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2-Betti Numbers of CW Complexes

  • Holger Kammeyer
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2247)

Abstract

We explain the concept of equivariant CW complexes and how the 2-chain complex of Hilbert modules arises from the cellular chain complex by completion. We give the definition of 2-Betti numbers and compute them in easy examples. After clarifying the relation to cohomological 2-Betti numbers, we discuss Atiyah’s question on possible values of 2-Betti numbers and expound how this is relevant for Kaplansky’s zero divisor conjecture. The chapter concludes with proofs that positive 2-Betti numbers obstruct self-coverings, mapping torus structures, and circle actions on even dimensional hyperbolic manifolds.

References

  1. 8.
    M. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, in Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974) (Society of Mathematics, Paris, 1976), pp. 43–72. Astérisque, No. 32-33. MR 0420729Google Scholar
  2. 9.
    M. Atiyah, F. Hirzebruch, Spin-manifolds and group actions, in Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham) (Springer, New York, 1970), pp. 18–28. MR 0278334CrossRefGoogle Scholar
  3. 10.
    T. Austin, Rational group ring elements with kernels having irrational dimension. Proc. Lond. Math. Soc. (3) 107(6), 1424–1448 (2013). MR 3149852MathSciNetCrossRefGoogle Scholar
  4. 24.
    N. Bourbaki, Elements of Mathematics. Algebra, Part I: Chapters 1–3 (Addison-Wesley, Reading, 1974). Translated from the French. MR 0354207Google Scholar
  5. 34.
    J. Dodziuk, de Rham-Hodge theory for L 2-cohomology of infinite coverings. Topology 16(2), 157–165 (1977). MR 0445560MathSciNetCrossRefGoogle Scholar
  6. 35.
    J. Dodziuk, L 2 Harmonic forms on rotationally symmetric Riemannian manifolds. Proc. Am. Math. Soc. 77(3), 395–400 (1979). MR 545603Google Scholar
  7. 59.
    Ł. Grabowski, On Turing dynamical systems and the Atiyah problem. Invent. Math. 198(1), 27–69 (2014). MR 3260857MathSciNetCrossRefGoogle Scholar
  8. 62.
    M. Gromov, Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56, 5–99 (1982). MR 686042Google Scholar
  9. 68.
    A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002). MR 1867354Google Scholar
  10. 78.
    S. Illman, The equivariant triangulation theorem for actions of compact Lie groups. Math. Ann. 262(4), 487–501 (1983). MR 696520Google Scholar
  11. 85.
    H. Kammeyer, Algebraic Topology I. Lecture Notes (2016). http://www.math.kit.edu/iag7/~kammeyer/
  12. 105.
    P.A. Linnell, Division rings and group von Neumann algebras. Forum Math. 5(6), 561–576 (1993). MR 1242889Google Scholar
  13. 109.
    C. Löh, Simplicial volume. Bull. Manifold Atlas (2011). http://www.boma.mpim-bonn.mpg.de/
  14. 114.
    W. Lück, L 2-Betti numbers of mapping tori and groups. Topology 33(2), 203–214 (1994). MR 1273782MathSciNetCrossRefGoogle Scholar
  15. 117.
    W. Lück, L 2-Invariants: Theory and Applications to Geometry and K-Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 44 (Springer, Berlin, 2002). MR 1926649Google Scholar
  16. 145.
    M. Pichot, T. Schick, A. Zuk, Closed manifolds with transcendental L 2-Betti numbers. J. Lond. Math. Soc. (2) 92(2), 371–392 (2015). MR 3404029MathSciNetCrossRefGoogle Scholar
  17. 167.
    T. tom Dieck, Algebraic Topology. EMS Textbooks in Mathematics (European Mathematical Society, Zürich, 2008). MR 2456045Google Scholar
  18. 169.
    C.T.C. Wall, Surgery on Compact Manifolds, 2nd edn. Mathematical Surveys and Monographs, vol. 69 (American Mathematical Society, Providence, 1999). Edited and with a foreword by A.A. Ranicki. MR 1687388Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Holger Kammeyer
    • 1
  1. 1.Institute for Algebra and GeometryKarlsruhe Institute of TechnologyKarlsruheGermany

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