Israel Journal of Mathematics

, Volume 117, Issue 1, pp 183–219 | Cite as

Introduction to ℓ2-methods in topology: Reduced ℓ2-homology, harmonic chains, ℓ2-betti numbers

  • Beno EckmannEmail author


Fundamental Group Betti Number Finite Index Hilbert Module Morse Inequality 
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  1. [1]
    M. Atiyah,Elliptic operators, discrete groups and von Neumann algebras, Astérisque32 (1976), 43–72.MathSciNetGoogle Scholar
  2. [2]
    M. Burger and A. Valette,Idempotents in complex group rings: theorems of Zalesskii and Bass revisited, Journal of Lie Theory8 (1998), 219–228.zbMATHMathSciNetGoogle Scholar
  3. [3]
    J. Cheeger and M. Gromov,L 2-cohomology and group cohomology, Topology25 (1986), 189–215.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J. Dodziuk,De Rham-Hodge theory for L 2-cohomology of infinite coverings, Topology16 (1977), 157–165.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    J. Dodziuk and V. Mathai,Approximating L 2 invariants of amenable covering spaces: a combinatorial approach, Journal of Functional Analysis154 (1998), 359–378.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    B. Eckmann,Harmonische Funktionen und Randwertaufgaben in einem Komplex, Commentarii Mathematici Helvetici17 (1944/45), 240–255.CrossRefMathSciNetGoogle Scholar
  7. [7]
    B. Eckmann,Coverings and Betti numbers, Bulletin of the American Mathematical Society55 (1949), 95–101.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    B. Eckmann,Amenable groups and Euler characteristics, Commentarii Mathematici Helvetici67 (1992), 383–393.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    B. Eckmann,Manifolds of even dimension with amenable fundamental group, Commentarii Mathematici Helvetici69 (1994), 501–511.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    B. Eckmann,4-manifolds, group invariants, and ℓ 2-Betti numbers, L’Enseignement Mathématiques43 (1997), 271–279.zbMATHMathSciNetGoogle Scholar
  11. [11]
    B. Eckmann and P. Linnell,Poincaré duality groups of dimension two, II, Commentarii Mathematici Helvetici58 (1983), 111–114.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    B. Eckmann and H. Müller,Poincaré duality groups of dimension two, Commentarii Mathematici Helvetici55 (1980), 510–520.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    M. S. Farber,Novikov-Shubin invariants and Morse inequalities, Geometric and Functional Analysis6 (1996), 628–665.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    E. Følner,On groups with full Banach mean value, Mathematica Scandinavica3 (1955), 336–354.Google Scholar
  15. [15]
    R. I. Grigorchuk,An example of a finitely presented amenable group that does not belong to the class EG, Matematicheskii Sbornik189 (1998), no. 1, 79–100.zbMATHMathSciNetGoogle Scholar
  16. [16]
    J.-C. Hausmann and S. Weinberger,Caractéristique d’Euler et groupes fondamentaux des variétés de dimension 4, Commentarii Mathematici Helvetici60 (1985), 139–144.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    P. Linnell,Division rings and group von Neumann algebras, Forum Mathematicum5 (1993), 561–576.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    W. Lück,L 2-Betti numbers of mapping tori and groups, Topology33 (1994), 203–214.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    W. Lück,Hilbert modules over finite von Neumann algebras and applications to L 2-invariants, Mathematische Annalen309 (1997), 247–285.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    F. J. Murray and J. von Neumann,On rings of operators, Annals of Mathematics (2)37 (1936), 116–229.CrossRefMathSciNetGoogle Scholar
  21. [21]
    S. P. Novikov and M. A. Shubin,Morse inequalities and von Neumann invariants of nonsimply connected manifolds, Uspekhi Matematicheskikh Nauk41 (1986), 222–223.Google Scholar
  22. [22]
    A. L. Paterson,Amenability, Mathematical Surveys and Monographs 29, American Mathematical Society, 1988.Google Scholar
  23. [23]
    A. E. Zalesskii,On a problem of Kaplansky, Soviet Mathematics13 (1972), 449–452.zbMATHGoogle Scholar

Copyright information

© Hebrew University 2000

Authors and Affiliations

  1. 1.Mathematical Research Institute, ETHZurichSwitzerland

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