2-Betti Numbers of CW Complexes

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


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.


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© 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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