## Interpreting the Bökstedt smash product as the norm

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- by Vigleik Angeltveit, Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill and Tyler Lawson PDF
- Proc. Amer. Math. Soc.
**144**(2016), 5419-5433

## Abstract:

This paper compares two models of the equivariant homotopy type of the smash powers of a spectrum, namely the “Bökstedt smash product” and the Hill-Hopkins-Ravenel norm.## References

- Andrew J. Blumberg and Michael A. Mandell,
*The homotopy theory of cyclotomic spectra*, Geom. Topol.**19**(2015), no. 6, 3105–3147. MR**3447100**, DOI 10.2140/gt.2015.19.3105 - Andrew J. Blumberg,
*Continuous functors as a model for the equivariant stable homotopy category*, Algebr. Geom. Topol.**6**(2006), 2257–2295. MR**2286026**, DOI 10.2140/agt.2006.6.2257 - M. Bökstedt,
*Topological Hochschild homology*, Preprint. - M. Bökstedt, W. C. Hsiang, and I. Madsen,
*The cyclotomic trace and algebraic $K$-theory of spaces*, Invent. Math.**111**(1993), no. 3, 465–539. MR**1202133**, DOI 10.1007/BF01231296 - Bjørn Ian Dundas,
*Relative $K$-theory and topological cyclic homology*, Acta Math.**179**(1997), no. 2, 223–242. MR**1607556**, DOI 10.1007/BF02392744 - Wojciech Chachólski and Jérôme Scherer,
*Homotopy theory of diagrams*, Mem. Amer. Math. Soc.**155**(2002), no. 736, x+90. MR**1879153**, DOI 10.1090/memo/0736 - A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May,
*Rings, modules, and algebras in stable homotopy theory*, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. MR**1417719**, DOI 10.1090/surv/047 - Thomas G. Goodwillie,
*Relative algebraic $K$-theory and cyclic homology*, Ann. of Math. (2)**124**(1986), no. 2, 347–402. MR**855300**, DOI 10.2307/1971283 - Lars Hesselholt and Ib Madsen,
*On the $K$-theory of finite algebras over Witt vectors of perfect fields*, Topology**36**(1997), no. 1, 29–101. MR**1410465**, DOI 10.1016/0040-9383(96)00003-1 - M. A. Hill, M. J. Hopkins, and D. C. Ravenel,
*On the non-existence of elements of Kervaire invariant one*, Preprint, arXiv:0908.3724. - John A. Lind,
*Diagram spaces, diagram spectra and spectra of units*, Algebr. Geom. Topol.**13**(2013), no. 4, 1857–1935. MR**3073903**, DOI 10.2140/agt.2013.13.1857 - Sverre Lunøe-Nielsen and John Rognes,
*The topological Singer construction*, Doc. Math.**17**(2012), 861–909. MR**3007679** - Ib Madsen,
*Algebraic $K$-theory and traces*, Current developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge, MA, 1994, pp. 191–321. MR**1474979** - M. A. Mandell and J. P. May,
*Equivariant orthogonal spectra and $S$-modules*, Mem. Amer. Math. Soc.**159**(2002), no. 755, x+108. MR**1922205**, DOI 10.1090/memo/0755 - M. A. Mandell, J. P. May, S. Schwede, and B. Shipley,
*Model categories of diagram spectra*, Proc. London Math. Soc. (3)**82**(2001), no. 2, 441–512. MR**1806878**, DOI 10.1112/S0024611501012692 - M. A. Mandell and B. Shipley,
*A telescope comparison lemma for THH*, Topology Appl.**117**(2002), no. 2, 161–174. MR**1875908**, DOI 10.1016/S0166-8641(00)00121-8 - Randy McCarthy,
*Relative algebraic $K$-theory and topological cyclic homology*, Acta Math.**179**(1997), no. 2, 197–222. MR**1607555**, DOI 10.1007/BF02392743 - Christian Schlichtkrull,
*Units of ring spectra and their traces in algebraic $K$-theory*, Geom. Topol.**8**(2004), 645–673. MR**2057776**, DOI 10.2140/gt.2004.8.645 - Brooke Shipley,
*Symmetric spectra and topological Hochschild homology*, $K$-Theory**19**(2000), no. 2, 155–183. MR**1740756**, DOI 10.1023/A:1007892801533 - R. Villarroel-Flores,
*The action by natural transformations of a group on a diagram of spaces*, Preprint, arxiv.org 0411502.

## Additional Information

**Vigleik Angeltveit**- Affiliation: Department of Mathematics, Australian National University, Canberra, ACT 0200, Australia
- MR Author ID: 769881
- Email: vigleik.angeltveit@anu.edu.au
**Andrew J. Blumberg**- Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
- MR Author ID: 648837
- Email: blumberg@math.utexas.edu
**Teena Gerhardt**- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
- MR Author ID: 852649
- Email: teena@math.msu.edu
**Michael A. Hill**- Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095
- MR Author ID: 822452
- ORCID: 0000-0001-8125-8107
- Email: mikehill@math.ucla.edu
**Tyler Lawson**- Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- MR Author ID: 709060
- Email: tlawson@math.umn.edu
- Received by editor(s): November 12, 2015
- Received by editor(s) in revised form: February 8, 2016
- Published electronically: June 17, 2016
- Additional Notes: The first author was supported in part by an NSF All-Institutes Postdoctoral Fellowship administered by the Mathematical Sciences Research Institute through its core grant DMS-0441170, NSF grant DMS-0805917, and an Australian Research Council Discovery Grant

The second author was supported in part by NSF grant DMS-0906105

The third author was supported in part by NSF DMS–1007083 and NSF DMS–1149408

The fourth author was supported in part by NSF DMS–0906285, DARPA FA9550-07-1-0555, and the Sloan Foundation

The fifth author was supported in part by NSF DMS–1206008 and the Sloan Foundation. - Communicated by: Michael A. Mandell
- © Copyright 2016 by the authors
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 5419-5433 - MSC (2010): Primary 55P91
- DOI: https://doi.org/10.1090/proc/13139
- MathSciNet review: 3556283