Novikov -shubin Signatures, I

Torsion objects of von Neumann categories describe the phenomenon ”spectrum near zero” discovered by S. Novikov and M. Shubin. In this paper we classify Hermitian forms on torsion objects of a finite von Neumann category. We prove that any such Hermitian form can be represented as the discriminant form of a degenerate Hermitian form on a projective module. We also find the relation between the Hermitian forms on projective modules which holds if and only if their discriminant forms are congruent. A notion of superfinite von Neumann category is introduced. It is proven that the classification of torsion Hermitian forms in a superfinite category can be completely reduced to the isomorphism types of their positive and the negative parts. §0. Introduction S. Novikov and M. Shubin [NS1], [NS2] discovered a new way of producing topological invariants of manifolds by studying the spectrum near zero of the Laplacian acting on L forms on the universal cover. It was proven by M. Gromov and M. Shubin [GS1], [GS2] that the Novikov Shubin invariants depend only on the homotopy type of the manifold. W. Lück and J. Lott [LL] computed the Novikov Shubin invariants for some 3-dimensional manifolds. Two different homological ”explanations” the Novikov Shubin invariants were suggested independently by W. Lück [Lu1], [Lu2],[Lu3] and by the author [Fa1], [Fa2]. The main idea of the approach developed in [Fa1 Fa2] was to view the ”spectrum near zero” as a torsion part of the L cohomology, understood in an extended sense. The main principles of [Fa1-Fa2] are the following: • the category of Hilbert representations of a given descrete group can be canonically embedded into an abelian category E ; • the extended abelian category E contains the chain complex l(π)⊗π C∗(M̃) of L forms on the universal covering M̃ for any compact polyhedron M ; • the homology of this chain complex (called the extended L cohomology) is a well defined object of E . • the extended cohomology naturally splits as a sum of its projective and torsion parts; • the theory of von Neumann dimension provides a tool to measure the size of the projective part of the extended cohomology; • the Novikov-Shubin invariants depend only of the torsion part of the extended cohomology. The research was supported by a grant from the Israel Academy of Sciencies and Humanities and by the Herman Minkowski Center for Geometry