ℓ 2-Betti Numbers of CW Complexes

Abstract

We explain the concept of equivariant CW complexes and how the l2-chain complex of Hilbert modules arises from the cellular chain complex by completion. We give the definition of l2-Betti numbers and compute them in easy examples. After clarifying the relation to cohomological l2-Betti numbers, we discuss Atiyah’s question on possible values of l2-Betti numbers and expound how this is relevant for Kaplansky’s zero divisor conjecture. The chapter concludes with proofs that positive l2-Betti numbers obstruct self-coverings, mapping torus structures, and circle actions on even dimensional hyperbolic manifolds.