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Annals of Global Analysis and Geometry

, Volume 19, Issue 3, pp 259–291 | Cite as

Novikov–Shubin Signatures, II

  • M. Farber
Article

Abstract

This paper continues the author's previous paper, published inAnn. Global Anal. Geom.18 (2000), 477–515. Here weconstruct a linking form on the torsion part of middle-dimensionalextended L2 homology and cohomology of odd-dimensionalmanifolds. We give a geometric necessary condition when this linkingform is hyperbolic. We compute this linking form in case, when themanifold bounds. We introduce and study new numerical invariants of thelinking form: the Novikov–Shubin signature and the torsion signature;we compute these invariants explicitly for manifolds withπ1 = Z in terms of the Blanchfield form. We develop anotion of excess for extensions of torsion modules and show how thisconcept can be used to guarantee vanishing of the torsion signature.

extended L2 cohomology Hermitian forms von Neumann categories 

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References

  1. 1.
    Atiyah, M. F.: Elliptic operators, discrete groups and von Neumann algebras, Asterisque 32-33 (1976), 43-72.Google Scholar
  2. 2.
    Bayer-Fluckiger, E. and Fainsilber, L.: Non-unimodular Hermitian forms, Invent. Math. 123 (1996), 233-240.Google Scholar
  3. 3.
    Blanchfield, R. C.: Intersection theory of manifolds with operators with applications to knot theory, Ann. Math. 65 (1957), 340-356.Google Scholar
  4. 4.
    Cheeger, J. and Gromov, M.: L 2-cohomology and group cohomology, Topology 25 (1986), 189-215.Google Scholar
  5. 5.
    Dixmier, J.: Von Neumann Algebras, North-Holland, Amsterdam, 1981.Google Scholar
  6. 6.
    Dodziuk, J.: De Rham-Hodge theory for L 2-cohomology of infinite coverings, Topology 16 (1977), 157-165.Google Scholar
  7. 7.
    Farber, M.: Abelian categories, Novikov-Shubin invariants, and Morse inequalities, C.R. Acad. Sci. Paris 321 (1995), 1593-1598.Google Scholar
  8. 8.
    Farber, M.: Homological algebra of Novikov-Shubin invariants and Morse inequalities, Geom. Funct. Anal. 6 (1996), 628-665.Google Scholar
  9. 9.
    Farber, M.: Von Neumann categories and extended L 2 cohomology, K-Theory 15 (1998), 347-405.Google Scholar
  10. 10.
    Farber, M.: Geometry of growth: Approximation theorems for L 2-invariants, Math. Anal. 311 (1998), 335-375.Google Scholar
  11. 11.
    Farber, M.: Novikov-Shubin signatures, I, Anal. Global Anal. Geom. 18 (2000), 477-515.Google Scholar
  12. 12.
    Farber, M. and Levine, J. L.: Jumps of the eta-invariant, Math. Z. 223 (1996), 197-246.Google Scholar
  13. 13.
    Gromov, M. and Shubin, M. A.: Von Neumann spectra near zero, Geom. Funct. Anal. 1 (1991), 375-404.Google Scholar
  14. 14.
    Lück, W.: Hilbert modules and modules over finite von Neumann algebras and applications to L 2invariants, Math. Anal. 309 (1997), 247-285.Google Scholar
  15. 15.
    Milgram, R. J.: Orientations for Poincaré Duality Spaces and Applications, Lecture Notes in Math. 1370, Springer, New York, 1989, pp. 293-324.Google Scholar
  16. 16.
    Milnor, J.: A duality theorem for Reidemeister torsion, Ann. of Math. 76 (1962), 137-147.Google Scholar
  17. 17.
    Novikov, S. P. and Shubin, M. A.: Morse inequalities and von Neumann II 1-factors, Dokl. Akad. Nauk SSSR 289 (1986), 289-292.Google Scholar
  18. 18.
    Quebbemann, H.-G., Scharlau, W. and Schulte, M.: Quadratic and Hermitian forms in additive and Abelian categories, J. Algebra 59 (1979), 264-289.Google Scholar
  19. 19.
    Ranicki, A.: Lower K-and L-Theory, London Math. Soc. Lecture Notes Ser. 178, Cambridge Univ. Press, 1992.Google Scholar
  20. 20.
    Vogel, P.: Localization in Algebraic L-Theory, Lecture Notes in Math. 778, Springer, New York, 1980, pp. 482-495.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • M. Farber
    • 1
  1. 1.Department of MathematicsTel Aviv UniversityTel AvivIsrael

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