Index, eta and rho invariants on foliated bundles

Benameur, Moulay-Tahar ; Piazza, Paolo

From probability to geometry (I) - Volume in honor of the 60th birthday of Jean-Michel Bismut, Astérisque, no. 327 (2009), pp. 201-287.

MR Zbl

@incollection{AST_2009__327__201_0,
     author = {Benameur, Moulay-Tahar and Piazza, Paolo},
     title = {Index, eta and rho invariants on foliated bundles},
     booktitle = {From probability to geometry (I) - Volume in honor of the 60th birthday of Jean-Michel Bismut},
     editor = {Dai Xianzhe and L\'eandre R\'emi and Xiaonan Ma and Zhang Weiping},
     series = {Ast\'erisque},
     pages = {201--287},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {327},
     year = {2009},
     mrnumber = {2642361},
     zbl = {1206.58006},
     language = {en},
     url = {http://archive.numdam.org/item/AST_2009__327__201_0/}
}

TY  - CHAP
AU  - Benameur, Moulay-Tahar
AU  - Piazza, Paolo
TI  - Index, eta and rho invariants on foliated bundles
BT  - From probability to geometry (I) - Volume in honor of the 60th birthday of Jean-Michel Bismut
AU  - Collectif
ED  - Dai Xianzhe
ED  - Léandre Rémi
ED  - Xiaonan Ma
ED  - Zhang Weiping
T3  - Astérisque
PY  - 2009
SP  - 201
EP  - 287
IS  - 327
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/AST_2009__327__201_0/
LA  - en
ID  - AST_2009__327__201_0
ER  -

%0 Book Section
%A Benameur, Moulay-Tahar
%A Piazza, Paolo
%T Index, eta and rho invariants on foliated bundles
%B From probability to geometry (I) - Volume in honor of the 60th birthday of Jean-Michel Bismut
%A Collectif
%E Dai Xianzhe
%E Léandre Rémi
%E Xiaonan Ma
%E Zhang Weiping
%S Astérisque
%D 2009
%P 201-287
%N 327
%I Société mathématique de France
%U http://archive.numdam.org/item/AST_2009__327__201_0/
%G en
%F AST_2009__327__201_0

Benameur, Moulay-Tahar; Piazza, Paolo. Index, eta and rho invariants on foliated bundles, dans From probability to geometry (I) - Volume in honor of the 60th birthday of Jean-Michel Bismut, Astérisque, no. 327 (2009), pp. 201-287. http://archive.numdam.org/item/AST_2009__327__201_0/

Bibliographie
Cité par

[1] M. F. Atiyah - "Elliptic operators, discrete groups and von Neumann algebras", Astérisque 32-33 (1976), p. 43-72. | Numdam | MR | Zbl

[2] M. F. Atiyah, V. K. Patodi & I. M. Singer - "Spectral asymmetry and Riemannian geometry. I", Math. Proc. Cambridge Philos. Soc. 77 (1975), p. 43-69. | DOI | MR | Zbl

[3] M. F. Atiyah, V. K. Patodi & I. M. Singer, "Spectral asymmetry and Riemannian geometry. II", Math. Proc. Cambridge Philos. Soc. 78 (1975), p. 405-432. | DOI | MR | Zbl

[4] P. Baum & A. Connes - "Geometric $K$ -theory for Lie groups and foliations", Enseign. Math. 46 (2000), p. 3-42. | MR | Zbl

[5] P. Baum, A. Connes & N. Higson - "Classifying space for proper actions and $K$ -theory of group $C^{*}$ -algebras", in $C^{*}$ -algebras: 1943-1993 (San Antonio, TX, 1993), Contemp. Math., vol. 167, Amer. Math. Soc., 1994, p. 240-291. | MR | Zbl

[6] P. Baum, N. Higson & T. Schick - "On the equivalence of geometric and analytic $K$ -homology", Pure Appl. Math. Q. 3 (2007), p. 1-24. | DOI | MR | Zbl

[7] M.-T. Benameur - "Triangulations and the stability theorem for foliations", Pacific J. Math. 179 (1997), p. 221-239. | DOI | MR | Zbl

[8] M.-T. Benameur & T. Fack - "Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras", Adv. Math. 199 (2006), p. 29-87. | DOI | MR | Zbl

[9] M.-T. Benameur & J. L. Heitsch - "The higher harmonic signature for foliations", preprint, submitted. | Zbl

[10] M.-T. Benameur & V. Nistor - "Homology of algebras of families of pseudodifferential operators", J. Funct. Anal. 205 (2003), p. 1-36. | DOI | MR | Zbl

[11] M.-T. Benameur & H. Oyono-Oyono - "Index theory for quasi-crystals. I. Computation of the gap-label group", J. Funct. Anal. 252 (2007), p. 137-170. | DOI | MR | Zbl

[12] M.-T. Benameur & I. Roy - "Leafwise homotopy invariance of the foliated rho invariant", preprint.

[13] J.-M. Bismut - "The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs", Invent. Math. 83 (1985), p. 91-151. | DOI | EuDML | MR | Zbl

[14] J.-M. Bismut & D. S. Freed - "The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem", Comm. Math. Phys. 107 (1986), p. 103-163. | MR | Zbl

[15] S. Chang & S. Weinberger - "On invariants of Hirzebruch and Cheeger-Gromov", Geom. Topol. 7 (2003), p. 311-319. | DOI | EuDML | MR | Zbl

[16] J. Cheeger & M. Gromov - "Bounds on the von Neumann dimension of $L^{2}$ -cohomology and the Gauss-Bonnet theorem for open manifolds", J. Differential Geom. 21 (1985), p. 1-34. | DOI | MR | Zbl

[17] A. Connes - "Sur la théorie non commutative de l'intégration", in Algèbres d'opérateurs (Sém., Les Plans-sur-Bex, 1978), Lecture Notes in Math., vol. 725, Springer, 1979, p. 19-143. | MR | Zbl

[18] A. Connes, Noncommutative geometry, Academic Press Inc., 1994. | MR | Zbl

[19] A. Connes & G. Skandalis - "The longitudinal index theorem for foliations", Publ. Res. Inst. Math. Sci. 20 (1984), p. 1139-1183. | DOI | MR | Zbl

[20] J. Dixmier - Les algèbres d'opérateurs dans l'espace hilbertien (algèbres de von Neumann), Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, 1996, reprint of the second (1969) edition. | MR

[21] J. Dixmier, Les $C^{*}$ -algèbres et leurs représentations, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, 1996, reprint of the second (1969) edition. | MR

[22] T. Fack & H. Kosaki - "Generalized $s$ -numbers of $τ$ -measurable operators", Pacific J. Math. 123 (1986), p. 269-300. | DOI | MR | Zbl

[23] A. Gorokhovsky & J. Lott - "Local index theory over étale groupoids", J. reine angew. Math. 560 (2003), p. 151-198. | MR | Zbl

[24] P. D. L. Harpe & G. Skandalis - "Déterminant associé à une trace sur une algébre de Banach", Ann. Inst. Fourier (Grenoble) 34 (1984), p. 241-260. | DOI | EuDML | Numdam | MR | Zbl

[25] J. L. Heitsch & C. Lazarov - "A Lefschetz theorem for foliated manifolds", Topology 29 (1990), p. 127-162. | DOI | MR | Zbl

[26] J. L. Heitsch & C. Lazarov, "Homotopy invariance of foliation Betti numbers", Invent. Math. 104 (1991), p. 321-347. | DOI | EuDML | MR | Zbl

[27] M. Hilsum & G. Skandalis - "Morphismes $K$ -orientés d'espaces de feuilles et fonctorialité en théorie de Kasparov (d'après une conjecture d'A. Connes)", Ann. Sci. École Norm. Sup. 20 (1987), p. 325-390. | DOI | EuDML | Numdam | MR | Zbl

[28] M. Hilsum & G. Skandalis, "Invariance par homotopie de la signature à coefficients dans un fibré presque plat", J. reine angew. Math. 423 (1992), p. 73-99. | EuDML | MR | Zbl

[29] N. Keswani - "Geometric K-homology and controlled paths", New York J. Math. 5 (1999), p. 53-81. | EuDML | MR | Zbl

[30] N. Keswani, "Relative eta-invariants and $C^{*}$ -algebra $K$ -theory", Topology 39 (2000), p. 957-983. | DOI | MR | Zbl

[31] N. Keswani, "Von Neumann eta-invariants and $C^{*}$ -algebra $K$ -theory", J. London Math. Soc. 62 (2000), p. 771-783. | DOI | MR | Zbl

[32] R. Lauter, B. Monthubert & V. Nistor - "Pseudodifferential analysis on continuous family groupoids", Doc. Math. 5 (2000), p. 625-655. | EuDML | MR | Zbl

[33] E. Leichtnam & P. Piazza - "Cut-and-paste on foliated bundles", in Spectral geometry of manifolds with boundary and decomposition of manifolds, Contemp. Math., vol. 366, Amer. Math. Soc., 2005, p. 151-192. | DOI | MR | Zbl

[34] E. Leichtnam & P. Piazza, "Étale groupoids, eta invariants and index theory", J. reine angew. Math. 587 (2005), p. 169-233. | DOI | MR | Zbl

[35] D. Lenz, N. Peyerimhoff & I. Veselić - "Groupoids, von Neumann algebras and the integrated density of states", Math. Phys. Anal. Geom. 10 (2007), p. 1-41. | DOI | MR | Zbl

[36] V. Mathai - "Spectral flow, eta invariants, and von Neumann algebras", J. Funct. Anal. 109 (1992), p. 442-456. | DOI | MR | Zbl

[37] R. B. Melrose - The Atiyah-Patodi-Singer index theorem, Research Notes in Math., vol. 4, A K Peters Ltd., 1993. | MR | Zbl

[38] C. C. Moore & C. L. Schochet - Global analysis on foliated spaces, second ed., Mathematical Sciences Research Institute Publications, vol. 9, Cambridge Univ. Press, 2006. | MR | Zbl

[39] H. Moriyoshi & T. Natsume - "The Godbillon-Vey cyclic cocycle and longitudinal Dirac operators", Pacific J. Math. 172 (1996), p. 483-539. | MR | Zbl

[40] H. Moscovici & F.-B. Wu - "Localization of topological Pontryagin classes via finite propagation speed", Geom. Funct. Anal. 4 (1994), p. 52-92. | DOI | EuDML | MR | Zbl

[41] W. D. Neumann - "Signature related invariants of manifolds. I. Monodromy and $γ$ -invariants", Topology 18 (1979), p. 147-172. | DOI | MR | Zbl

[42] V. Nistor, A. Weinstein & P. Xu - "Pseudodifferential operators on differential groupoids", Pacific J. Math. 189 (1999), p. 117-152. | DOI | MR | Zbl

[43] G. Peric - "Eta invariants of Dirac operators on foliated manifolds", Trans. Amer. Math. Soc. 334 (1992), p. 761-782. | DOI | MR | Zbl

[44] J. Phillips - "The holonomic imperative and the homotopy groupoid of a foliated manifold", Rocky Mountain J. Math. 17 (1987), p. 151-165. | DOI | MR | Zbl

[45] P. Piazza & T. Schick - "Bordism, rho-invariants and the Baum-Connes conjecture", J. Noncommut. Geom. 1 (2007), p. 27-111. | DOI | MR | Zbl

[46] P. Piazza & T. Schick, "Groups with torsion, bordism and rho invariants", Pacific J. Math. 232 (2007), p. 355-378. | DOI | MR | Zbl

[47] M. Ramachandran - "von Neumann index theorems for manifolds with boundary", J. Differential Geom. 38 (1993), p. 315-349. | DOI | MR | Zbl

[48] J. Raven - "An equivariant bivariant Chern character", Ph.D. Thesis, Pennsylvania State University, 2004, http://etda.libraries.psu.edu/theses/approved/WorldWideFiles/ETD-723/dissertation.pdf. | MR

[49] J. Renault - A groupoid approach to $C^{*}$ -algebras, Lecture Notes in Math., vol. 793, Springer, 1980. | MR | Zbl

[50] J. Roe - "Finite propagation speed and Connes' foliation algebra", Math. Proc. Cambridge Philos. Soc. 102 (1987), p. 459-466. | DOI | MR | Zbl

[51] J. Roe, "Partitioning noncompact manifolds and the dual Toeplitz problem", in Operator algebras and applications, Vol. 1, London Math. Soc. Lecture Note Ser., vol. 135, Cambridge Univ. Press, 1988, p. 187-228. | MR | Zbl

[52] J. Roe, Elliptic operators, topology and asymptotic methods, second ed., Pitman Research Notes in Math. Series, vol. 395, Longman, 1998. | MR | Zbl

[53] M. A. Shubin - Pseudodifferential operators and spectral theory, Springer Series in Soviet Mathematics, Springer, 1987. | MR | Zbl

[54] M. E. Taylor - Pseudodifferential operators, Princeton Mathematical Series, vol. 34, Princeton Univ. Press, 1981. | MR | Zbl

[55] S. Vassout - "Feuilletages et résidu non commutatif longitudinal", Ph.D. Thesis, Université Paris 6, 2001, http://people.math.jussieu.fr/~vassout/these.pdf.

[56] S. Vassout, "Unbounded pseudodifferential calculus on Lie groupoids", J. Funct. Anal. 236(2006), p. 161-200. | DOI | MR | Zbl

[57] S. Weinberger - "Homotopy invariance of $η$ -invariants", Proc. Nat. Acad. Sci. U.S.A. 85 (1988), p. 5362-5363. | DOI | MR | Zbl