Metric without nonzero \(\mathbb Z/2\)-harmonic one-forms on non-Haken hyperbolic rational homology spheres

Prove that every closed connected oriented non-Haken hyperbolic rational homology three-sphere admits a Riemannian metric for which there is no nonzero \(\mathbb Z/2\)-harmonic one-form.

Background

The paper establishes a metric-existence result for irreducible non-Haken Seifert fibered rational homology three-spheres: suitable connection metrics admit no nonzero $\mathbb Z/2$-harmonic one-form. It also recalls that rational homology spheres which are reducible or Haken admit such a form for every Riemannian metric.

After applying geometrization, the authors identify closed non-Haken hyperbolic rational homology three-spheres as the remaining unresolved class in the proposed topological dichotomy. The conjecture asks whether the Seifert-manifold nonexistence phenomenon extends to this hyperbolic class.