Journal of Mathematical Sciences

, Volume 146, Issue 1, pp 5530–5551 | Cite as

Theory of spectral sequences. II

  • Ju. T. Lisica


In this paper, we continue to discuss the theory of spectral sequences in Abelian categories. In this connection, attention is paid to the manifestation of different dualities in the theory of spectral sequences. The duality in locally convex, topological vector spaces is of special interest. We use results of functional analysis for solving (co)homology problems (theorems on nondegenerate pairing (co)homology) of manifolds.


Spectral Sequence Chain Complex Topological Vector Space Inductive Limit Strong Topology 
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.
    F. J. Almgren, “The homotopy groups of integral cycle groups,” Topology, 1, 257–299 (1962).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J. M. Boardman, Conditionally Convergent Spectral Sequences, Preprint, Johns Hopkins Univ., Baltimore (1981).Google Scholar
  3. 3.
    R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer, New York (1982).MATHGoogle Scholar
  4. 4.
    N. Bourbaki, Espaces Vectoriels Topologiques, Hermann, Paris (1955).Google Scholar
  5. 5.
    G. E. Bredon, Sheaf Theory, McGraw-Hill, New York (1967).MATHGoogle Scholar
  6. 6.
    I. Bucur and A. Deleanu, Introduction to the Theory of Categories and Functors, John Wiley, London (1968).MATHGoogle Scholar
  7. 7.
    H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton (1956).MATHGoogle Scholar
  8. 8.
    G. S. Chogoshvili, “On duality laws for arbitrary sets,” Mat. Sb., 28, 89–118 (1951).Google Scholar
  9. 9.
    P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Springer, Berlin (1967).MATHGoogle Scholar
  10. 10.
    S. I. Gelfand and Ju. I. Manin, Methods of Homological Algebra, Vol. I, Introduction to Cohomology Theory and Derived Functors [in Russian], Nauka, Moscow (1988).Google Scholar
  11. 11.
    R. Godement, Topologie Algébraique et Théorie des Faisceaux, Hermann, Paris (1958).Google Scholar
  12. 12.
    P. A. Griffiths and J. W. Morgan, Rational Homotopy Theory and Differential Forms, Birkhauser, Boston (1981).MATHGoogle Scholar
  13. 13.
    A. Dold, Lectures on Algebraic Topology, Springer, Berlin (1972).MATHGoogle Scholar
  14. 14.
    A. Dold and R. Thom, “Quasifaserungen und unendliche symmetrische Produkte,” Ann. Math., 67, 230–281 (1958).MathSciNetCrossRefGoogle Scholar
  15. 15.
    V. G. Drinfeld, “On quasitriangular and quasi-Hopf algebras and a group connected with \((\bar {\mathbb{Q}}/{\mathbb{Q}})\),” Algebra Analiz, 2, No. 4, 149–181 (1989).MathSciNetGoogle Scholar
  16. 16.
    S. Eilenberg and J. C. Moore, “Limits and spectral sequences,” Topology, 1, 1–23 (1962).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    H. Federer, Geometric Measure Theory, Springer, New York (1969).MATHGoogle Scholar
  18. 18.
    R. Hartshorne, Algebraic Geometry, Springer, New York (1977).MATHGoogle Scholar
  19. 19.
    M. Kashiwara and P. Schapira, Sheaves on Manifolds, Springer, Berlin (1994).Google Scholar
  20. 20.
    C. Kassel, Quantum Groups, Springer, New York (1994).Google Scholar
  21. 21.
    Ju. T. Lisica, “On the compact \(\underrightarrow {\lim } - functor\) in the category of compact groups,” Glas. Mat., 32, 301–314 (1997).MATHMathSciNetGoogle Scholar
  22. 22.
    Ju. T. Lisica, “On the cocommutative chains problem,” Int. Conf. on Topology and Its Appl., Abstracts, Skopje, 2004, pp. 19–22.Google Scholar
  23. 23.
    Ju. T. Lisica, “On (co)homological locally connected spaces,” Usp. Mat. Nauk, 58, No. 6, 153–154 (2004).Google Scholar
  24. 24.
    Ju. T. Lisica, “Theory of spectral sequences. I,” Proc. Steklov Inst. Math., 247, 115–134 (2004)].Google Scholar
  25. 25.
    Ju. T. Lisica, “Strong bonding homology and cohomology,” Topology Appl., 153, 394–447 (2005).MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    S. Mac Lane, Categories for the Working Mathematician, Springer (1998).Google Scholar
  27. 27.
    W. S. Massey, Homology and Cohomology Theory, Marcel Dekker, New York (1978).MATHGoogle Scholar
  28. 28.
    J. Milnor, “On axiomatic homology theory,” Pacific J. Math., 12, 337–341 (1962).MATHMathSciNetGoogle Scholar
  29. 29.
    S. A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, Cambridge Univ. Press, Cambridge (1977).MATHGoogle Scholar
  30. 30.
    A. Polishchuk, Abelian Varieties, Theta Functions and the Fourier Transform, Cambridge Tracts Math., Vol. 153, Cambridge Univ. Press (2003).Google Scholar
  31. 31.
    D. Quillen, “Rational homotopy theory,” Ann. Math., 90, 205–295 (1969).CrossRefMathSciNetGoogle Scholar
  32. 32.
    D. A. Raikov, “Some linear-topological properties of spaces D and D′,” in: A. P. Robertson and W. Robertson, Topological Vector Spaces [Russian translation], Mir, Moscow (1967), pp. 250–298.Google Scholar
  33. 33.
    G. de Rham, Variétés Différentiables, Hermann, Paris (1955).MATHGoogle Scholar
  34. 34.
    A. P. Robertson and W. Robertson, Topological Vector Spaces, Cambridge Univ. Press, Cambridge (1964).MATHGoogle Scholar
  35. 35.
    M. Sato, “Theory of hyperfunctions. I, II,” Fac. Sci. Univ. Tokyo, 8, 139–193, 387–436 (1959–1960).MATHGoogle Scholar
  36. 36.
    L. Schwartz, Théorie des Distribution. I, Hermann, Paris (1950).Google Scholar
  37. 37.
    E. G. Sklyarenko, “Homology and cohomology of general spaces,” Itogi Nauki i Tekhn., Sovr. Probl. Mat., 50, All Union Institute for Scientific and Technical Information, Moscow (1989), pp. 129–266.Google Scholar
  38. 38.
    E. G. Sklyarenko, “Homology and cohomology of set binding. Homology and cohomology of environments of closed sets,” Usp. Mat. Nauk, 56, 1040–1071 (1992).MATHGoogle Scholar
  39. 39.
    E. G. Sklyarenko, “Hyper(co)homology for left exact covariant functors and a homology theory for topological spaces,” Russ. Math. Surv., 50, 575–611 (1995).MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    E. G. Sklyarenko, “Homology dual to the de Rham cohomology,” Proc. Steklov Inst. Math., 247, 217–226 (2004).MathSciNetGoogle Scholar
  41. 41.
    D. Sullivan, “Infinitesimal computations in topology,” Publ. IHES, 47, 269–331 (1978).Google Scholar
  42. 42.
    R. M. Switzer, Algebraic Topology—Homotopy and Homology, Springer, Berlin (1975).MATHGoogle Scholar
  43. 43.
    H. Whitney, Geometric Integration Theory, Princeton Univ. Press, Princeton (1957).MATHGoogle Scholar
  44. 44.
    S.-T. Yau, ed., Essays on Mirror Manifolds, International Press, Hong Kong (1992).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • Ju. T. Lisica
    • 1
  1. 1.Peoples’ Friendship University of RussiaRussia

Personalised recommendations