Hodge theory for Riemannian solenoids

TL;DR

This paper develops Hodge theory for Riemannian measured solenoids, establishing a harmonic forms framework that computes their De Rham $L^2$-cohomology and proves a Poincaré duality, extending classical geometric analysis.

Contribution

It introduces a harmonic forms theory for Riemannian measured solenoids and demonstrates its equivalence to De Rham $L^2$-cohomology, including a Poincaré duality result.

Findings

Harmonic forms characterize De Rham $L^2$-cohomology of solenoids.

Established Poincaré duality for measured solenoids.

Extended Hodge theory to a new class of laminated spaces.

Abstract

A measured solenoid is a compact laminated space endowed with a transversal measure. The De Rham $L^2$-cohomology of the solenoid is defined by using differential forms which are smooth in the leafwise directions and $L^2$ in the transversal direction. We develop the theory of harmonic forms for Riemannian measured solenoids, and prove that this computes the De Rham $L^2$-cohomology of the solenoid. This implies in particular a Poincare duality result.