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Jumps of the eta-invariant. (With an appendix by Shmuel Weinberger: Rationality of \(\rho\)-invariants). (English) Zbl 0867.57027
We study the \(\eta\)-invariant, defined by M. F. Atiyah, V. K. Patodi and I. M. Singer [Math. Proc. Camb. Philos. Soc. 77, 43-69 (1975; Zbl 0297.58008); 78, 405-432 (1975; Zbl 0314.58016); 79, 71-99 (1976; Zbl 0325.58015)], a real-valued invariant of an oriented odd-dimensional Riemannian manifold equipped with a unitary representation of its fundamental group. When the representation varies analytically, the corresponding eta-invariant may have an integral jump, known also as the spectral flow. The main result of the paper establishes a formula for this spectral jump in terms of the signatures of some homological forms, defined naturally by the path of representations. These signatures may also be computed by means of a spectral sequence of Hermitian forms, defined by the deformation data. Our theorem on the spectral jump has a generalization to arbitrary analytic families of selfadjoint elliptic operators.
As an application we consider the problem of homotopy invariance of the \(\rho\)-invariant. We give an intrinsic homotopy theoretic definition of the \(\rho\)-invariant, up to indeterminacy in the form of a locally constant function on the space of unitary representations. In an Appendix, written by S. Weinberger, it is shown (using the results of this paper) that the difference in the \(\rho\)-invariants of homotopy-equivalent manifolds is always rational.

57R57 Applications of global analysis to structures on manifolds
58J20 Index theory and related fixed-point theorems on manifolds
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