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Delocalized eta invariants, cyclic cohomology and higher rho invariants

Xiaoman Chen, Jinmin Wang, Zhizhang Xie and Guoliang Yu

Vol. 8 (2023), No. 4, 531–608
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-algebraic secondary invariants for word hyperbolic groups. Equivalently, we give an explicit formula for the pairing between C-algebraic secondary invariants and delocalized cyclic cocycles of the group algebra. When the C-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-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-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.

Keywords
higher eta invariants, higher rho invariants, cyclic cohomology, higher index, higher secondary invariants, Connes–Chern characters, higher index theorem for manifolds with boundary
Mathematical Subject Classification
Primary: 19D55, 19K56
Milestones
Received: 3 April 2023
Accepted: 10 September 2023
Published: 6 December 2023
Authors
Xiaoman Chen
School of Mathematical Sciences
Fudan University
Shanghai
China
Jinmin Wang
Shanghai Center for Mathematical Sciences
Fudan University
Shanghai
China
Zhizhang Xie
Department of Mathematics
Texas A&M University
College Station, TX
United States
Guoliang Yu
Department of Mathematics
Texas A&M University
College Station, TX
United States