## Algebraic & Geometric Topology

### On the Adams isomorphism for equivariant orthogonal spectra

#### Abstract

We give a natural construction and a direct proof of the Adams isomorphism for equivariant orthogonal spectra. More precisely, for any finite group $G$, any normal subgroup $N$ of $G$, and any orthogonal $G$–spectrum $X$, we construct a natural map $A$ of orthogonal $G ∕ N$–spectra from the homotopy $N$–orbits of $X$ to the derived $N$–fixed points of $X$, and we show that $A$ is a stable weak equivalence if $X$ is cofibrant and $N$–free. This recovers a theorem of Lewis, May and Steinberger in the equivariant stable homotopy category, which in the case of suspension spectra was originally proved by Adams. We emphasize that our Adams map $A$ is natural even before passing to the homotopy category. One of the tools we develop is a replacement-by-$Ω$–spectra construction with good functorial properties, which we believe is of independent interest.

#### Article information

Source
Algebr. Geom. Topol., Volume 16, Number 3 (2016), 1493-1566.

Dates
Revised: 22 July 2015
Accepted: 21 September 2015
First available in Project Euclid: 28 November 2017

https://projecteuclid.org/euclid.agt/1511895855

Digital Object Identifier
doi:10.2140/agt.2016.16.1493

Mathematical Reviews number (MathSciNet)
MR3523048

Zentralblatt MATH identifier
1350.55015

#### Citation

Reich, Holger; Varisco, Marco. On the Adams isomorphism for equivariant orthogonal spectra. Algebr. Geom. Topol. 16 (2016), no. 3, 1493--1566. doi:10.2140/agt.2016.16.1493. https://projecteuclid.org/euclid.agt/1511895855

#### References

• J,F Adams, Prerequisites (on equivariant stable homotopy) for Carlsson's lecture, from: “Algebraic topology”, (I Madsen, B Oliver, editors), Lecture Notes in Math. 1051, Springer, Berlin (1984) 483–532
• M B ökstedt, W,C Hsiang, I Madsen, The cyclotomic trace and algebraic $K$–theory of spaces, Invent. Math. 111 (1993) 465–539
• T Br öcker, T tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics 98, Springer, New York (1985)
• G Carlsson, A survey of equivariant stable homotopy theory, Topology 31 (1992) 1–27
• T tom Dieck, Transformation groups and representation theory, Lecture Notes in Math. 766, Springer, Berlin (1979)
• T tom Dieck, Transformation groups, de Gruyter Studies in Mathematics 8, de Gruyter, Berlin (1987)
• T tom Dieck, Algebraic topology, European Mathematical Society, Zürich (2008)
• B,I Dundas, T,G Goodwillie, R McCarthy, The local structure of algebraic $K$–theory, Algebra and Applications 18, Springer, London (2013)
• L Hesselholt, I Madsen, On the $K$–theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997) 29–101
• L Hesselholt, I Madsen, On the $K$–theory of local fields, Ann. of Math. (2) 158 (2003) 1–113
• M,A Hill, M,J Hopkins, D,C Ravenel, On the non-existence of elements of Kervaire invariant one, Ann. of Math. (2) 184 (2016) 1–262
• J Hollender, R,M Vogt, Modules of topological spaces, applications to homotopy limits and $E\sb \infty$ structures, Arch. Math. $($Basel$)$ 59 (1992) 115–129
• L,G Lewis, Jr, J,P May, M Steinberger, J,E McClure, Equivariant stable homotopy theory, Lecture Notes in Math. 1213, Springer, Berlin (1986)
• W Lück, H Reich, The Baum–Connes and the Farrell–Jones conjectures in $K$– and $L$–theory, from: “Handbook of $K$–theory, Volume 2”, (E,M Friedlander, D,R Grayson, editors), Springer, Berlin (2005) 703–842
• W Lück, H Reich, J Rognes, M Varisco, Algebraic K-theory of group rings and the cyclotomic trace map, preprint (2015)
• S Mac Lane, Categories for the working mathematician, 2nd 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. 755, Providence, RI (2002)
• M,A Mandell, J,P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441–512
• J,P May, A concise course in algebraic topology, University of Chicago Press (1999)
• S Schwede, Lectures on equivariant stable homotopy theory, preprint (February 7, 2016) \setbox0\makeatletter\@url http://www.math.uni-bonn.de/people/schwede/equivariant.pdf {\unhbox0
• N,P Strickland, The category of CGWH spaces, preprint (2009) Available at \setbox0\makeatletter\@url http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf {\unhbox0
• F Waldhausen, Algebraic $K$–theory of spaces, from: “Algebraic and geometric topology”, (A Ranicki, N Levitt, F Quinn, editors), Lecture Notes in Math. 1126, Springer, Berlin (1985) 318–419
• S Waner, Equivariant homotopy theory and Milnor's theorem, Trans. Amer. Math. Soc. 258 (1980) 351–368
• K Wirthmüller, Equivariant homology and duality, Manuscripta Math. 11 (1974) 373–390