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Weinberger’s conjecture on homotopy invariance of APS rho invariants (torsion-free case)

Determine whether the Atiyah–Patodi–Singer rho invariant of the signature operator is an oriented homotopy invariant for all torsion-free finitely generated countable discrete groups.

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Background

The paper reviews historical developments on rho invariants and their rigidity properties. After early homotopy invariance results in special cases, Weinberger proposed a general conjecture asserting homotopy invariance of the APS rho invariant for the signature operator across all torsion-free finitely generated groups. This conjecture remains a central open problem in the area and motivates the analytic framework developed in the paper to paper rigidity via Higson–Roe sequences.

References

Some years later, Weinberger proved the same homotopy invariance for a large class of torsion-free groups and conjectured that the APS rho invariant for the signature operator should be an oriented homotopy invariant for all torsion-free finitely generated countable discrete groups.

— Admissible Higson-Roe sequences for transformation groupoids (2411.00182 - Benameur et al., 31 Oct 2024) in Introduction