The Higson–Roe exact sequence and ℓ2ℓ2 eta invariants

The goal of this paper is to solve the problem of existence of an ℓ2ℓ2 relative eta morphism on the Higson–Roe structure group. Using the Cheeger–Gromov ℓ2ℓ2 eta invariant, we construct a group morphism from the Higson–Roe maximal structure group constructed in [35] to the reals. When we apply this morphism to the structure class associated with the spin Dirac operator for a metric of positive scalar curvature, we get the spin ℓ2ℓ2 rho invariant. When we apply this morphism to the structure class associated with an oriented homotopy equivalence, we get the difference of the ℓ2ℓ2 rho invariants of the corresponding signature operators. We thus get new proofs for the classical ℓ2ℓ2 rigidity theorems of Keswani obtained in [41].