# Topology of Pollicott-Ruelle resonant states

## TL;DR

This paper establishes a deep connection between the topology of certain flows on manifolds and the spectral properties of associated dynamical resonances, leading to new insights into Morse inequalities and Reidemeister torsion.

## Contribution

It proves an isomorphism between twisted De Rham cohomology and invariant Pollicott-Ruelle resonant states, extending Morse inequalities and relating resonances to Reidemeister torsion.

## Key findings

- Isomorphism between twisted De Rham cohomology and Pollicott-Ruelle resonant states
- Generalized Morse inequalities for flows with these resonances
- Reconstruction of Reidemeister torsion from dynamical resonances on the imaginary axis

## Abstract

We prove that the twisted De Rham cohomology of a flat vector bundleover some smooth manifold is isomorphic to the cohomology of invariant Pollicott--Ruelleresonant states associated with Anosov and Morse--Smale flows. As a consequence, weobtain generalized Morse inequalities for such flows. In the case of Morse--Smale flows,we relate the resonances lying on the imaginary axis with the twisted Fuller measuresused by Fried in his work on Reidemeister torsion. In particular, when V is a nonsingularMorse-Smale flow, we show that the Reidemeister torsion can be recovered from the onlyknowledge of dynamical resonances on the imaginary axis by expressing the torsion as azeta regularized infinite product of these resonances.