Transactions of the American Mathematical Society

Author(s): Robert Lauter; Sergiu Moroianu
Journal: Trans. Amer. Math. Soc. 355 (2003), 3009-3046.
MSC (2000): Primary 58J42, 58J20
Posted: April 24, 2003
Retrieve article in: PDF DVI PostScript

Abstract: The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals that generate the

-dimensional Hochschild cohomology groups, the index of a fully elliptic fibered cusp operator is expressed as the sum of a local contribution of Atiyah-Singer type and a global term on the boundary. We announce a result relating this boundary term to the adiabatic limit of the eta invariant in a particular case.

1.: B. Ammann, R. Lauter, V. Nistor, and A. Vasy. Complex powers and non-compact manifolds, math.OA/0211305, preprint, November 2002.
2.: M. Benameur and V. Nistor, Homology of complete symbols and noncommutative geometry, Landsman, N. P. (ed.) et al., Quantization of singular symplectic quotients, Basel, Birkhäuser. Prog. Math. 198 (2001), 21-46.
3.: J.-L. Brylinski, A differential complex for Poisson manifolds, J. Differential Geometry 28 (1988), 93-114. MR 89m:58006
4.: J.-L. Brylinski and E. Getzler, The homology of algebras of pseudodifferential operators and the noncommutative residue, -Theory 1 (1987), 385-403. MR 89j:58135
5.: B. Bucicovschi, An extension of the work of V. Guillemin on complex powers and zeta functions of elliptic pseudodifferential operators, Proc. Amer. Math. Soc. 127 (1999), 3081-3090. MR 2000a:58085
6.: A. Connes, Noncommutative Geometry, Academic Press, New York - London (1994). MR 95j:46063
7.: A. Connes, Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys. 182 (1996), 155-176. MR 98f:58024
8.: H. O. Cordes, On a class of $C^{*}$ -algebras, Math. Annalen 170 (1967), 283-313. MR 35:749
9.: Yu. V. Egorov and B.-W. Schulze, Pseudodifferential operators, Singularities, Applications, volume 93 of Operator Theory and Applications. Birkäuser, Basel (1997). MR 98e:35181
10.: C. L. Epstein, R. B. Melrose, and G. A. Mendoza, Resolvent of the Laplacian on strictly pseudoconvex domains, Acta Math. 167 (1991), 1-106. MR 92i:32016
11.: B. Gramsch. Relative Inversion in der Störungstheorie von Operatoren und $\Psi$ -Algebren, Math. Annalen 269 (1984), 27-71. MR 86j:47065
12.: V. Guillemin, A new proof of Weyl's formula on the asymptotic distribution of eigenvalues, Adv. in Math. 55 (1985), 131-160. MR 86i:58135
13.: R. Lauter and S. Moroianu. An index formula on manifolds with fibered cusp ends, preprint, 2002.
14.: R. Lauter and S. Moroianu, Homology of pseudo-differential operators on manifolds with fibered boundaries, Journal Reine Angew. Math. 547 (2002), 207-234.
15.: R. Lauter and S. Moroianu, The index of cusp operators on manifolds with corners, Ann. Global Anal. Geom. 21, nr. 1 (2002), 31-49. MR 2003e:58033
16.: R. Lauter and S. Moroianu, On the index formula of Melrose and Nistor. Preprint Nr. 3, IMAR, Bucharest, March 2000.
17.: R. Lauter and S. Moroianu, Fredholm theory for degenerate pseudodifferential operators on manifolds with fibered boundaries, Comm. Partial Differential Equations 26 (2001), 233-283. MR 2002e:58052
18.: R. Lauter and V. Nistor, Analysis of geometric operators on open manifolds: a groupoid approach, In N.P. Landsman, M. Pflaum, and M. Schlichenmaier, ed., Quantization of Singular Symplectic Quotients, vol. 198 of Progress in Mathematics, pp. 181-229. Birkhäuser, Basel - Boston - Berlin, 2001.
19.: M. Lesch and M. J. Pflaum, Traces on algebras of parameter dependent pseudodifferential operators and the eta-invariant, Trans. Amer. Math. Soc. 352 (2000), 4911-4936. MR 2001b:58042
20.: J.-L. Loday, Cyclic homology, volume 301 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin - Heidelberg - New York (1992). MR 94a:19004
21.: F. Mantlik, Norm closure and extension of the symbolic calculus for the cone algebra, Ann. Global Anal. and Geometry 13 (1995), 339-376. MR 97a:58183
22.: R. R. Mazzeo, Elliptic theory of differential edge operators I, Comm. Partial Differential Equations 16 (1991), 1615-1664. MR 93d:58152
23.: R. R. Mazzeo and R. B. Melrose, Pseudodifferential operators on manifolds with fibred boundaries, Asian J. Math. 2 (1998), 833-866. MR 2000m:58046
24.: J. McCleary, User's guide to spectral sequences, volume 12 of Mathematical Lecture Series, Publish or Perish, Wilmington (1985). MR 87f:55014
25.: R. B. Melrose, Analysis on manifolds with corners, in preparation.
26.: R. B. Melrose, Pseudodifferential operators, corners and singular limits, In Proceeding of the International Congress of Mathematicians, Kyoto, Springer-Verlag, Berlin - Heidelberg - New York (1990), 217-234. MR 93j:58131
27.: R. B. Melrose, The Atiyah-Patodi-Singer index theorem, volume 4 of Research Notes in Mathematics, A. K. Peters, Wellesley, Massachusetts (1993). MR 96g:58180
28.: R. B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidean space, In M. Ikawa (ed.), Spectral and Scattering Theory, volume 162 of Lecture Notes in Pure and Applied Mathematics, pages 85-130, New York, 1994. Marcel Dekker Inc. Proceedings of the Taniguchi International Workshop held in Sanda, November 1992. MR 95k:58168
29.: R. B. Melrose, The eta invariant and families of pseudodifferential operators, Math. Res. Letters 2 (1995), 541-561. MR 96h:58169
30.: R. B. Melrose, Geometric scattering theory, Cambridge University Press, 1995. MR 96k:35129
31.: R. B. Melrose, Fibrations, compactifications and algebras of pseudodifferential operators, In L. Hörmander and A. Mellin, editors, Partial Differential Equations and Mathematical Physics, Birkhäuser, Boston, 1996, 246-261. MR 98j:58117
32.: R. B. Melrose, Geometric optics and the bottom of the spectrum, In F. Colombini and N. Lerner, editors, Geometrical optics and related topics, volume 32 of Progress in nonlinear differential equations and their applications, Birkhäuser, Basel - Boston - Berlin (1997).
33.: R. B. Melrose and V. Nistor, Higher index and $\eta$ -invariants for suspended algebras of pseudodifferential operators, unfinished manuscript.
34.: R. B. Melrose and V. Nistor, Homology of pseudodifferential operators I. Manifolds with boundary, to appear in Amer. Math. J., Preprint, May 1996.
35.: S. Moroianu, Higher residues on the algebra of adiabatic pseudodifferential operators, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts (1999).
36.: W. Müller, Manifolds with cusps of rank one. Spectral theory and $L\sp 2$ -index theorem, Lecture Notes in Mathematics 1244, Springer-Verlag, Berlin (1987). MR 89g:58196
37.: R. Nest and E. Schrohe, Hochschild homology of Boutet de Monvel's algebra, in preparation.
38.: T. M. W. Nye and M. A. Singer, An ${L}\sp 2$ -index theorem for Dirac operators on ${S}\sp 1\times{\mathbb{R} }\sp 3$ , J. Funct. Anal. 177 (2000), 203-218. MR 2002a:58020
39.: V. Nistor, Groupoids and the integration of Lie algebroids, J. Math. Soc. Japan 52 (2000), 847-868. MR 2002e:58035
40.: S. Rempel and B.-W. Schulze, Complete Mellin and Green symbolic calculus in spaces with conormal asymptotics, Ann. Global Anal. Geometry 4 (1986), 137-224. MR 89f:58132
41.: C. E. Rickart, Banach algebras with an adjoint operation, Ann. Math. 47 (1946), 528-550. MR 8:159b
42.: B. Vaillant, Index- and spectral theory for manifolds with generalized fibered cusps, Ph.D. thesis, University of Bonn (2001).
43.: M. Wodzicki, Cyclic homology of differential operators, Duke Math. J. 54 (1987), 641-647. MR 88k:32035
44.: M. Wodzicki, Noncommutative residue. I. Fundamentals, In -theory, arithmetic and geometry (Moscow, 1984-1986), Springer, Berlin (1987), 320-399. MR 90a:58175
45.: M. Wodzicki, Cyclic homology of pseudodifferential operators and noncommutative Euler class, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 321-325. MR 89h:58189
46.: J. Wunsch, Propagation of singularities and growth for Schrödinger operators, Duke Math. J. 98 (1999), 137-186. MR 2000h:58054

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 58J42, 58J20

Robert Lauter
Affiliation: Fachbereich 17 - Mathematik, Universität Mainz, D-55099 Mainz, Germany
Email: lauter@mathematik.uni-mainz.de

Sergiu Moroianu
Affiliation: Institutul de Matematica al Academiei Române, P.O. Box 1-764, RO-70700 Bucharest, Romania
Email: moroianu@alum.mit.edu

DOI: 10.1090/S0002-9947-03-03294-X
PII: S 0002-9947(03)03294-X
Received by editor(s): July 15, 2002
Received by editor(s) in revised form: January 16, 2003
Posted: April 24, 2003
Additional Notes: Moroianu was partially supported by a DFG-grant (436-RUM 17/7/01) and by the European Commission RTN HPRN-CT-1999-00118 \emph{Geometric Analysis}.
Copyright of article: Copyright 2003, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS