The eta invariant (some recent developments)

Werner Müller

Séminaire Bourbaki (1993-1994)

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

How to cite

top

Müller, Werner. "The eta invariant (some recent developments)." Séminaire Bourbaki 36 (1993-1994): 335-364. <http://eudml.org/doc/110190>.

@article{Müller1993-1994,
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 = {http://eudml.org/doc/110190},
volume = {36},
year = {1993-1994},
}

TY - JOUR
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 - http://eudml.org/doc/110190
ER -

References

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.