Abstract
We give a survey on η-invariants including methods of computation and applications in differential topology.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Astey, L., Micha, E., Pastor, G.: Homeomorphism and diffeomorphism types of Eschenburg spaces. Diff. Geom. Appl. 7, 41–50 (1997)
Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry, Bull. London Math. Soc. 5, 229–234 (1973)
Atiyah, M.F., Patodi, V.K., Singer, I.M., Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc. 77, 43–69 (1975)
Atiyah, M.F., Patodi, V.K., Singer, I.M., Spectral asymmetry and Riemannian geometry, II, Math. Proc. Cambridge Philos. Soc. 78, 405–432 (1975)
Atiyah, M.F., Patodi, V.K., Singer, I.M., Spectral asymmetry and Riemannian geometry, III, Math. Proc. Cambridge Philos. Soc. 79, 71–99 (1976)
Atiyah, M.F., Singer, I.M.: The index of elliptic operators, III, Ann. of Math. 87, 546–604 (1968)
Bär, C.: Dependence of the Dirac spectrum on the Spin structure, in Global analysis and harmonic analysis (Marseille-Luminy, 1999), pp. 17–33, Smin. Congr., 4, Soc. Math. France, Paris (2000)
Bechtluft-Sachs, S.: The computation of η-invariants on manifolds with free circle action, J. Funct. Anal. 174, 251–263 (2000)
Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators, Springer, Berlin (1992)
Bismut, J.-M.: The infinitesimal Lefschetz formulas: A heat equation proof. J. Funct. Anal. 62, 435–457 (1985)
Bismut, J.-M., Cheeger, J.: η-invariants and their adiabatic limits. J. Amer. Math. Soc. 2 33–70 (1989)
Bismut, J.-M., Cheeger, J., Families index for manifolds with boundary, superconnections, and cones, I, Families of manifolds with boundary and Dirac operators. J. Funct. Anal. 89, 313–363 (1990)
Bismut, J.-M., Cheeger, J., Families index for manifolds with boundary, superconnections and cones, II, The Chern character. J. Funct. Anal. 90, 306–354 (1990)
Bismut, J.-M., Cheeger, J., Remarks on the index theorem for families of Dirac operators on manifolds with boundary. In: Lawson, H.B., Tenenblat, K. (eds.) Differential geometry, pp. 59–83, Pitman Monogr. Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow (1991)
Bismut, J.M., Freed, D.S.: The Analysis of Elliptic Families II, Dirac Operators, Êta Invariants, and the Holonomy Theorem, Comm. Math. Phys. 107, 103–163 (1986)
Bismut, J.-M., Lott, J.: Flat vector bundles, direct images and higher real analytic torsion. J. Amer. Math. Soc. 8, 291–363 (1995)
Bismut, J.-M., Zhang, W.: Real embeddings and eta invariants. Math. Ann. 295, 661–684 (1993)
Botvinnik, B., Gilkey, P.B.: The eta invariant and metrics of positive scalar curvature, Math. Ann. 302, 507–517 (1995)
Botvinnik, B., Gilkey, P.B., Metrics of positive scalar curvature on spherical space forms, Canad. J. Math 48, 64–80 (1996)
Bunke, U.: On the gluing problem for the η-invariant. J. Diff. Geom. 41, 397–448 (1995)
Bunke, U., Naumann, N.: The f-invariant and index theory. Manuscr. Math. 132, 365–397 (2010)
Cappell, S.E., Lee, R., Miller, E.Y.: On the Maslov index. Comm. Pure Appl. Math. 47, 121–186 (1994)
Cheeger, J., Simons, J.: Differential characters and geometric invariants. In: Geometry and topology (College Pack, 1983/84), 50–80, Lecture Notes in Math. 1167, Springer, Berlin (1985)
Cisneros-Molina, J.L.: The η-invariant of twisted Dirac operators of S 3 ∕ Γ, Geom. Dedicata 84, 207–228 (2001)
Cisneros-Molina, J.L., A note on torsion in K 3of the real numbers, Bol. Soc. Mat. Mexicana (3) 10, 117–128 (2004)
Crowley, D.: The classification of highly connected manifolds in dimensions 7 and 15, preprint, arXiv:math/0203253.
Crowley, D., Escher, C.: Classification of S 3-bundles over S 4, Diff. Geom. Appl. 18, 363–380 (2003)
Crowley, D., Goette, S.: Kreck-Stolz invariants for quaternionic line bundles, preprint, arXiv:1012.5237
Dahl, M.: Dependence on the Spin structure of the eta and Rokhlin invariants. Topology Appl. 118, 345–355 (2002)
Dai, X.: Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence, J. Amer. Math. Soc. 4, 265–321 (1991); Ann. Global Anal. Geom. 27, 333–340 (2005)
Dai, X., Zhang, W.: Circle bundles and the Kreck-Stolz invariant. Trans. Amer. Math. Soc. 347, 3587–3593 (1995)
Dearricott, O.: A 7-manifold with positive curvature, to appear in Duke Math. J.
Degeratu, A.: Eta-invariants from Molien series, Q. J. Math. 60, 303–311 (2009)
Deninger, C., Singhof, W.: The e-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups, Invent. math. 78, 101–112 (1984)
Donnelly, H.: Spectral geometry and invariants from differential topology. Bull. London Math. Soc. 7, 147–150 (1975)
Donnelly, H., Eta invariants for G-spaces, Indiana Univ. Math. J. 27, 889–918 (1978)
Eells, J. Jr., Kuiper, N.: An invariant for certain smooth manifolds. Ann. Mat. Pura Appl. (4) 60, 93–110 (1962)
Farsi, C.: Orbifold η-invariants. Indiana Univ. Math. J. 56, 501–521 (2007)
Feder, S., Gitler, S.: Mappings of quaternionic projective spaces, Bol. Soc. Mat. Mexicana (2) 18, 33–37 (1973)
Gilkey, P.B., Miatello, R.J., Podestá, R.A.: The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order. Ann. Glob. Anal. Geom. 37, 275–306 (2010)
Goette, S.: Equivariant η-invariants on homogeneous spaces. Math. Z. 232, 1–42 (1999)
Goette, S., Equivariant eta-Invariants and eta-Forms. J. reine angew. Math. 526, 181–236 (2000)
Goette, S., Eta invariants of homogeneous spaces. Pure Appl. Math. Q. 5, 915–946 (2009)
Goette, S., Adiabatic limits of Seifert fibrations, Dedekind sums, and the diffeomorphism types of certain 7-manifolds, in preparation
Goette, S., Kitchloo, N., Shankar, K.: Diffeomorphism type of the Berger space SO(5)/ SO(3). Am. J. Math. 126, 395–416 (2004)
Gromov, M., Lawson, H.B.: The classification of simply connected manifolds of positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math. IHES 58, 83–196 (1983)
Grove, K., Verdiani, L., Ziller, W.: A Positively Curved Manifold Homeomorphic to T 1 S 4, to appear in Geom. Funct. Anal.; arXiv:0809.2304
Grove, K., Wilking, B., Ziller, W.: Positively curved cohomogeneity one manifolds and 3-sasakian geometry. J. Diff. Geo. 78, 33–111 (2008); arXiv:math/0511464
Habel, M.: Die η-Invariante der Berger-Sphäre, Diplomarbeit Universität Hamburg (2000)
Hepworth, R.: Generalized Kreck-Stolz invariants and the topology of certain 3-Sasakian 7-manifolds, thesis, University of Edinburgh, Scotland, UK (2005)
Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)
Jones, J.D., Westbury, B.W.: Algebraic K-theory, homology spheres, and the η-invariant. Topology 34, 929–957 (1995)
Keswani, N.: Relative eta-invariants and C ∗ -algebra K-theory. Topology 39, 957–983 (2000)
Kitchloo, N., Shankar, K.: On complexes equivalent to S 3-bundles over S 4. Int. Math. Research Notices 8, 381–394 (2001)
Kostant, B.: A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups. Duke Math. J. 100, 447–501 (1999)
Kreck, M., Stolz, S.: A diffeomorphism classification of 7-dimensional homogeneous Einstein manifolds with SU(3) ×SU(2) ×U(1)-symmetry. Ann. of Math. 127, 373–388 (1988)
Kreck, M., Stolz, S., Some nondiffeomorphic homeomorphic homogeneous 7-manifolds with positive sectional curvature. J. Differ. Geom. 33, 465–486 (1991)
Kreck, M., Stolz, S., Nonconnected moduli spaces of positive sectional curvature metrics. J. Amer. Math. Soc. 6, 825–850 (1993)
Kruggel, B.: Homeomorphism and diffeomorphism classification of Eschenburg spaces. Q. J. Math. 56, 553–577 (2005)
Lawson, H.B., Michelsohn, M.-L.: Spin geometry, p. xii + 427. Princeton University Press, Princeton, NJ (1989)
Leichtnam, E., Piazza, P.: On higher eta-invariants and metrics of positive scalar curvature, K-Theory 24, 341–359 (2001)
Lesch, M., Wojciechowski, K.P.: On the η-invariant of generalized Atiyah-Patodi-Singer boundary value problems. Illinois J. Math. 40, 30–46 (1996)
Loya, P., Moroianu, S., Park, J.: Adiabatic limit of the eta invariant over cofinite quotients of PSL(2, R). Compos. Math. 144, 1593–1616 (2008)
Mathai, V.: Spectral flow, eta invariants, and von Neumann algebras. J. Funct. Anal. 109, 442–456 (1992)
Melrose, R.B., Piazza, P.: Families of Dirac operators, boundaries and the b-calculus. J. Differ. Geom. 46, 99–180 (1997)
Miatello, R.J., Podestá, R.A.: Eta invariants and class numbers. Pure Appl. Math. Q. 5, 729–753 (2009)
Miller, E.Y., Lee, R.: Some invariants of spin manifolds. Topology Appl. 25, 301–311 (1987)
Millson, J.J.: Closed geodesics and the eta-invariant. Ann. Math. 108, 1–39 (1978)
Moscovici, H., Stanton, R.J.: Eta invariants of Dirac operators on locally symmetric spaces. Invent. Math. 95, 629–666 (1989)
Neumann, W.: Signature related invariants of manifolds I: monodromy and γ-invariants. Topology 18, 147–172 (1979)
Park, J.: Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps. Amer. J. Math. 127, 493–534 (2005)
Pfäffle, F.: The Dirac spectrum of Bieberbach manifolds. J. Geom. Phys. 35, 367–385 (2000)
Piazza, P., Schick, T.: Bordism, rho-invariants and the Baum-Connes conjecture. J. Noncommut. Geom. 127–111 (2007)
Piazza, P., Schick, T., Groups with torsion, bordism and rho invariants. Pacific J. Math. 232, 355–378 (2007)
Sadowski, M., Szczepánski, A.: Flat manifolds, harmonic spinors and eta invariants. Adv. Geom. 6, 287–300 (2006)
Seade, J.A.: On the η-function of the Dirac operator on Γ ∖ S 3. An. Inst. Mat. Univ. Nac. Autónoma México 21, 129–147 (1981)
Seade, J., Steer, B.: A note on the eta function for quotients of PSL 2(R) by co-compact Fuchsian groups. Topology 26, 79–91 (1987)
Stolz, S.: A note on the bP-component of (4n − 1)-dimensional homotopy spheres. Proc. Amer. Math. Soc. 99, 581–584 (1987)
Tsuboi, K.: Eta invariants and conformal immersions. Publ. Res. Inst. Math. Sci. 17, 201–214 (1981)
Verdiani, L., Ziller, W.: Obstructions to positive curvature, preprint (2010); arXiv: 1012.2265
Weinberger, S.: Homotopy invariance of eta invariants. Proc. Nat. Acad. Sci. 85, 5362–5365 (1988)
Wojciechowski, K.P.: The additivity of the η-invariant: The case of an invertible tangential operator. Houston J. Math. 20, 603–621 (1994)
Wojciechowski, K.P., The additivity of the η-invariant: The case of a singular tangential operator. Comm. Math. Phys. 169, 315–327 (1995)
Zhang, W.: Circle bundles, η-invariants and Rokhlin congruences. Ann. Inst. Fourier (Grenoble) 44, 249–270 (1994)
Acknowledgements
Supported in part by DFG special programme “Global Differential Geometry”
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Goette, S. (2012). Computations and Applications of η Invariants. In: Bär, C., Lohkamp, J., Schwarz, M. (eds) Global Differential Geometry. Springer Proceedings in Mathematics, vol 17. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22842-1_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-22842-1_13
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22841-4
Online ISBN: 978-3-642-22842-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)