L ²-Invariants of Symmetric Spaces

Abstract

In this chapter we state the values of the L ²-Betti numbers, the NovikovShubin invariants and the L ²-torsion for universal coverings of closed locally symmetric spaces. We give a brief survey about locally symmetric and symmetric spaces in Section 5.1 and state the values in Section 5.2 and 5.3. These computations will give evidence for various general conjectures about L ²-invariants such as Conjecture 2.82 about the positivity and rationality of Novikov-Shubin invariants, the Strong Atiyah Conjecture 10.2, the Singer Conjecture 11.1, Conjecture 11.3 about the parity of the L ²-torsion of the universal covering of an aspherical closed manifold and the zero-in-the-spectrum Conjecture 12.1.