Mathematische Annalen

, Volume 328, Issue 1–2, pp 27–57 | Cite as

Identifying assembly maps in K- and L-theory

  • Ian Hambleton
  • Erik K. PedersenEmail author


In this paper we prove the equivalence of various algebraically or geometrically defined assembly maps used in formulating the main conjectures in K- and L-theory, and C * -theory.


Main Conjecture 
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.
    Anderson, D.R., Connolly, F., Ferry, S.C., Pedersen, E.K.: Algebraic K-theory with continuous control at infinity. J. Pure Appl. Algebra 94, 25–47 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Adams, J.F.: Infinite Loop Spaces. Annals of Mathematics Studies, Vol. 90, Princeton Univ. Press, 1978Google Scholar
  3. 3.
    Baum, P., Connes, A., Higson, N.: Classifying Space for Proper Actions and K-theory of Group C *-algebras. C *-algebras: 1943–1993 (San Antonio, TX, 1993), Contemp. Math., Vol. 167, Am. Math. Soc., Providence, RI, 1994, pp. 240–291Google Scholar
  4. 4.
    Bartels, A., Farrell, F.T., Jones, L.E., Reich, H.: On the isomorphism conjecture in algebraic k-theory. Preprint, 2001Google Scholar
  5. 5.
    Carlsson, G., Pedersen, E.K.: Controlled algebra and the Novikov conjectures for K- and L-theory. Topology 34, 731–758 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Cárdenas, M., Pedersen, E.K.: On the Karoubi filtration of a category. K-theory 12, 165–191 (1997)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Carlsson, G., Pedersen, E.K., Vogell, W.: Continuously controlled algebraic K-theory of spaces and the Novikov conjecture. Math. Ann. 310, 169–182 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Davis, J.F., Lück, W.: Spaces over a category and assembly maps in isomorphism conjectures in K- and L-Theory. K-theory 15, 201–252 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Farrell, F.T., Jones, L.E.: Isomorphism conjectures in algebraic K-theory. J. Am. Math. Soc. 6, 249–297 (1993)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Farrell, F.T., Jones, L.E. Lück, W.: A caveat on the isomorphism conjecture in l-theory. Forum Math. To appearGoogle Scholar
  11. 11.
    Ferry, S.C., Ranicki, A.A., Rosenberg, J.J.: A history and survey of the Novikov conjecture. Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), Cambridge Univ. Press, Cambridge, 1995, pp. 7–66Google Scholar
  12. 12.
    Gersten, S.: On the spectrum of algebraic K-theory. Bull. Amer. Math. Soc. (N.S.) 78, 216–219 (1972)zbMATHGoogle Scholar
  13. 13.
    Grayson, D.: Higher algebraic K-theory: II (after D. Quillen). Lecture Notes in Mathematics, Vol. 551, Springer, 1976Google Scholar
  14. 14.
    Higson, N.: Algebraic K-theory of stable C *-algebras. Adv. Math., 1–140 (1988)Google Scholar
  15. 15.
    Hambleton, I., Pedersen, E.K.: Bounded surgery and dihedral group actions on spheres. J. Am. Math. Soc. 4, 105–126 (1991)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Higson, N., Pedersen, E.K., Roe, J.: C *-algebras and controlled Topology. K-theory 11, 209–239 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Hsiang, W.C.: Geometric applications of Algebraic K-theory. Proc. I. C. M. 1983, Warsaw, North Holland, 1984, pp. 99–118Google Scholar
  18. 18.
    Karoubi, M.: Foncteur dérivés et K-théorie. Lecture Notes in Mathematics, Vol. 136, Springer, 1970Google Scholar
  19. 19.
    Kasparov, G.G.: Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91, 147–201 (1988)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lewis, L.G. Jr., May, J.P., Steinberger, M., McClure, J.E.: Equivariant stable homotopy theory. Springer-Verlag, Berlin, 1986, With contributions by J. E. McClureGoogle Scholar
  21. 21.
    Loday, J.-L.: K-théorie algébrique et représentations de groupes. Ann. Sci. École Norm. Sup. (4) 9, 309–377 (1976)Google Scholar
  22. 22.
    May, J.P.: A ring spaces and algebraic K-theory. Geometric applications of homotopy theory. II (Evanston, Ill., 1977), Lecture Notes in Mathematics, Vol. 658, Springer, Berlin, 1978, pp. 240–315Google Scholar
  23. 23.
    May, J.P.: Pairings of categories and spectra. J. Pure Appl. Algebra 19, 299–346 (1980)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Nicas, A.J.: Induction theorems for groups of manifold homotopy structure sets. Memoirs, Vol. 267, Am. Math. Soc., Providence, RI, 1982Google Scholar
  25. 25.
    Pedersen, E.K.: Continuously controlled surgery theory. Surveys on surgery theory, Vol. 1, Princeton Univ. Press, Princeton, NJ, 2000, pp. 307–321Google Scholar
  26. 26.
    Pedersen, E.K. Weibel, C.: A nonconnective delooping of algebraic K-theory. Algebraic and Geometric Topology, (Rutgers, 1983), Lecture Notes in Mathematics, vol. 1126, Springer, Berlin, 1985, pp. 166–181Google Scholar
  27. 27.
    Pedersen, E.K. Weibel, C.: K-theory homology of spaces. Algebraic Topology, (Arcata, 1986), Lecture Notes in Mathematics, Vol. 1370, Springer, Berlin, 1989, pp. 346–361Google Scholar
  28. 28.
    Quinn, F.: A geometric formulation of surgery. Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, 1970, pp. 500–511Google Scholar
  29. 29.
    Quillen, D.G.: Higher algebraic K-theory I. Algebraic K-theory, I: Higher K-theories, (Battelle Memorial Inst., Seattle, Washington, 1972), Lecture Notes in Mathematics, Vol. 341, Springer, Berlin, 1973, pp. 85–147Google Scholar
  30. 30.
    Quinn, F.: Ends of maps, II. Invent. Math. 68, 353–424 (1982)zbMATHGoogle Scholar
  31. 31.
    Quillen, D.G.: Assembly maps in bordism-type theories. Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., Vol. 226, Cambridge Univ. Press, Cambridge, 1995, pp. 201–271Google Scholar
  32. 32.
    Ranicki, A.A.: The algebraic theory of finiteness obstructions. Math. Scand. 57, 105–126 (1985)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Ranicki, A.A.: Additive L-theory. K-theory 3, 163–195 (1989)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Ranicki, A.A.: Algebraic L-theory and Topological Manifolds. Cambridge Tracts in Math., Vol. 102, Cambridge Univ. Press, 1992Google Scholar
  35. 35.
    Ranicki, A.A.: Lower K- and L-theory. London Math. Soc. Lecture Notes, Vol. 178, Cambridge Univ. Press, 1992Google Scholar
  36. 36.
    Staffeldt, R.: On fundamental theorems on algebraic K-theory. K-theory 1, 511–532 (1989)MathSciNetGoogle Scholar
  37. 37.
    Thomason, R.W.: First quadrant spectral sequences in algebraic K-theory via homotopy colimits. Comm. Algebra 10, 1589–1668 (1982)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Wagoner, J.: Delooping classifying spaces in algebraic K-theory. Topology 11, 349–370 (1972)CrossRefzbMATHGoogle Scholar
  39. 39.
    Wall, C.T.C.: Surgery on Compact Manifolds. Academic Press, New York, 1970Google Scholar
  40. 40.
    Waldhausen, F.: Algebraic K-theory of spaces. Algebraic and Geometric Topology, (Rutgers, 1983), Lecture Notes in Mathematics, Vol. 1126, Springer, Berlin, 1985, pp. 318–419Google Scholar
  41. 41.
    Weibel, C.A.: A survey of products in algebraic K-theory. Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Mathematics, Vol. 854, Springer, Berlin, 1981, pp. 466–493Google Scholar
  42. 42.
    Weiss, M., Williams, B.: Assembly. Novikov Conjectures, Rigidity and Index Theorems Vol. 2, (Oberwolfach, 1993), London Math. Soc. Lecture Notes, Vol. 227, Cambridge Univ. Press, Cambridge, 1995, pp. 332–352Google Scholar
  43. 43.
    Yamasaki, M.: L-groups of crystallographic groups. Invent. Math. 88, 571–602 (1987)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsMcMaster UniversityON L8S 4K1Canada
  2. 2.Department of Mathematical SciencesSUNY at BinghamtonNYUSA

Personalised recommendations