The Higson-Roe exact sequence and ${\ell }^2$ eta invariants

Abstract

The goal of this paper is to solve the problem of existence of an ${\ell }^2$ relative eta morphism on the Higson-Roe structure group. Using the Cheeger-Gromov ${\ell }^2$ eta invariant, we construct a group morphism from the Higson-Roe maximal structure group constructed in [HiRo:10] 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 ${\ell }^2$ rho invariant. When we apply this morphism to the structure class associated with an oriented homotopy equivalence, we get the difference of the ${\ell }^2$ rho invariants of the corresponding signature operators. We thus get new proofs for the classical ${\ell }^2$ rigidity theorems of Keswani obtained in [Ke:00].

arXiv comment40 pages; typos fixed, references added

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Cited by in corpus (2)

• Realizing the analytic surgery group of Higson and Roe geometrically, Part II: Relative eta-invariants
• Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants