The b-pseudodifferential calculus on Galois coverings and a higher Atiyah-Patodi-Singer index theorem (1997)


Leichtnam, Éric; Piazza, Paolo
Mémoires de la Société Mathématique de France, Tome 68 (1997) 130 p doi : 10.24033/msmf.382
URL stable : http://www.numdam.org/item?id=MSMF_1997_2_68__1_0

Bibliographie

[A1] M.F. Atiyah, K-theory, Benjamin, 1967. MR 36 #7130 | Zbl 0159.53302

[A2] M.F. Atiyah: Elliptic operators, discrete groups and von Neumann algebras, Astérisque, 32 (1976) p.43. MR 54 #8741 | Zbl 0323.58015

[ABP] Atiyah-Bott-Patodi: On The Heat Equation and The Index Theorem, Inv. Math. 19, (1968), p. 113. Zbl 0257.58008

[APS 1,3] M.F. Atiyah-V.Patodi-I.Singer: Spectral Asymmetry and Riemannian Geometry I, III, Math. proc. Cambridge Philo. Soc. 77, (1975), p.43; 79, (1976), p.71. Zbl 0297.58008

[AS 1,3] M.F. Atiyah-I.Singer: The Index of Elliptic Operators I, III, Annals of Maths, 87 (1968) p. 484; p. 546. MR 38 #5245 | Zbl 0164.24001

[BJ] S.Baaj-P.Julg: Théorie Bivariante de Kasparov et Opérateurs Non Bornés dans les C*-Modules Hilbertiens, C.R.Acad. Sci. Paris 296, Ser. I, (1883) p. 875. Zbl 0551.46041

[BGV] N.Berline-E.Getzler-M.Vergne: Heat Kernels and Dirac Operators, Springer Verlag (1992). Zbl 0744.58001

[BV] N.Berline-M.Vergne: The Equivariant Chern Character and Index of G-Invariant Operators, In D-modules, Representation Theory and Quantum Groups, Venezia 1992. Lecture Notes in Math. 1565, (1992). Zbl 0793.58032

[B] J-M.Bismut: The Atiyah-Singer Index Theorem for Families of dirac Operators: Two Heat Equations Proofs, Inv. Math. 83, (1986) p.91. MR 87g:58117 | Zbl 0592.58047

[BC 1,2] J-M.Bismut-J.Cheeger: Families Index For Manifolds with Boundaries, Superconnections and Cones,I and II, Journ. Funct. Anal. 89 (1990), p. 313, 90 (1991), p.306. Zbl 0696.53021

[BC 3] J-M.Bismut-J.Cheeger: Remarks on The Index Theorem for Families of Dirac Operators on Manifolds with Boundary, Differential Geometry, B. Lawson and K.Teneblat (eds), Longman Scientific, (1992).

[Br] Breuer: Fredholm Theories in von Neumann algebras I, II, Math. Ann. 178 (1968), p 243; 180, (1969), p 313. MR 38 #2611 | Zbl 0162.18701

[CG] J.Cheeger-M.Gromov: Bounds on the von Neumann dimension of L2-cohomology and the Gauss-Bonnet theorem on open manifolds, J. Diff. Geom. 21 (1985) p.1. Zbl 0614.53034

[CGT] J.Cheeger-M.Gromov-M.Taylor: Finite Propagation Speed, Kernels Estimates for Functions of the Laplace Operator and the Geometry of Complete Riemannian Manifolds, J. Diff. Geom. 17 (1982) p. 15. MR 84b:58109 | Zbl 0493.53035

[C] A.Connes: Noncommutative geometry, Academic Press, (1994). MR 95j:46063 | Zbl 0818.46076

[CM] A.Connes-H.Moscovici: Cyclic Cohomology, The Novikov Conjecture and Hyperbolic Groups, Topology, 29 No 3, (1990) p.345. MR 92a:58137 | Zbl 0759.58047

[DZ] X.Dai-W.Zhang: Higher spectral flow, Preprint (1996).

[ES] D.Efremov-M.Shubin: Spectrum Distribution Function and variational principle for automorphic operators on hyperbolic space, Seminaire E.D.P Ecole Polytechnique (1988-1989). Numdam | Zbl 0698.35168

[Ge] E.Getzler: Pseudodifferential Operators on Supermanifolds and the Atiyah-Singer Index Theorem, Comm. Math. Phys. 92 (1983) p. 163. MR 86a:58104 | Zbl 0543.58026

[G] P.Gilkey: Invariance Theory, The Heat Equation, and The Atiyah-Singer Index Theorem, Publish or Perish, (1984). Zbl 0565.58035

[K] G.Kasparov: Equivariant K-Theory and the Novikov Conjecture, Inventiones Math. 91, (1988) p. 147. MR 88j:58123 | Zbl 0647.46053

[Ka] M.Karoubi: Homologie Cyclique et K-Théorie, Astérisque 149 (1987). MR 89c:18019 | Zbl 0648.18008

[L1] J.Lott: Superconnections and higher index theory, Geom. Funct. An. 2 (1992) p.421. MR 93j:58128 | Zbl 0777.58038

[L2] J.Lott: Higher eta invariants, K-Theory 6 (1992) p. 191. MR 93j:58129 | Zbl 0773.58026

[Lu] G.Lustzig: Novikov's Higher Signature and Families of Elliptic Operators, J.Diff. Geom 7, (1971), p. 229. Zbl 0265.57009

[Me] R.Melrose: The Atiyah-Patodi-Singer Index Theorem, A. K Peters 1993. MR 96g:58180 | Zbl 0796.58050

[Me2] R.Melrose: Differential Analysis on Manifolds with Corners, in preparation.

[MeNi] R.Melrose-V.Nistor: Homology of Pseudo-differential Operators I. Manifolds with Boundary, Preprint (1996).

[MP 1] R.Melrose-P.Piazza: Family of Dirac Operators, Boundaries and the b-Calculus, J. Diff. Geom. 46 (1997) p1. Zbl 0955.58020

[MP 2] R.Melrose-P.Piazza: An Index Theorem For Families of Dirac Operators on Odd-dimensional Manifolds with Boundary, J. Diff. Geom. To appear. Zbl 0920.58053

[MF] A.Mishenko-A.Fomenko: The Index of Elliptic Operators over C*-Algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 43, (1979) p.831. MR 81i:46075 | Zbl 0416.46052

[Pa] W.Paschke: Inner Product Modules Over B*-Algebras, Trans AMS, 182, (1973). MR 50 #8087 | Zbl 0239.46062

[P1] P.Piazza: On The Index of Elliptic Operators on Manifolds with Boundary, Journ. Funct. Anal. 117, (1993), p.308. MR 94h:58160 | Zbl 0793.58035

[Q] D.Quillen: Superconnections and the Chern Character, Topology 24 (1985) p.89. MR 86m:58010 | Zbl 0569.58030

[R] M.Ramachandran: von Neumann Index Theorems for Manifolds With Boundary, J.Diff. Geom. (1992).

[S] I.Singer: Some remarks on operator theory and index theory, Lect. Notes Math 575. MR 57 #7699 | Zbl 0444.47040

[Sh 1] M.Shubin: Spectral Theory of Elliptic pseudodifferential operators, Springer Verlag 1987.

[Sh 2] M.Shubin: L2-Riemann-Roch theorem for elliptic operators, GAFA (1995). MR 96f:58162 | Zbl 0840.58045

[Wu] F.Wu: Higher noncommutative spectral flow, Preprint (1996).