Abstract
We explain the concept of equivariant CW complexes and how the ℓ 2-chain complex of Hilbert modules arises from the cellular chain complex by completion. We give the definition of ℓ 2-Betti numbers and compute them in easy examples. After clarifying the relation to cohomological ℓ 2-Betti numbers, we discuss Atiyah’s question on possible values of ℓ 2-Betti numbers and expound how this is relevant for Kaplansky’s zero divisor conjecture. The chapter concludes with proofs that positive ℓ 2-Betti numbers obstruct self-coverings, mapping torus structures, and circle actions on even dimensional hyperbolic manifolds.
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Kammeyer, H. (2019). ℓ 2-Betti Numbers of CW Complexes. In: Introduction to ℓ²-invariants. Lecture Notes in Mathematics, vol 2247. Springer, Cham. https://doi.org/10.1007/978-3-030-28297-4_3
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DOI: https://doi.org/10.1007/978-3-030-28297-4_3
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