Homological Algebra of Novikov-shubin Invariants and Morse Inequalities

It is shown in this paper that the topological phenomenon ”zero in the continuous spectrum”, discovered by S.P.Novikov and M.A.Shubin, can be explained in terms of a homology theory on the category of finite polyhedra with values in certain abelian category. This approach implies homotopy invariance of the Novikov-Shubin invariants. Its main advantage is that it allows to use the standard homological techniques, such as spectral sequences, derived functors, universal coefficients etc., while studying the Novikov-Shubin invariants. It also leads to some new quantitative invariants, measuring the Novikov-Shubin phenomenon in a different way, which are used in the present paper in order to strengthen the Morse type inequalities of S.P. Novikov and M.A. Shubin [NS1]. §0. Introduction This paper suggests a conceptually new approach, which unites the L cohomology theory and the Novikov-Shubin invariants. It is shown here that these theories are two different parts of a unique cohomology theory with values in an abelian category. This abelian category, denoted E(A), contains the familiar additive category of Hilbertian modules over a von Neumann algebra A as a full subcategory of projectives. An important abelian subcategory of E(A) is formed by torsion virtual Hilbertian modules. It turns out that any object of E(A) has canonically defined torsion and projective parts and coincides with their direct sum. The von Neumann dimension is an invariant of the projective part; similarly, the Novikov-Shubin number is an invariant of the torsion part. There are natural homology and cohomology theories with values in the abelian category E(A). I denote these theories by Hi(X,M) and Hi(X,M) correspondingly and call extended L homology and cohomology. Here X is a CW complex having finitely many cells in every dimension, and M is a Hilbertian (A− π)or (π −A)bimodule (cf. §6 below), and π = π1(X) is the fundamental group of X . These theories are homotopy invariant. The projective part of Hi(X,M) coincides with The research was supported by a grant from US Israel binational Science Foundation