The eta invariant (some recent developments)

Werner Müller

Séminaire Bourbaki (1993-1994)

  • Volume: 36, page 335-364
  • ISSN: 0303-1179

How to cite


Müller, Werner. "The eta invariant (some recent developments)." Séminaire Bourbaki 36 (1993-1994): 335-364. <>.

author = {Müller, Werner},
journal = {Séminaire Bourbaki},
keywords = {-index theorems; survey; eta-invariant; manifolds with boundary},
language = {eng},
pages = {335-364},
publisher = {Société Mathématique de France},
title = {The eta invariant (some recent developments)},
url = {},
volume = {36},
year = {1993-1994},

AU - Müller, Werner
TI - The eta invariant (some recent developments)
JO - Séminaire Bourbaki
PY - 1993-1994
PB - Société Mathématique de France
VL - 36
SP - 335
EP - 364
LA - eng
KW - -index theorems; survey; eta-invariant; manifolds with boundary
UR -
ER -


  1. [Ar] J. Arthur - Characters, harmonic analysis, and an l2-Lefschetz formula, Proc. Sympos. Pure Math., Vol. 48, Amer. Math. Soc., Providence, R.I. (1987), 167-179. Zbl0671.22007MR974334
  2. [At] M.F. Atiyah - The logarithm of the Dedekind η-function, Math. Annalen278, 335-380 (1987). Zbl0648.58035
  3. [APS1] M.F. Atiyah, V.K. Patodi, and I.M. Singer - Spectral asymmetry and Riemannian geometry, I, Math. Proc. Camb. Phil. Soc.77, 43-69 (1975). Zbl0297.58008MR397797
  4. [APS2] M.F. Atiyah, V.K. Patodi, and I.M. Singer - Spectral asymmetry and Riemannian geometry, II, Math. Proc. Camb. Phil. Soc.78, 405-432 (1975). Zbl0314.58016MR397798
  5. [APS3] M.F. Atiyah, V.K. Patodi, and I.M. Singer - Spectral asymmetry and Riemannian geometry, III, Math. Proc. Camb. Phil. Soc.79, 71-99 (1976). Zbl0325.58015MR397799
  6. [AS] M.F. Atiyah and I.M. Singer - The index of elliptic operators, IV, Ann. of Math.93, 119-138 (1971). Zbl0212.28603MR279833
  7. [Ba] H. Baumgärtel and M. Wollenberg - Mathematical scattering theory, Akademie-Verlag, Berlin1983. Zbl0536.47007MR781534
  8. [Bi] J.-M. Bismut - The Atiyah-singer index theorem for families of Dirac operators : Two heat equation proofs, Invent. math.83, 91-151 (1986). Zbl0592.58047MR813584
  9. [BC1] J.-M. Bismut and J. Cheeger - Eta invariants and their adiabatic limits, J. Amer. Math. Soc.2, 33-70 (1989). Zbl0671.58037MR966608
  10. [BC2] J.-M. Bismut and J. Cheeger - Transgressed Euler classes of SL(n,Z) vector bundles, adiabatic limits of eta invariants and special values of L-functions, Ann. scient. Ec. Norm. Sup., 4e série, t. 25, 335-391 (1992). Zbl0768.58048MR1186908
  11. [BC3] J.-M. Bismut and J. Cheeger - Families index for manifolds with boundary, superconnections and cones, I, Journal Funct. Anal.89, 313-363 (1990). Zbl0696.53021MR1042214
  12. [BC4] J.-M. Bismut and J. Cheeger - Families index for manifolds with boundary, superconnections and cones, II, Journal Funct. Anal.90, 306-354 (1990). Zbl0711.53023MR1052337
  13. [BC5] J.-M. Bismut and J. Cheeger - Remarks on the index theorem for families of Dirac operators on manifolds with boundary, in: Differential Geometry, 59- 83, A symposium in honor of Manfredo do Carmo. ed. B. Lawson and K. Tenenblat, Longman Scientific & Technical, 1991. Zbl0727.58045MR1173033
  14. [BF1] J.-M. Bismut and D.S. Freed - The analysis of elliptic families, I, Commun. Math. Phys.106, 159-176 (1986). Zbl0657.58037MR853982
  15. [BF2] J.-M. Bismut and D.S. Freed - The analysis of elliptic families, II, 107, 103-163 (1986). Zbl0657.58038MR861886
  16. [Bo] A. Borel - Stable real cohomology of arithmetic groups, Ann. Sci. Éc. Norm. Super., 4e série, 7, 235-272 (1974). Zbl0316.57026MR387496
  17. [BS] A. Borel and J.-P. Serre - Corners and arithmetic groups, Comment. Math. Helv.48, 244-297 (1974). Zbl0274.22011MR387495
  18. [BMS] N.V. Borisov, W. Müller and R. Schrader - Relative index theorems and supersymmetric scattering theory, Commun. Math. Phys.114, 475-513 (1988). Zbl0663.58032MR929141
  19. [BG] T. Branson and P. Gilkey - Residues of the η-function for operators of Dirac type, J. Funct. Anal.108, 47-87 (1992). Zbl0756.58048
  20. [Br1] J. Bruning - L2-index theorems on certain complete manifolds, J. Diff. Geom.32, 491-532 (1990). Zbl0722.58043MR1072916
  21. [Br2] J. Brüning - On L2-index theorems for complete manifolds of rank-one type, Duke Math. J.66, 257-309. Zbl0765.58030MR1162191
  22. [C1] J. Cheeger - On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A.76, 2103-2106 (1979). Zbl0411.58003MR530173
  23. [C2] J. Cheeger - Spectral geometry of singular Riemannian spaces, J. Diff. Geom.18, 575-657 (1983). Zbl0529.58034MR730920
  24. [C3] J. Cheeger - η-invariants, the adiabatic approximation and conical singularities, J. Diff. Geom.26, 175-221 (1987). Zbl0623.58021
  25. [Da] X. Dai - Adiabatic limits, the non-multiplicativity of signature and Leray spectral sequence, J. Amer. Math. Soc.4, 265-321 (1991). Zbl0736.58039MR1088332
  26. [DW] R.D. Douglas and K.P. Wojciechowski - Adiabatic limits of the η-invariants. The odd-dimensional Atiyah-Patodi-Singer problem, Commun. Math. Phys.142, 139-168 (1991). Zbl0746.58074
  27. [Gi1] P.B. Gilkey - The residue of the global eta function at the origin, Advances in Math.40, 290-307 (1981). Zbl0469.58015MR624667
  28. [Gi2] P.B. Gilkey - The boundary integrand in the formula for the signature and the Euler characteristic of a Riemannian manifold with boundary, Advances in Math.15, 334-360 (1975). Zbl0306.53042MR368084
  29. [GS] P.B. Gilkey and L. Smith - The eta invariant for a class of elliptic boundary value problems, Commun. Pure Appl. Math.36, 85-132 (1983). Zbl0512.58035MR680084
  30. [GM] M. Goresky, R. MacPherson - Topological trace formula (To appear). MR1992803
  31. [He] E. Hecke - Darstellung von Klassenzahlen als Perioden von Integralen 3. Gat- tung aus dem Gebiet der elliptischen Modulfunktionen, Mathematische Werke, Göttingen, Vandenhoeck & Ruprecht, 1959, 405-417. MR104550JFM52.0378.01
  32. [Hi1] F. Hirzebruch - Topological methods in algebraic geometry, Springer-Verlag, Berlin, 1966. Zbl0138.42001MR202713
  33. [Hi2] F. Hirzebruch - Hilbert modular surfaces, Enseignement Math.19, 183-281 (1973). Zbl0285.14007MR393045
  34. [K1] T. Kato - Perturbation theory for linear operators, Berlin, Heidelberg, New York, Springer1980. Zbl0148.12601
  35. [K2] T. Kato - Wave operators and unitary equivalence, Pacific J Math.15, 171-180 (1965). Zbl0125.34902MR180138
  36. [La] J.-P. Labesse - Signature des variétés modulaires de Hilbert et représentations diédrales, Lect. Notes in Math.1447, 249-260, Springer-Verlag, Berlin1990. Zbl0723.11019MR1082968
  37. [Me] R. Melrose - The Atiyah-Patodi-Singer index theorem, A.K. Peters, Boston, 1993. Zbl0796.58050MR1348401
  38. [MP] R. Melrose and P. Piazza - Families of Dirac operators, boundaries and the b-calculus, Preprint, MIT, 1994. MR1472895
  39. [Mo] H. Moscovici - L2-index of elliptic operators on locally symmetric spaces of finite volume, Contemporary Math.10, 129-137 (1982). Zbl0495.58026MR658511
  40. [Mü1] W. Müller - Spectral theory for noncompact Riemannian manifolds with cusps and a related trace formula, I.H.E.S.-Preprint, Nr. 46, 1980. 
  41. [Mü2] W. Müller - Signature defects of cusps of Hilbert modular varieties and values of L-series at s = 1, J. Diff. Geom.20, 55-119 (1984). Zbl0575.10023MR772126
  42. [Mü3] W. Müller - Manifolds with cusps of rank one, Lect. Notes in Math.1244, Springer-Verlag, Berlin, 1987. Zbl0632.58001MR891654
  43. [Mü4] W. Müller - L2-index theory, eta invariants, and values of L-functions, Contemporary Math.105, 145-189 (1990). Zbl0705.58048MR1047280
  44. [Mü5] W. Müller. Eta invariants and manifolds with boundary. MPI-Preprint, 93-50, Bonn, 1993. MR1231418
  45. [Mü6] W. Müller - L2-index and resonances, in: Analysis on manifolds (Katata 1987), pp. 203-211, Lect. Notes in Math.1339, Springer, Berlin-New York, 1988. Zbl0657.58044MR961482
  46. [O] S. Ogata - Hirzebruch conjecture on cusp singularities, Math. Annalen296, 69-86 (1993). Zbl0810.14002MR1213372
  47. [S] I. Satake - On numerical invariants of arithmetic varieties of Q-rank one, in: "Automorphic forms of several variables", Taniguchi Symposium, Katata, 1983 (I. Satake and Y. Morita eds.), Adv. Studies in Pure Math.46, Birkhäuser, Basel, 1984, 353-369. Zbl0549.10020MR763022
  48. [SO] I. Satake and S. Ogata - Zeta functions associated to cones and their special values, in: "Automorphic forms and geometry of arithmetic varieties" (Y. Namikawa and K. Hashimodo eds), Adv. Studies in Pure Math.15, Kinokuniya, Tokyo and North-Holland, Amsterdam, New York, Oxford, 1989, 1-27. Zbl0712.14009MR1040602
  49. [Sh] H. Shimizu - On discontinuous groups operating on the product of upper halfplanes, Ann. of Math.77, 33-71 (1963). Zbl0218.10045MR145106
  50. [St1] M. Stern - L2-index theorems on warped products, Princeton University Ph.D. thesis, October 1984. 
  51. [St2] M. Stern - L2-index theorems on locally symmetric spaces, Invent. math.96, 231-282 (1989). Zbl0694.58039MR989698
  52. [St3] M. Stern - Eta invariants and hermitian locally symmetric spaces, J. Diff. Geom.31, 771-789 (1990). Zbl0715.53039MR1053345
  53. [St4] M. Stern - Lefschetz formulae for arithmetic varieties, Invent. math.115, 241-296 (1994). Zbl0847.58071MR1258905
  54. [W1] E. Witten - Global gravitational anomalies, Commun. Math. Phys.100, 197-229 (1985). Zbl0581.58038MR804460
  55. [W2] E. Witten - Global anomalies in string theory, in: Proc. of the Symposium on Anomalies, Geometry, Topology of Chicago (1985), Bardeen, W.A., White, A.R. (eds.), Singapore, World Scientific, 1985, pp. 61-99. Zbl0643.53071MR850846

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