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