On the Whitehead Spectrum of the Circle

Part of the Abel Symposia book series (ABEL, volume 4)


The seminal work of Waldhausen, Farrell and Jones, Igusa, and Weiss andWilliams shows that the homotopy groups in low degrees of the space of homeomorphisms of a closed Riemannian manifold of negative sectional curvature can be expressed as a functor of the fundamental group of the manifold. To determine this functor, however, it remains to determine the homotopy groups of the topological Whitehead spectrum of the circle. The cyclotomic trace of Bökstedt, Hsiang, and Madsen and a theorem of Dundas, in turn, lead to an expression for these homotopy groups in terms of the equivariant homotopy groups of the homotopy fiber of the map from the topological Hochschild T-spectrum of the sphere spectrum to that of the ring of integers induced by the Hurewicz map. We evaluate the latter homotopy groups, and hence, the homotopy groups of the topologicalWhitehead spectrum of the circle in low degrees. The result extends earlier work by Anderson and Hsiang and by Igusa and complements recent work by Grunewald, Klein, and Macko.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    K. Igusa, The stability theorem for smooth pseudoisotopies, K-Theory 2 (1988), 1–355.CrossRefMathSciNetGoogle Scholar
  2. 2.
    M. Weiss and B. Williams, Automorphisms of manifolds and algebraic K-theory: I, K-Theory 1 (1988), 575–626.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    F. Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology (New Brunswick, N. J., 1983), Lecture Notes in Math., vol. 1126, Springer, New York, 1985, pp. 318–419.Google Scholar
  4. 4.
    F. Waldhausen, B. Jahren, and J. Rognes, Spaces of PL manifolds and categories of simple maps, Preprint 2008.Google Scholar
  5. 5.
    W. Vogell, The canonical involution on the algebraic K-theory of spaces, Algebraic and geometric topology (New Brunswick, N. J., 1983), Lecture Notes in Math., vol. 1126, Springer, New York, 1985.Google Scholar
  6. 6.
    F. T. Farrell and L. E. Jones, Rigidity in geometry and topology, Proceedings of the International Congress of Mathematicians, vol. 1 (Kyoto, 1990), Springer, Berlin, 1991, pp. 653–663.Google Scholar
  7. 7.
    P. Petersen, Riemannian geometry. Second edition, Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006.MATHGoogle Scholar
  8. 8.
    D. H. Gottlieb, A certain subgroup of the fundamental group, Am. J. Math. 87 (1965), 840–856.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    W. Lück and H. Reich, The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory, Handbook of K-theory, vol. 2, Springer, New York, 2005, pp. 703–842.Google Scholar
  10. 10.
    D. R. Anderson and W.-c. Hsiang, The functors K i and pseudo-isotopies of polyhedra, Ann. Math. 105 (1977), 201–223.CrossRefMathSciNetGoogle Scholar
  11. 11.
    K. Igusa, On the algebraic K-theory of A -spaces, Algebraic K-theory, Part II (Oberwolfach, 1980), Lecture Notes in Math., vol. 967, Springer, Berlin, 1982, pp. 146–194.CrossRefGoogle Scholar
  12. 12.
    J. Grunewald, J. R. Klein, and T. Macko, Operations on the A-theoretic Nil-terms, J. Topol. 1 (2008), 317–341.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    D. Quillen, Higher algebraic K-theory I, Algebraic K-theory I: Higher K-theories (Battelle Memorial Inst., Seattle, Washington, 1972), Lecture Notes in Math., vol. 341, Springer, New York, 1973.Google Scholar
  14. 14.
    A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257–281.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    M. Bökstedt, W.-c. Hsiang, and I. Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 465–540.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    B. I. Dundas, Relative K-theory and topological cyclic homology, Acta Math. 179 (1997), 223–242.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    L. Hesselholt and I. Madsen, On the de Rham-Witt complex in mixed characteristic, Ann. Sci. École Norm. Sup. 37 (2004), 1–43.MATHMathSciNetGoogle Scholar
  18. 18.
    L. Hesselholt and I. Madsen, On the K-theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997), 29–102.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    M. Hovey, B. Shipley, and J. Smith, Symmetric spectra, J. Am. Math. Soc. 13 (2000), 149–208.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    I. Madsen, Algebraic K-theory and traces, Current Developments in Mathematics, 1995, International, Cambridge, MA, 1996, pp. 191–321.Google Scholar
  21. 21.
    L. Hesselholt, On the p-typical curves in Quillen's K-theory, Acta Math. 177 (1997), 1–53.CrossRefMathSciNetGoogle Scholar
  22. 22.
    L. Hesselholt and I. Madsen, On the K-theory of local fields, Ann. Math. 158 (2003), 1–113.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    B. I. Dundas and R. McCarthy, Topological Hochschild homology of ring functors and exact categories, J. Pure Appl. Algebra 109 (1996), 231–294.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    B. I. Dundas, The cyclotomic trace for S-algebras, London Math. Soc. 70 (2004), 659–677.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    T. Geisser and L. Hesselholt, Topological cyclic homology of schemes, K-theory (Seattle, 1997), Proc. Symp. Pure Math. 67, 1999, 41–87.MathSciNetGoogle Scholar
  26. 26.
    J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, vol. 113, Members of American Mathematical Society, no. 543, American Mathematical Society, Providence, RI, 1995.Google Scholar
  27. 27.
    R. E. Mosher, Some stable homotopy of complex projective space, Topology 7 (1968), 179–193.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    G. W. Whitehead, Recent advances in homotopy theory, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, vol. 5, American Mathematical Society, Providence, RI, 1970.MATHGoogle Scholar
  29. 29.
    M. Bökstedt, Topological Hochschild homology of \(\mathbb{F}_p\) and \(\mathbb{Z}\), Preprint, Bielefeld University, 1985.Google Scholar
  30. 30.
    A. Lindenstrauss and I. Madsen, Topological Hochschild homology of number rings, Trans. Am. Math. Soc. (2000), 2179–2204.Google Scholar
  31. 31.
    M. Bökstedt and I. Madsen, Topological cyclic homology of the integers, K-theory (Strasbourg, 1992), Astérisque 226 (1994), 57–143.Google Scholar
  32. 32.
    J. Rognes, Topological cyclic homology of the integers at two, J. Pure Appl. Algebra 134 (1999), 219–286.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    J. Rognes, Algebraic K-theory of the two-adic integers, J. Pure Appl. Algebra 134 (1999), 287–326.MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    J. Rognes, Two-primary algebraic K-theory of pointed spaces, Topology 41 (2002), 873–926.MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    M. Bökstedt, The rational homotopy type of Ω WhDiff (*), Algebraic topology (Aarhus, 1982), Lecture Notes in Math., vol. 1051, Springer, New York, 1984, pp. 25–37.Google Scholar
  36. 36.
    V. Costeanu, On the 2-typical de Rham-Witt complex, Doc. Math. (2009) (to appear).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.Nagoya UniversityNagoyaJapan

Personalised recommendations