Homological algebra of Novikov-Shubin invariants and Morse inequalities

TL;DR

This paper introduces a homology theory framework to explain the zero in the continuous spectrum phenomenon related to Novikov-Shubin invariants, enabling the use of classical homological tools and deriving new invariants to strengthen Morse inequalities.

Contribution

It provides a homological approach to Novikov-Shubin invariants, establishing homotopy invariance and introducing new invariants to enhance Morse inequalities.

Findings

Homotopy invariance of Novikov-Shubin invariants established.

Homological techniques applied to spectral invariants.

New quantitative invariants proposed for spectral phenomena.

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.