Mathai’s conjecture on homotopy invariance of Cheeger–Gromov (ℓ2) rho invariants (torsion-free case)

Determine whether the Cheeger–Gromov ℓ2 rho invariant of the signature operator is an oriented homotopy invariant for any torsion-free discrete countable group.

Background

Building on computations and partial results for specific classes (e.g., Bieberbach groups), Mathai formulated a conjecture about the oriented homotopy invariance of the ℓ2 (Cheeger–Gromov) rho invariant. The conjecture parallels Weinberger’s conjecture for the APS rho invariant and is likewise a key open problem connected to index-theoretic rigidity addressed via Higson–Roe analytic methods.