Jumps of the eta invariant
| dc.creator | Farber, Michael S. | |
| dc.creator | Levine, Jerome P. | |
| dc.date | 1994-07-20 | |
| dc.date.accessioned | 2026-07-07T09:12:23Z | |
| dc.date.available | 2026-07-07T09:12:23Z | |
| dc.description | 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. | |
| dc.description | 46 pages | |
| dc.identifier | https://arxiv.org/abs/dg-ga/9407008 | |
| dc.identifier | http://arxiv.org/abs/dg-ga/9407008 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/152005 | |
| dc.subject | Differential Geometry | |
| dc.title | Jumps of the eta invariant | |
| dc.type | text |