Von Neumann categories and extended $L^2$ cohomology
Abstract
In this paper we suggest a new general formalism for studying the invariants of polyhedra and manifolds comming from the theory of von Neumann algebras. First, we examine generality in which one may apply the construction of the extended abelian category, which was suggested in the previous publications of the author, using the ideas of P.Freyd. This leads to the notions of a finite von Neumann category and of a trace on such category. Given a finite von Neumann category, we study the extended homology and cohomology theories with values in the abelian extension. Any trace on the initial category produces numerical invariants  the von Neumann dimension and the Novikov  Shubin numbers. Thus, we obtain the local versions of the Novikov  Shubin invariants, localized at different traces. In the "abelian" case this localization can be made more geometric: we show that any torsion object determines a "divisor"  a closed subspace of the space of the parameters. The divisors of torsion objects together with the information produced by the local Novikov  Shubin invariants may be used to study multiplicities of intersections of algebraic and analytic varieties (we discuss here only simple examples demonstrating this possibility). We compute explicitly the divisors and the von Neumann dimensions of the extended cohomology in the real analytic situation. We also give general formulae for the extended cohomology of a mapping torus. Finally, we show how one can define a De Rham version of the extended cohomology and prove a De Rham type theorem.
 Publication:

eprint arXiv:dgga/961001
 Pub Date:
 October 1996
 arXiv:
 arXiv:dgga/9610016
 Bibcode:
 1996dg.ga....10016F
 Keywords:

 Mathematics  Differential Geometry
 EPrint:
 AMSTex, 53 pages