The eta invariant (some recent developments)

Müller, Werner

Séminaire Bourbaki : volume 1993/94, exposés 775-789, Astérisque, no. 227 (1995), Exposé no. 787, 30 p.

[Ar] J. Arthur - Characters, harmonic analysis, and an $l^2$-Lefschetz formula, Proc. Sympos. Pure Math., Vol. 48, Amer. Math. Soc., Providence, R.I. (1987), 167-179. | Zbl | MR | DOI

[At] M. F. Atiyah - The logarithm of the Dedekind $\eta$-function, Math. Annalen 278, 335-380 (1987). | Zbl | EuDML | DOI

[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). | Zbl | MR | DOI

[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). | Zbl | MR | DOI

[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). | Zbl | MR | DOI

[AS] M. F. Atiyah and I. M. Singer - The index of elliptic operators, IV, Ann. of Math. 93, 119-138 (1971). | Zbl | MR | DOI

[Ba] H. Baumgärtel and M. Wollenberg - Mathematical scattering theory, Akademie-Verlag, Berlin 1983. | Zbl | MR | DOI

[Bi] J.-M. Bismut - The Atiyah-singer index theorem for families of Dirac operators : Two heat equation proofs, Invent. math. 83, 91-151 (1986). | Zbl | MR | EuDML | DOI

[BC1] J.-M. Bismut and J. Cheeger - Eta invariants and their adiabatic limits, J. Amer. Math. Soc. 2, 33-70 (1989). | Zbl | MR

[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). | Numdam | Zbl | MR | EuDML | DOI

[BC3] J.-M. Bismut and J. Cheeger - Families index for manifolds with boundary, superconnections and cones, I, Journal Funct. Anal. 89, 313-363 (1990). | Zbl | MR | DOI

[BC4] J.-M. Bismut and J. Cheeger - Families index for manifolds with boundary, superconnections and cones, II, Journal Funct. Anal. 90, 306-354 (1990). | Zbl | MR | DOI

[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. | Zbl | MR

[BF1] J.-M. Bismut and D. S. Freed - The analysis of elliptic families, I, Commun. Math. Phys. 106, 159-176 (1986). | Zbl | MR

[BF2] J.-M. Bismut and D. S. Freed - The analysis of elliptic families, II, 107, 103-163 (1986). | Zbl | MR

[Bo] A. Borel - Stable real cohomology of arithmetic groups, Ann. Sci. Éc. Norm. Super., 4e série, 7, 235-272 (1974). | Numdam | Zbl | MR | EuDML | DOI

[BS] A. Borel and J.-P. Serre - Corners and arithmetic groups, Comment. Math. Helv. 48, 244-297 (1974). | Zbl | MR | EuDML

[BMS] N. V. Borisov, W. Müller and R. Schrader - Relative index theorems and supersymmetric scattering theory, Commun. Math. Phys. 114, 475-513 (1988). | Zbl | MR

[BG] T. Branson and P. Gilkey - Residues of the $\eta$-function for operators of Dirac type, J. Funct. Anal. 108, 47-87 (1992). | Zbl

[Br1] J. Bruning - $L^2$-index theorems on certain complete manifolds, J. Diff. Geom. 32, 491-532 (1990). | Zbl | MR

[Br2] J. Brüning - On $L^2$-index theorems for complete manifolds of rank-one type, Duke Math. J. 66, 257-309. | Zbl | MR

[C1] J. Cheeger - On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76, 2103-2106 (1979). | Zbl | MR

[C2] J. Cheeger - Spectral geometry of singular Riemannian spaces, J. Diff. Geom. 18, 575-657 (1983). | Zbl | MR

[C3] J. Cheeger - $\eta$-invariants, the adiabatic approximation and conical singularities, J. Diff. Geom. 26, 175-221 (1987). | Zbl

[Da] X. Dai - Adiabatic limits, the non-multiplicativity of signature and Leray spectral sequence, J. Amer. Math. Soc. 4, 265-321 (1991). | Zbl | MR

[DW] R. D. Douglas and K. P. Wojciechowski - Adiabatic limits of the $\eta$-invariants. The odd-dimensional Atiyah-Patodi-Singer problem, Commun. Math. Phys. 142, 139-168 (1991). | Zbl

[Gi1] P. B. Gilkey - The residue of the global eta function at the origin, Advances in Math. 40, 290-307 (1981). | Zbl | MR

[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). | Zbl | MR

[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). | Zbl | MR

[GM] M. Goresky, R. Macpherson - Topological trace formula (To appear). | MR

[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. | MR | JFM

[Hi1] F. Hirzebruch - Topological methods in algebraic geometry, Springer-Verlag, Berlin, 1966. | Zbl | MR

[Hi2] F. Hirzebruch - Hilbert modular surfaces, Enseignement Math. 19, 183-281 (1973). | Zbl | MR

[K1] T. Kato - Perturbation theory for linear operators, Berlin, Heidelberg, New York, Springer 1980. | Zbl

[K2] T. Kato - Wave operators and unitary equivalence, Pacific J Math. 15, 171-180 (1965). | Zbl | MR

[La] J.-P. Labesse - Signature des variétés modulaires de Hilbert et représentations diédrales, Lect. Notes in Math. 1447, 249-260, Springer-Verlag, Berlin 1990. | Zbl | MR

[Me] R. Melrose - The Atiyah-Patodi-Singer index theorem, A.K. Peters, Boston, 1993. | Zbl | MR

[MP] R. Melrose and P. Piazza - Families of Dirac operators, boundaries and the $b$-calculus, Preprint, MIT, 1994. | MR

[Mo] H. Moscovici - $L^2$-index of elliptic operators on locally symmetric spaces of finite volume, Contemporary Math. 10, 129-137 (1982). | Zbl | MR

[Mü1] W. Müller - Spectral theory for noncompact Riemannian manifolds with cusps and a related trace formula, I.H.E.S.-Preprint, Nr. 46, 1980.

[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). | Zbl | MR

[Mü3] W. Müller - Manifolds with cusps of rank one, Lect. Notes in Math. 1244, Springer-Verlag, Berlin, 1987. | Zbl | MR

[Mü4] W. Müller - $L^2$-index theory, eta invariants, and values of $L$-functions, Contemporary Math. 105, 145-189 (1990). | Zbl | MR | DOI

[Mü5] W. Müller. Eta invariants and manifolds with boundary. MPI-Preprint, 93-50, Bonn, 1993. | MR | Zbl

[Mü6] W. Müller - $L^2$-index and resonances, in: Analysis on manifolds (Katata 1987), pp. 203-211, Lect. Notes in Math. 1339, Springer, Berlin-New York, 1988. | Zbl | MR

[O] S. Ogata - Hirzebruch conjecture on cusp singularities, Math. Annalen 296, 69-86 (1993). | Zbl | MR | EuDML | DOI

[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. | Zbl | MR

[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. | Zbl | MR

[Sh] H. Shimizu - On discontinuous groups operating on the product of upper halfplanes, Ann. of Math. 77, 33-71 (1963). | Zbl | MR | DOI

[St1] M. Stern - $L^2$-index theorems on warped products, Princeton University Ph.D. thesis, October 1984.

[St2] M. Stern - $L^2$-index theorems on locally symmetric spaces, Invent. math. 96, 231-282 (1989). | Zbl | MR | EuDML | DOI

[St3] M. Stern - Eta invariants and hermitian locally symmetric spaces, J. Diff. Geom. 31, 771-789 (1990). | Zbl | MR | DOI

[St4] M. Stern - Lefschetz formulae for arithmetic varieties, Invent. math. 115, 241-296 (1994). | Zbl | MR | EuDML | DOI

[W1] E. Witten - Global gravitational anomalies, Commun. Math. Phys. 100, 197-229 (1985). | Zbl | MR | DOI

[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. | Zbl | MR