\(L^2\)-invariants of regular coverings of compact manifolds and CW-complexes. (Q2776343)

The author gives a survey on \(L^2\)-cohomology and its geometric applications.NEWLINENEWLINE The contents of the paper are as follows: 0. Introduction. 1. \(L^2\)-Betti numbers for CW-complexes of finite type. 2. Basic conjectures. 3. Low-dimensional manifolds. 4. Aspherical manifolds and amenability. 4. Approximating \(L^2\)-Betti numbers by ordinary Betti numbers. 6. \(L^2\)-Betti numbers and groups. 7. Kähler hyperbolic manifolds. 8. Novikov-Shubin invariants. 9. \(L^2\)-torsion. 10. Algebraic dimension theory of finite von Neumann algebras. 11. The zero in-the-spectrum conjecture. 12. Miscellaneous.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00029].