Homological algebra of Novikov-Shubin invariants and Morse inequalities

Abstract

It is shown 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 order to strengthen the Morse type inequalities of Novikov and Shubin.