The Novikov-Bott inequalities

TL;DR

This paper extends Novikov inequalities to include non-isolated critical points and incorporates twisting by flat bundles, also providing an $L^2$ version using advanced spectral analysis techniques.

Contribution

It introduces generalized Novikov inequalities accommodating non-isolated critical points and flat bundle twistings, with an $L^2$ version based on spectral estimates.

Findings

Generalized Novikov inequalities for non-isolated critical points

Inclusion of flat bundle twistings in the inequalities

Development of an $L^2$ version using spectral analysis

Abstract

We generalize the Novikov inequalities for 1-forms in two different directions: first, we allow non-isolated critical points (assuming that they are non-degenerate in the sense of R.Bott), and, secondly, we strengthen the inequalities by means of twisting by an arbitrary flat bundle. We also obtain an $L^2$ version of these inequalities with finite von Neumann algebras. The proof of the main theorem uses Bismut's modification of the Witten deformation of the de Rham complex; it is based on an explicit estimate on the lower part of the spectrum of the corresponding Laplacian.