2 1 Fe b 20 07 Various L 2 - signatures and a topological L 2 - signature theorem

For a normal covering over a closed oriented topological manifold we give a proof of the L-signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the L-theory isomorphism conjecture as well as the C max-version of the Baum-Connes conjecture imply the L signature theorem for a normal covering over a Poincaré space, provided that the group of deck transformations is torsion-free. We discuss the various possible definitions of L-signatures (using the signature operator, using the cap product of differential forms, using a cap product in cellular L-cohomology, . . . ) in this situation, and prove that they all coincide.