Homological algebra of Novikov-Shubin invariants and morse inequalities

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Abstract

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 a certain abelian category. This approach implies homotopy invariance of the Novikov-Shubin invariants. Its main advantage is that it allows the use of 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 Novikov and Shubin [NSh2].

Original languageEnglish
Pages (from-to)628-665
Number of pages38
JournalGeometric and Functional Analysis
Volume6
Issue number4
DOIs
StatePublished - 1996

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