Abstract

We prove that the convergence of Lott’s delocalized eta invariant holds for all self-adjoint first order elliptic differential operators with a sufficiently large spectral gap at zero. Furthermore, to each delocalized cyclic cocycle, we define a higher analogue of Lott’s delocalized eta invariant and prove its convergence when the delocalized cyclic cocycle has at most exponential growth. As an application, for each cyclic cocycle of at most exponential growth, we prove a formal higher Atiyah–Patodi–Singer index theorem on manifolds with boundary, under the condition that the operator on the boundary has a sufficiently large spectral gap at zero.

Our second main result is to obtain an explicit formula of the delocalized Connes–Chern character of all $C^{\ast }$-algebraic secondary invariants for word hyperbolic groups. Equivalently, we give an explicit formula for the pairing between $C^{\ast }$-algebraic secondary invariants and delocalized cyclic cocycles of the group algebra. When the $C^{\ast }$-algebraic secondary invariant is a $K$-theoretic higher rho invariant of an invertible differential operator, we show this pairing is precisely the higher analogue of Lott’s delocalized eta invariant alluded to above. Our work uses Puschnigg’s smooth dense subalgebra for word hyperbolic groups in an essential way. We emphasize that our construction of the delocalized Connes–Chern character is at the $C^{\ast }$-algebra $K$-theory level. This is of essential importance for applications to geometry and topology. As a consequence, we compute the pairing between delocalized cyclic cocycles and $C^{\ast }$-algebraic Atiyah–Patodi–Singer index classes for manifolds with boundary, when the fundamental group of the given manifold is hyperbolic. In particular, this improves the formal delocalized higher Atiyah–Patodi–Singer theorem from above and removes the condition that the spectral gap of the operator on the boundary is sufficiently large.

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