Vietnam Journal of Mathematics

, Volume 42, Issue 1, pp 73–82 | Cite as

Category of Noncommutative CW-Complexes. II

  • Do Ngoc DiepEmail author


We introduce in this paper the notion of noncommutative Serre fibration (shortly, NCSF) and show that up to homotopy, every morphism between NCCW-complexes is some noncommutative Serre fibration. We then associate a six-term exact sequence with the periodic cyclic homology and for K-theory of an arbitrary noncommutative Serre fibration. We also show how to use this technique to compute K-groups and cyclic theory groups of some noncommutative quotients. This paper is a follow-up of ideas in Diep (K-Theory Archiv 153, 2007, Vietnam J. Math. 38:363–371, 2010).


Noncommutative geometry C*-Algebra Noncommutative CW-complex Noncommutative mapping cylinder Noncommutative mapping cone 

Mathematics Subject Classification (2000)

46L87 58B34 58B32 



The author expresses his deep and sincere thanks to Professor C. Schochet for some e-mail discussion messages and in particular for Refs. [12, 13, 14, 15]. The author thanks the referee for carefully reading the text and correcting some misprints.


  1. 1.
    Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994) zbMATHGoogle Scholar
  2. 2.
    Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994) zbMATHGoogle Scholar
  3. 3.
    Diep, D.N.: On the structure of C*-algebras of type I. Vestnik MSU 2, 81–87 (1978) Google Scholar
  4. 4.
    Diep, D.N.: Methods of Noncommutative Geometry for Groups C*-Algebras. Chapman & Hall/CRC Research Notes in Mathematics Series, vol. 416. Chapman & Hall, Boca Raton (1999), 365 pp. Google Scholar
  5. 5.
    Diep, D.N.: Category of noncommutative CW complexes. K-Theory Archiv 153 (2007).
  6. 6.
    Diep, D.N.: Vietnam J. Math. 38, 363–371 (2010) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Diep, D.N., Kuku, A.O., Tho, N.Q.: Noncommutative Chern characters of group C*-algebras of compact Lie groups. K-Theory 17, 195–208 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Diep, D.N., Kuku, A.O., Tho, N.Q.: Noncommutative Chern characters of compact quantum groups. J. Algebra 226, 311–331 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Eilers, S., Loring, T.A., Pedersen, G.K.: Stability of anticommutation relations: an application of noncommutative CW complexes. J. Reine Angew. Math. 99, 101–143 (1998) MathSciNetGoogle Scholar
  10. 10.
    Loday, J.-L.: Cyclic Homology. Springer, Berlin (1992) CrossRefzbMATHGoogle Scholar
  11. 11.
    Pedersen, G.K.: Pullback and pushout constructions in C*-algebras theory. J. Funct. Anal. 167, 243–344 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Schochet, C.: Topological methods for C*-algebras I: spectral sequences. Pac. J. Math. 96, 193–211 (1981) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Schochet, C.: II: Geometric resolutions and Künneth formula. Pac. J. Math. 98, 443–458 (1982) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Schochet, C.: III: Axiomatic homology. Pac. J. Math. 114, 399–445 (1984) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Schochet, C.: IV: Mod p homology. Pac. J. Math. 114, 447–467 (1984) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyCau Giay DistrictVietnam

Personalised recommendations