×

zbMATH — the first resource for mathematics

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.

MSC:
57R57 Applications of global analysis to structures on manifolds
58J20 Index theory and related fixed-point theorems on manifolds
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Atiyah, M.F., Bott, R., Patodi, V.K.: On the heat equation and the index theorem, Invent. Math.19, 279–330 (1973) · Zbl 0257.58008 · doi:10.1007/BF01425417
[2] Atiyah, M.F., Patodi, V.K. and Singer, I.M.: Spectral asymmetry and Riemannian Geometry. I, II, III, Math. Proc. Camb. Phil. Soc. vol.77, 43–69 (1975); vol.78, 405–432 (1975); vol.79, 71–99 (1976) · Zbl 0297.58008 · doi:10.1017/S0305004100049410
[3] Bismut, J.-M., Cheeger, J.:\(\eta\)-invariants and their adiabatic limits, Journal of the Amer. Math. Soc.2, 33–70 (1989) · Zbl 0671.58037
[4] Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators, Springer-Verlag, 1991 · Zbl 1037.58015
[5] Blanchfield, R.C.: Intersection theory of manifolds with operators with applications to knot theory, Ann. Math.65, 340–356 (1957) · Zbl 0080.16601 · doi:10.2307/1969966
[6] Bochnak, J., Coste, M. and Roy, M-R.: Géométrie algébraique réelle, Springer-Verlag, 1987
[7] Bott, R. and Tu, L.: Differential forms in Algebraic topology, Springer-Verlag, 1982 · Zbl 0496.55001
[8] Chern, S.S. and Simons, J.: Characteristic forms and geometric invariants, Ann. of Math.99, 48–69 (1974) · Zbl 0283.53036 · doi:10.2307/1971013
[9] Dai, X.: Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence, J. of AMS4, 265–321 (1991) · Zbl 0736.58039
[10] Farber, M.: Singularities of the analytic torsion, J. of Diff. Geom.41, 528–572 (1995) · Zbl 0838.58038
[11] Farber, M. and Levine, J.: A topological interpretation of the Atiyah-Patidi-Singer invariant, Contemporary Math.164, 9–16 (1994) · Zbl 0812.58081 · doi:10.1090/conm/164/01581
[12] Fine, B., Kirk, P., Klassen, E.: A local analytic splitting of the holonomy map on flat connections, Mathematische Annalen299, 171–189 (1994) · Zbl 0797.53027 · doi:10.1007/BF01459778
[13] Gantmacher, F.R.: The theory of matrices, vol. 2, Chelsea Publishing Company, 1974 · Zbl 0085.01001
[14] Gilkey, P.B.: Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Publish or Perish, 1984 · Zbl 0565.58035
[15] Godement, R.: Topologie algebrique et theorie des faisceaux, Hermann and Cie, Paris, 1964 · Zbl 0202.41103
[16] Kato, T.: Perturbation theory for linear operators, Springer-Verlag, 1966 · Zbl 0148.12601
[17] Kirk, P., Klassen, E.: Computing Spectral Flow via Cup Products, Journal of Diff. Geometry40, 505–562 (1994) · Zbl 0816.58038
[18] Kirk, P. and Klassen, E.: The spectral flow of the odd signature operator on a manifold with boundary, Preprint (1994) · Zbl 0996.58010
[19] Kirk P. and Klassen, E.: The spectral flow of the odd signature operator and higher Massey products, Preprint (1994) · Zbl 0816.58038
[20] Kobayashi, S.: Differential geometry of complex vector bundles, Iwanami Shoten and Princeton Univ. Press, 1987 · Zbl 0708.53002
[21] Levine, J.: Metabolic and hyperbolic forms from knot theory, J. of Pure and Applied Algebra58, 442–456 (1989) · Zbl 0683.57009 · doi:10.1016/0022-4049(89)90040-6
[22] Levine, J.: Knots 90: Proceedings of the Osaka knot theory conference, 1992
[23] Levine, J.: Link invariants via the eta-invariant, Commentarii Math. Helv.69, 82–119 (1994) · Zbl 0831.57012 · doi:10.1007/BF02564475
[24] Mathai, V.: Spectral flow, Eta invariants, and von Neuman algebras, J. of functional analysis109, 442–456 (1992) · Zbl 0783.57015 · doi:10.1016/0022-1236(92)90022-B
[25] Mazzeo, R.R. and Melrose, R.B.: The adiabatic limit, Hodge cohomology and Lerey’s sequence for a fibration, J. of Diff. Geom.31, 185–213 (1990) · Zbl 0702.58007
[26] Neumann, W.: Signature related invariants of manifolds – I. Monodromy and\(\alpha\)-invariants, Topology18, 147–172 (1979) · Zbl 0416.57013 · doi:10.1016/0040-9383(79)90033-8
[27] Novikov, S.: Manifolds with free abelian fundamental groups and their applications, Izv. Akad. Nauk SSSR, Ser. Math.30, 207–246 (1966), AMS Translations (2),71, p. 1–42 (1968)
[28] Palais, R.S.: Seminar on the Atiyah-Singer index theorem, Annals of Math. Studies, N 57, Princeton Univ. Press, 1965 · Zbl 0137.17002
[29] Rudin, W.: Functional analysis, McGraw-Hill Book Company, 1973 · Zbl 0253.46001
[30] Wall, C.T.C.: Surgery on compact manifolds, Academic Press, 1971
[31] F. Warner, Foundations of differentiable manifolds and Lie groups, Springer-Verlag, 1983 · Zbl 0516.58001
[32] Weinberger, S.: Homotopy invariance of\(\eta\)-invariants, Proc. Nat. Acad. Sci. USA85, 5362–5363 (1988) · Zbl 0659.57016 · doi:10.1073/pnas.85.15.5362
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.