equivalences”. One of the versions only makes sense for pro–spectra. In the end we use the theory to study homotopy fixed points of pro–$G$–spectra.

#### Article information

Source
Geom. Topol., Volume 12, Number 1 (2008), 103-176.

Dates
Revised: 16 April 2007
Accepted: 23 July 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800016

Digital Object Identifier
doi:10.2140/gt.2008.12.103

Mathematical Reviews number (MathSciNet)
MR2377247

Zentralblatt MATH identifier
1135.55005

Subjects
Secondary: 18G55: Homotopical algebra

#### Citation

Fausk, Halvard. Equivariant homotopy theory for pro–spectra. Geom. Topol. 12 (2008), no. 1, 103--176. doi:10.2140/gt.2008.12.103. https://projecteuclid.org/euclid.gt/1513800016

#### References

• J F Adams, Lectures on Lie groups, W. A. Benjamin, New York-Amsterdam (1969)
• M Artin, B Mazur, Etale homotopy, Lecture Notes in Mathematics 100, Springer, Berlin (1986) Reprint of the 1969 original
• A A Beĭlinson, J Bernstein, P Deligne, Faisceaux pervers, from: “Analysis and topology on singular spaces, I (Luminy, 1981)”, Astérisque 100, Soc. Math. France, Paris (1982) 5–171
• A K Bousfield, The localization of spaces with respect to homology, Topology 14 (1975) 133–150
• A K Bousfield, On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001) 2391–2426
• G Carlsson, Structured stable homotopy theory and the descent problem for the algebraic $K$–theory of fields (2003) http://math.stanford.edu/\char'176gunnar/PDFpage.html
• R L Cohen, J D S Jones, G B Segal, Floer's infinite-dimensional Morse theory and homotopy theory, from: “The Floer memorial volume”, Progr. Math. 133, Birkhäuser, Basel (1995) 297–325
• D G Davis, Iterated homotopy fixed points for the Lubin–Tate spectrum
• D G Davis, The Lubin–Tate spectrum and its homotopy fixed point spectra, PhD thesis, Northwestern University (2003)
• D G Davis, The $E\sb 2$–term of the descent spectral sequence for continuous $G$–spectra, New York J. Math. 12 (2006) 183–191
• D G Davis, Homotopy fixed points for $L\sb {K(n)}(E\sb n\wedge X)$ using the continuous action, J. Pure Appl. Algebra 206 (2006) 322–354
• E S Devinatz, Small ring spectra, J. Pure Appl. Algebra 81 (1992) 11–16
• E S Devinatz, A Lyndon–Hochschild–Serre spectral sequence for certain homotopy fixed point spectra, Trans. Amer. Math. Soc. 357 (2005) 129–150
• E S Devinatz, M J Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004) 1–47
• W G Dwyer, Homology decompositions for classifying spaces of finite groups, Topology 36 (1997) 783–804
• W G Dwyer, E M Friedlander, Algebraic and etale $K$–theory, Trans. Amer. Math. Soc. 292 (1985) 247–280
• H Fausk, Artin and Brauer induction for compact Lie groups
• H Fausk, Atiyah–Segal completion for profinite groups in preparation
• H Fausk, $T$–model structures on chain complexes of presheaves
• H Fausk, D C Isaksen, Model structures on pro-categories, Homology, Homotopy Appl. 9 (2007) 367–398
• H Fausk, D C Isaksen, $t$–model structures, Homology, Homotopy Appl. 9 (2007) 399–438
• J P C Greenlees, Equivariant connective $K$–theory for compact Lie groups, J. Pure Appl. Algebra 187 (2004) 129–152
• J P C Greenlees, J P May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995) viii+178
• P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society, Providence, RI (2003)
• K H Hofmann, S A Morris, Projective limits of finite-dimensional Lie groups, Proc. London Math. Soc. $(3)$ 87 (2003) 647–676
• M Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathematical Society, Providence, RI (1999)
• M Hovey, J H Palmieri, N P Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997) x+114
• S Illman, The equivariant triangulation theorem for actions of compact Lie groups, Math. Ann. 262 (1983) 487–501
• D C Isaksen, A model structure on the category of pro-simplicial sets, Trans. Amer. Math. Soc. 353 (2001) 2805–2841
• D C Isaksen, Strict model structures for pro-categories, from: “Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001)”, Progr. Math. 215, Birkhäuser, Basel (2004) 179–198
• U Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988) 207–245
• L G Lewis, Jr, Change of universe functors in equivariant stable homotopy theory, Fund. Math. 148 (1995) 117–158
• L G Lewis, Jr, J P May, M Steinberger, J E McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics 1213, Springer, Berlin (1986)
• S Mac Lane, Categories for the working mathematician, second edition, Graduate Texts in Mathematics 5, Springer, New York (1998)
• M A Mandell, J P May, Equivariant orthogonal spectra and $S$–modules, Mem. Amer. Math. Soc. 159 (2002) x+108
• M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. $(3)$ 82 (2001) 441–512
• J P May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC (1996)
• O Renaudin, Spectres en diagramme dans les catégories modèles, Bull. Belg. Math. Soc. Simon Stevin 13 (2006) 1–30
• S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. $(3)$ 80 (2000) 491–511
• T tom Dieck, Transformation groups, de Gruyter Studies in Mathematics 8, Walter de Gruyter & Co., Berlin (1987)