L 2 -topological invariants of 3-manifolds

Abstract

We give results on the L 2 -Betti numbers and Novikov-Shubin invariants of compact manifolds, especially 3-manifolds. We first study the Betti numbers and Novikov-Shubin invariants of a chain complex of Hilbert modules over a finite von Neumann algebra. We establish inequalities among the Novikov-Shubin invariants of the terms in a short exact sequence of chain complexes. Our algebraic results, along with some analytic results on geometric 3-manifolds, are used to compute the L 2 -Betti numbers of compact 3-manifolds which satisfy a weak form of the geometrization conjecture, and to compute or estimate their Novikov-Shubin invariants.