Advertisement

Cyclic homology in a special world

  • Bjørn Ian DundasEmail author
Chapter

Abstract

In work of Connes and Consani, Γ-spaces have taken a new importance. Segal introduced Γ-spaces in order to study stable homotopy theory, but the new perspective makes it apparent that also information about the unstable structure should be retained. Hence, the question naturally presents itself: to what extent are the commonly used invariants available in this context? We offer a quick survey of (topological) cyclic homology and point out that the categorical construction is applicable also in an \({\mathbb N}\)-algebra (aka. semi-ring or rig) setup.

Keywords

Cyclic homology Ring spectra Topological cyclic homology Special gamma spaces Unstable homology Group completion 

Mathematical Subject Classification (2010)

Primary: 13D03; Secondary: 18G60 19D55 55P92 

Notes

Acknowledgements

Apart from obvious input from Alain Connes this tribute has benefitted from enlightening conversations with M. Brun, K. Consani, L. Hesselholt, M. Hill, and C. Schlichtkrull. Also, the preprint [8] by de Brito and Moerdijk and the papers of Santhanam [55] and Mandell [46] were inspirational. The author also wants to thank an anonymous referee for correcting an unfortunate misconception about events in the early 1980s.

References

  1. 1.
    Vigleik Angeltveit. On the K-theory of truncated polynomial rings in non-commuting variables. Bull. Lond. Math. Soc., 47(5):731–742, 2015.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Vigleik Angeltveit, Teena Gerhardt, Michael A. Hill, and Ayelet Lindenstrauss. On the algebraic K-theory of truncated polynomial algebras in several variables. J. K-Theory, 13(1):57–81, 2014.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Christian Ausoni. On the algebraic K-theory of the complex K-theory spectrum. Invent. Math., 180(3):611–668, 2010.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Christian Ausoni and John Rognes. Algebraic K-theory of topological K-theory. Acta Math., 188(1):1–39, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    S. Bloch. On the tangent space to Quillen K-theory. In AlgebraicK-theory, I: HigherK-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 205–210. Lecture Notes in Math., Vol. 341. Springer, Berlin, 1973.Google Scholar
  6. 6.
    Andrew J. Blumberg, David Gepner, and Gonçalo Tabuada. A universal characterization of higher algebraic K-theory. Geom. Topol., 17(2):733–838, 2013.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Andrew J. Blumberg, David Gepner, and Gonçalo Tabuada. Uniqueness of the multiplicative cyclotomic trace. Adv. Math., 260:191–232, 2014.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    P. Boavida de Brito and I. Moerdijk. Dendroidal spaces, Γ-spaces and the special Barratt-Priddy-Quillen theorem. ArXiv e-prints, January 2017.Google Scholar
  9. 9.
    M. Bökstedt, G. Carlsson, R. Cohen, T. Goodwillie, W. C. Hsiang, and I. Madsen. On the algebraic K-theory of simply connected spaces. Duke Math. J., 84(3):541–563, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    M. Bökstedt, W. C. Hsiang, and I. Madsen. The cyclotomic trace and algebraic K-theory of spaces. Invent. Math., 111(3):465–539, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    A. K. Bousfield and E. M. Friedlander. Homotopy theory of Γ-spaces, spectra, and bisimplicial sets. In Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, volume 658 of Lecture Notes in Math., pages 80–130. Springer, Berlin, 1978.Google Scholar
  12. 12.
    M. Brun, B. I. Dundas, and M. Stolz. Equivariant Structure on Smash Powers. ArXiv e-prints, April 2016.Google Scholar
  13. 13.
    D. Clausen, A. Mathew, and M. Morrow. K-theory and topological cyclic homology of henselian pairs. ArXiv e-prints, March 2018.Google Scholar
  14. 14.
    Alain Connes. C algèbres et géométrie différentielle. C. R. Acad. Sci. Paris Sér. A-B, 290(13):A599–A604, 1980.zbMATHGoogle Scholar
  15. 15.
    Alain Connes. Spectral sequence and homology of currents for operator algebras. Math. Forschungsinstitut Oberwolfach Tagungsbericht, 42/81, 1981.Google Scholar
  16. 16.
    Alain Connes. Cohomologie cyclique et foncteurs Extn. C. R. Acad. Sci. Paris Sér. I Math., 296(23):953–958, 1983.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Alain Connes. Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math., (62):257–360, 1985.Google Scholar
  18. 18.
    Alain Connes and Caterina Consani. Absolute algebra and Segal’s Γ-rings: au dessous de \(\overline {\mathrm {Spec}(\mathbb {Z})}\). J. Number Theory, 162:518–551, 2016.Google Scholar
  19. 19.
    Alain Connes and Max Karoubi. Caractère multiplicatif d’un module de Fredholm. C. R. Acad. Sci. Paris Sér. I Math., 299(19):963–968, 1984.MathSciNetzbMATHGoogle Scholar
  20. 20.
    Guillermo Cortiñas. The obstruction to excision in K-theory and in cyclic homology. Invent. Math., 164(1):143–173, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Daniel Dugger. Replacing model categories with simplicial ones. Trans. Amer. Math. Soc., 353(12):5003–5027, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Bjørn Ian Dundas, Thomas G. Goodwillie, and Randy McCarthy. The local structure of algebraic K-theory, volume 18 of Algebra and Applications. Springer-Verlag London, Ltd., London, 2013.Google Scholar
  23. 23.
    Bjørn Ian Dundas and Harald Øyen Kittang. Excision for K-theory of connective ring spectra. Homology Homotopy Appl., 10(1):29–39, 2008.Google Scholar
  24. 24.
    Bjørn Ian Dundas and Harald Øyen Kittang. Integral excision for K-theory. Homology Homotopy Appl., 15(1):1–25, 2013.Google Scholar
  25. 25.
    Bjørn Ian Dundas, Oliver Röndigs, and Paul Arne Østvær. Enriched functors and stable homotopy theory. Doc. Math., 8:409–488, 2003.Google Scholar
  26. 26.
    Bjørn Ian Dundas. Relative K-theory and topological cyclic homology. Acta Math., 179(2):223–242, 1997.Google Scholar
  27. 27.
    Bjørn Ian Dundas and Randy McCarthy. Stable K-theory and topological Hochschild homology. Ann. of Math. (2), 140(3):685–701, 1994.Google Scholar
  28. 28.
    Thomas Geisser and Lars Hesselholt. Bi-relative algebraic K-theory and topological cyclic homology. Invent. Math., 166(2):359–395, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Thomas Geisser and Lars Hesselholt. On the K-theory and topological cyclic homology of smooth schemes over a discrete valuation ring. Trans. Amer. Math. Soc., 358(1):131–145, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Thomas Geisser and Lars Hesselholt. On the vanishing of negative K-groups. Math. Ann., 348(3):707–736, 2010.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Thomas G. Goodwillie. Relative algebraic K-theory and cyclic homology. Ann. of Math. (2), 124(2):347–402, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Lars Hesselholt. On the K-theory of planar cuspical curves and a new family of polytopes. In Algebraic topology: applications and new directions, volume 620 of Contemp. Math., pages 145–182. Amer. Math. Soc., Providence, RI, 2014.Google Scholar
  33. 33.
    Lars Hesselholt. The big de Rham-Witt complex. Acta Math., 214(1):135–207, 2015.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Lars Hesselholt. Topological Hochschild homology and the Hasse-Weil zeta function. In An alpine bouquet of algebraic topology, volume 708 of Contemp. Math., pages 157–180. Amer. Math. Soc., Providence, RI, 2018.Google Scholar
  35. 35.
    Lars Hesselholt and Ib Madsen. On the K-theory of finite algebras over Witt vectors of perfect fields. Topology, 36(1):29–101, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Lars Hesselholt and Ib Madsen. On the K-theory of local fields. Ann. of Math. (2), 158(1):1–113, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Michael A. Hill, Michael J. Hopkins, and Douglas C. Ravenel. On the non-existence of elements of Kervaire invariant one. In Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, pages 1219–1243. Kyung Moon Sa, Seoul, 2014.Google Scholar
  38. 38.
    John D. S. Jones. Cyclic homology and equivariant homology. Invent. Math., 87(2):403–423, 1987.Google Scholar
  39. 39.
    T. A. Kro. Involutions on S[ ΩM]. ArXiv Mathematics e-prints, October 2005.Google Scholar
  40. 40.
    M. Land and G. Tamme. On the K-theory of pullbacks. ArXiv e-prints, 2018.Google Scholar
  41. 41.
    Ayelet Lindenstrauss and Randy McCarthy. On the Taylor tower of relative K-theory. Geom. Topol., 16(2):685–750, 2012.MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Jean-Louis Loday and Daniel Quillen. Cyclic homology and the Lie algebra homology of matrices. Comment. Math. Helv., 59(4):569–591, 1984.MathSciNetzbMATHGoogle Scholar
  43. 43.
    Wolfgang Lück, Holger Reich, John Rognes, and Marco Varisco. Algebraic K-theory of group rings and the cyclotomic trace map. Adv. Math., 304:930–1020, 2017.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Manos Lydakis. Smash products and Γ-spaces. Math. Proc. Cambridge Philos. Soc., 126(2):311–328, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Ib Madsen. Algebraic K-theory and traces. In Current developments in mathematics, 1995 (Cambridge, MA), pages 191–321. Internat. Press, Cambridge, MA, 1994.Google Scholar
  46. 46.
    Michael A. Mandell. An inverse K-theory functor. Doc. Math., 15:765–791, 2010.MathSciNetzbMATHGoogle Scholar
  47. 47.
    Randy McCarthy. Relative algebraic K-theory and topological cyclic homology. Acta Math., 179(2):197–222, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    T. Nikolaus and P. Scholze. On topological cyclic homology. ArXiv e-prints, July 2017.Google Scholar
  49. 49.
    Crichton Ogle. On the homotopy type of A( ΣX). J. Pure Appl. Algebra, 217(11):2088–2107, 2013.MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Daniel Quillen. Cohomology of groups. In Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pages 47–51. Gauthier-Villars, Paris, 1971.Google Scholar
  51. 51.
    J. Rognes and C. Weibel. Two-primary algebraic K-theory of rings of integers in number fields. J. Amer. Math. Soc., 13(1):1–54, 2000. Appendix A by Manfred Kolster.Google Scholar
  52. 52.
    John Rognes. Algebraic K-theory of the two-adic integers. J. Pure Appl. Algebra, 134(3): 287–326, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    John Rognes. Two-primary algebraic K-theory of pointed spaces. Topology, 41(5):873–926, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    John Rognes. The smooth Whitehead spectrum of a point at odd regular primes. Geom. Topol., 7:155–184 (electronic), 2003.Google Scholar
  55. 55.
    Rekha Santhanam. Units of equivariant ring spectra. Algebr. Geom. Topol., 11(3):1361–1403, 2011.MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Roland Schwänzl, Ross Staffeldt, and Friedhelm Waldhausen. Stable K-theory and topological Hochschild homology of A rings. In AlgebraicK-theory (Poznań, 1995), volume 199 of Contemp. Math., pages 161–173. Amer. Math. Soc., Providence, RI, 1996.Google Scholar
  57. 57.
    Stefan Schwede. Stable homotopical algebra and Γ-spaces. Math. Proc. Cambridge Philos. Soc., 126(2):329–356, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Graeme Segal. Categories and cohomology theories. Topology, 13:293–312, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Kazuhisa Shimakawa. A note on ΓG-spaces. Osaka J. Math., 28(2):223–228, 1991.MathSciNetzbMATHGoogle Scholar
  60. 60.
    R. W. Thomason. Symmetric monoidal categories model all connective spectra. Theory Appl. Categ., 1:No. 5, 78–118 (electronic), 1995.Google Scholar
  61. 61.
    B. L. Tsygan. Homology of matrix Lie algebras over rings and the Hochschild homology. Uspekhi Mat. Nauk, 38(2(230)):217–218, 1983.Google Scholar
  62. 62.
    Friedhelm Waldhausen. Algebraic K-theory of spaces, concordance, and stable homotopy theory. In Algebraic topology and algebraicK-theory (Princeton, N.J., 1983), volume 113 of Ann. of Math. Stud., pages 392–417. Princeton Univ. Press, Princeton, NJ, 1987.Google Scholar
  63. 63.
    Charles A. Weibel. Nil K-theory maps to cyclic homology. Trans. Amer. Math. Soc., 303(2):541–558, 1987.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of BergenBergenNorway

Personalised recommendations