Mathematische Annalen

, Volume 336, Issue 1, pp 191–238 | Cite as

Transfer maps and the cyclotomic trace

  • Christian SchlichtkrullEmail author


We analyze the equivariant restriction (or transfer) maps in topological Hochschild homology associated to inclusions of group rings of the form R[H]→R[G], where R is a symmetric ring spectrum, G is a discrete group and HG is a subgroup of finite index. This leads to a complete description of the associated restriction (or transfer) maps in topological cyclic homology

Open image in new window

in terms of the well-known stable transfers in equivariant stable homotopy theory. More generally, we analyze the restriction maps encountered in connection with monoid rings such as polynomial rings and truncated polynomial rings. As a first application of these results we prove a conjecture by Bökstedt, Hsiang and Madsen on how the transfer maps in Waldhausen's algebraic K-theory of spaces relate to the transfers in the stable equivariant homotopy category of a finite cyclic group. As a second application we calculate the subgroup of transfer invariant homotopy classes

Open image in new window

and we show that the TC-analogue of the lower K-groups vanish below degree −1.


Conjugacy Class Homotopy Group Smash Product Topological Realization Symmetric Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, J.F.: Stable homotopy and generalized homology. Chicago Lectures in Mathematics, University of Chicago University Press, (1974)Google Scholar
  2. 2.
    Adams, J.F.: Infinite loop spaces. Ann. of Math. Stud. 90, Princeton University Press, (1978)Google Scholar
  3. 3.
    Bass, H.: Algebraic K-theory. W. A. Benjamin, New York, (1968)Google Scholar
  4. 4.
    Barratt, M., Eccles, P.: Γ+-structures-I: A free group functor for stable homotopy theory. Topology 13, 25–45 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Betley, S., Schlichtkrull, C.: The cyclotomic trace and curves on K-theory. Topology 44, 845–874 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bousfield, A.K., Friedlander, E.M.: Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, from: ``Geometric applications of homotopy theory (Proc. Conf. Evanstone, Ill, 1977) II'', (M. G. Barratt, M. E. Mahowald editors) Springer LNM 658, Springer, Berlin. 80–130 (1978)Google Scholar
  7. 7.
    Bökstedt, M.: Topological Hochschild homology. Preprint (1985), BielefeldGoogle Scholar
  8. 8.
    Bökstedt, M., Hsiang, W.C., Madsen, I.: The cyclotomic trace and algebraic K-theory of spaces. Invent. Math. 111, 865–940 (1993)CrossRefGoogle Scholar
  9. 9.
    Dundas, B.: Relative K-theory and topological cyclic homology. Acta. Math. 179, 223–242 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Dundas, B.: The cyclotomic trace for symmetric monoidal categories, from: ``Geometry and topology: Aarhus (1998)'', Contemp. Math. 258, Amer. Math. Soc. 121–143 (2000)Google Scholar
  11. 11.
    Dundas, B., McCarthy, R.: Topological Hochschild homology of ring functors and exact categories. J. Pure Appl. Algebra 109, 231–294 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Goodwillie, T.: Notes on the cyclotomic trace. Lecture notes, (1990)Google Scholar
  13. 13.
    Hauschild, H.: Zerspaltung äquivarianter Homotopiemengen. Math. Ann. 230, 279–292 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Hesselholt, L., Madsen, I.: On the K-theory of finite algebras over Witt vectors of finite fields. Topology 36 1, 29–101 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Hovey, M., Shipley, B., Smith, J.: Symmetric spectra. J. Amer. Math. Soc. 13, 149–208 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Lewis, L.G., May, J.P., Steinberger, M.: Equivariant stable homotopy theory. Springer LNM 1213, Springer Verlag, (1986)Google Scholar
  17. 17.
    Lydakis, M.: Free loop spaces and equivariant classifying spaces. Arch. Math. 77(2), 181–186 (2001)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mac Lane, S.: Categories for the working mathematician. Graduate Texts in Mathematics 5, Springer Verlag, New York-Berlin, (1986)Google Scholar
  19. 19.
    McCarthy, R.: The cyclic homology of an exact category. J. Pure Appl. Algebra 93, 251–296 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    McCarthy, R.: Relative algebraic K-theory and topological cyclic homology. Acta. Math. 179, 197–222 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Madsen, I.: Algebraic K-theory and traces, from: ``Current Developments in Mathematics, 1995, (Cambridge, MA)''. International Press, Cambridge, MA, 191–321 (1996)Google Scholar
  22. 22.
    Mandell, M.A., May, J.P., Schwede, S., Shipley, B.: Model categories of diagram spectra. Proc. London Math. Soc. (3) 82, 441–512 (2001)Google Scholar
  23. 23.
    May, J.P.: The geometry of iterated loop spaces. Springer LNM 271, Springer Verlag, (1972)Google Scholar
  24. 24.
    May, J.P.: E -spaces, group completion and permutative categories, from ``New developments in Topology (Proc. Sympos. Algebraic Topology, Oxford, 1972)'', London Math. Soc. Lecture Notes 11, Cambridge University Press, 61–93Google Scholar
  25. 25.
    Ranicki, A.: Lower K- and L-Theory. London Math. Soc. Lecture Notes 178, Cambridge University Press, (1992)Google Scholar
  26. 26.
    Schlichtkrull, C.: The transfer map in topological Hochschild homology. J. Pure Appl. Algebra 133, 289–316 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Schlichtkrull, C.: A discrete model of equivariant stable homotopy theory for cyclic groups, Math. Scand. 85, 5–29 (1999)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Schlichtkrull, C.: Cyclic K-theory and the cyclotomic trace, In preparation, see
  29. 29.
    Segal, G.: Categories and cohomology theories. Topology 13, 293–312 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Waldhausen, F.: Algebraic K-theory of topological spaces II. Springer LNM 763, Springer Verlag, 356–394 (1979)Google Scholar
  31. 31.
    Waldhausen, F.: Operations in the algebraic K-theory of spaces. Springer LNM 967, Springer Verlag, 390–409 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway

Personalised recommendations