Jumps of the eta invariant

TL;DR

This paper investigates how the eta-invariant changes under variations of representations, providing a formula for spectral jumps and exploring implications for the homotopy invariance of the rho-invariant.

Contribution

It establishes a formula for the spectral jump of the eta-invariant in terms of signatures of homological forms and generalizes this to families of elliptic operators.

Findings

Derived a formula linking spectral jumps to homological signatures.

Showed the spectral jump can be computed via a spectral sequence of Hermitian forms.

Proved the difference in rho-invariants of homotopy-equivalent manifolds is rational.

Abstract

We study the eta-invariant, defined by Atiyah-Patodi-Singer 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 self-adjoint 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.