Skip to main content
SpringerLink
Log in
Book cover

The Local Structure of Algebraic K-Theory pp 227–279Cite as

Topological Cyclic Homology

  • Chapter

  • 1925 Accesses

Part of the book series: Algebra and Applications ((AA,volume 18))

Abstract

This chapter introduces the (integral) topological cyclic homology via a pullback diagram involving invariants of topological Hochschild homology. A lift of the trace to topological cyclic homology is given. Considerations of what occurs when completing at a prime is given special attention, as are connections to more algebraic counterparts like negative cyclic homology.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   99.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. J.F. Adams. On the groups J(X), IV. Topology, 5:21–71, 1966.

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Ausoni. On the algebraic K-theory of the complex K-theory spectrum. Invent. Math., 180(3):611–668, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Ausoni and J. Rognes. Algebraic K-theory of topological K-theory. Acta Math., 188(1):1–39, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  4. A.J. Blumberg. A discrete model of S 1-homotopy theory. J. Pure Appl. Algebra, 210(1):29–41, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Bökstedt, W.C. Hsiang, and I. Madsen. The cyclotomic trace and algebraic K-theory of spaces. Invent. Math., 111(3):465–539, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Bökstedt and I. Madsen. Topological cyclic homology of the integers. Astérisque, 226(7–8):57–143, 1994. K-theory (Strasbourg, 1992).

    Google Scholar 

  7. M. Bökstedt and I. Madsen. Algebraic K-theory of local number fields: the unramified case. In Prospects in Topology, Princeton, NJ, 1994, volume 138 of Ann. of Math. Stud., pages 28–57. Princeton University Press, Princeton, NJ, 1995.

    Google Scholar 

  8. D. Burghelea, Z. Fiedorowicz, and W. Gajda. Power maps and epicyclic spaces. J. Pure Appl. Algebra, 96(1):1–14, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  9. G. Carlsson. Equivariant stable homotopy and Segal’s Burnside ring conjecture. Ann. Math. (2), 120(2):189–224, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Connes. Cohomologie cyclique et foncteurs Extn. C. R. Acad. Sci. Paris Sér. I Math., 296(23):953–958, 1983.

    MathSciNet  MATH  Google Scholar 

  11. A. Connes. Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math., 62:257–360, 1985.

    MathSciNet  Google Scholar 

  12. W.G. Dwyer, M.J. Hopkins, and D.M. Kan. The homotopy theory of cyclic sets. Trans. Am. Math. Soc., 291(1):281–289, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  13. T.G. Goodwillie. Cyclic homology, derivations, and the free loopspace. Topology, 24(2):187–215, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  14. T.G. Goodwillie. Relative algebraic K-theory and cyclic homology. Ann. Math. (2), 124(2):347–402, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  15. T.G. Goodwillie. Letter to F. Waldhausen, August 10 1987.

    Google Scholar 

  16. T.G. Goodwillie. Notes on the cyclotomic trace. Lecture notes for a series of seminar talks at MSRI, Spring 1990, December 1991.

    Google Scholar 

  17. J.P.C. Greenlees and J.P. May. Generalized Tate cohomology. Mem. Am. Math. Soc., 113(543):178, 1995.

    MathSciNet  Google Scholar 

  18. L. Hesselholt. On the p-typical curves in Quillen’s K-theory. Acta Math., 177(1):1–53, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  19. L. Hesselholt. Witt vectors of non-commutative rings and topological cyclic homology. Acta Math., 178(1):109–141, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Hesselholt. Correction to: “Witt vectors of non-commutative rings and topological cyclic homology” [Acta Math. 178 (1997), no. 1, 109–141]. Acta Math., 195:55–60, 2005.

    Article  MathSciNet  Google Scholar 

  21. L. Hesselholt and I. Madsen. Cyclic polytopes and the K-theory of truncated polynomial algebras. Invent. Math., 130(1):73–97, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Hesselholt and I. Madsen. On the K-theory of finite algebras over Witt vectors of perfect fields. Topology, 36(1):29–101, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  23. L. Hesselholt and I. Madsen. On the K-theory of local fields. Ann. Math. (2), 158(1):1–113, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  24. J.D.S. Jones. Cyclic homology and equivariant homology. Invent. Math., 87(2):403–423, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  25. L.G. Lewis Jr., J.P. May, M. Steinberger, and J.E. McClure. Equivariant Stable Homotopy Theory, volume 1213 of Lecture Notes in Mathematics. Springer, Berlin, 1986. With contributions by J.E. McClure.

    MATH  Google Scholar 

  26. J.-L. Loday. Cyclic Homology, 2nd edition, volume 301 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, 1998. Appendix E by María O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili.

    MATH  Google Scholar 

  27. J.-L. Loday and D. Quillen. Cyclic homology and the Lie algebra homology of matrices. Comment. Math. Helv., 59(4):569–591, 1984.

    MathSciNet  Google Scholar 

  28. I. Madsen. Algebraic K-theory and traces. In Current Developments in Mathematics, Cambridge, MA, 1995, pages 191–321. International Press, Cambridge, 1994.

    Google Scholar 

  29. M.A. Mandell and J.P. May. Equivariant orthogonal spectra and S-modules. Mem. Am. Math. Soc., 159(755):108, 2002.

    MathSciNet  Google Scholar 

  30. J. Rognes. Algebraic K-theory of the two-adic integers. J. Pure Appl. Algebra, 134(3):287–326, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Rognes. The smooth Whitehead spectrum of a point at odd regular primes. Geom. Topol., 7:155–184, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Schwede and B. Shipley. Equivalences of monoidal model categories. Algebr. Geom. Topol., 3:287–334, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  33. G.B. Segal. Equivariant stable homotopy theory. In Actes du Congrès International des Mathématiciens, Tome 2, Nice, 1970, pages 59–63. Gauthier-Villars, Paris, 1971.

    Google Scholar 

  34. J.-P. Serre. Corps locaux. In Publications de l’Institut de Mathématique de l’Université de Nancago, VIII, volume 1296 of Actualités Scientifiques Et Industrielles. Hermann, Paris, 1962.

    Google Scholar 

  35. J. Spaliński. Strong homotopy theory of cyclic sets. J. Pure Appl. Algebra, 99(1):35–52, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  36. A.A. Suslin and M. Wodzicki. Excision in algebraic K-theory. Ann. Math. (2), 136(1):51–122, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  37. T.T. Dieck. Orbittypen und äquivariante Homologie, I. Arch. Math. (Basel), 23:307–317, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  38. T.T. Dieck. Orbittypen und äquivariante Homologie, II. Arch. Math. (Basel), 26(6):650–662, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  39. S. Tsalidis. On the topological cyclic homology of the integers. Am. J. Math., 119(1):103–125, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  40. S. Tsalidis. Topological Hochschild homology and the homotopy descent problem. Topology, 37(4):913–934, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  41. B.L. Tsygan. Homology of matrix Lie algebras over rings and the Hochschild homology. Usp. Mat. Nauk, 38(2(230)):217–218, 1983.

    MathSciNet  MATH  Google Scholar 

  42. M. Weiss and B. Williams. Automorphisms of manifolds and algebraic K-theory, II. J. Pure Appl. Algebra, 62(1):47–107, 1989.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bergen, Bergen, Norway

    Bjørn Ian Dundas

  2. Mathematics Department, Brown University, Providence, USA

    Thomas G. Goodwillie

  3. Department of Mathematics, University of Illinois, Urbana, IL, USA

    Randy McCarthy

Authors
  1. Bjørn Ian Dundas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas G. Goodwillie
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Randy McCarthy
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag London

About this chapter

Cite this chapter

Dundas, B.I., Goodwillie, T.G., McCarthy, R. (2013). Topological Cyclic Homology. In: The Local Structure of Algebraic K-Theory. Algebra and Applications, vol 18. Springer, London. https://doi.org/10.1007/978-1-4471-4393-2_6

Download citation

Publish with us

Policies and ethics

Search

Navigation