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On Carvalho's  -theoretic formulation of the cobordism invariance of the index
Author(s):
Sergiu
Moroianu
Journal:
Proc. Amer. Math. Soc.
134
(2006),
3395-3404.
MSC (2000):
Primary 58J20, 58J42
Posted:
May 11, 2006
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Abstract:
We give an analytic proof of the fact that the index of an elliptic operator on the boundary of a compact manifold vanishes when the principal symbol comes from the restriction of a  -theory class from the interior. The proof uses non-commutative residues inside the calculus of cusp pseudodifferential operators of Melrose.
References:
-
- 1.
- M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43-69. MR 0397797 (53:1655a)
- 2.
- M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422-433. MR 0157392 (28:626)
- 3.
- M. F. Atiyah and I. M. Singer, The index of elliptic operators. I, Ann. of Math. 87 (1968), 484-530. MR 0236950 (38:5243)
- 4.
- M. Braverman, New proof of the cobordism invariance of the index, Proc. Amer. Math. Soc. 130 (2002), 1095-1101. MR 1873784 (2002j:58035)
- 5.
- C. Carvalho, Pseudodifferential operators and applications to Index Theory on noncompact manifolds, Ph.D. thesis, Trinity College, University of Oxford (2003).
- 6.
- C. Carvalho, A
 -theory proof of the cobordism invariance of the index, preprint math.KT/0408260 (2004). - 7.
- N. Higson, A note on the cobordism invariance of the index, Topology 30 (1991), 439-443. MR 1113688 (92f:58171)
- 8.
- R. Lauter and S. Moroianu, The index of cusp operators on manifolds with corners, Ann. Global Anal. Geom. 21 (2002), 31-49. MR 1889248 (2003e:58033)
- 9.
- R. Lauter and S. Moroianu, An index formula on manifolds with fibered cusp ends, J. Geom. Anal. 15 (2005), 261-283. MR 2152483
- 10.
- M. Lesch, Deficiency indices for symmetric Dirac operators on manifolds with conic singularities, Topology 32 (1993), 611-623.MR 1231967 (94e:58133)
- 11.
- P. Loya, Tempered operators and the heat kernel and complex powers of elliptic pseudodifferential operators, Comm. Partial Differential Equations 26 (2001), 1253-1321. MR 1855279 (2002h:58048)
- 12.
- R. B. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics 4, A. K. Peters, Wellesley, MA (1993). MR 1348401 (96g:58180)
- 13.
- R. B. Melrose, Pseudodifferential operators, corners and singular limits, in Proc. Int. Congress of Mathematicians, Kyoto, Springer-Verlag, Berlin - Heidelberg - New York (1990), 217-234. MR 1159214 (93j:58131)
- 14.
- R. B. Melrose, The eta invariant and families of pseudodifferential operators, Math. Res. Letters 2 (1995), 541-561. MR 1359962 (96h:58169)
- 15.
- R. B. Melrose and V. Nistor, Homology of pseudodifferential operators I. Manifolds with boundary, preprint funct-an/9606005.
- 16.
- R. B. Melrose and P. Piazza, Families of Dirac operators, boundaries and the
 -calculus, J. Differ. Geom. 46 (1997), 99-180. MR 1472895 (99a:58144) - 17.
- S. Moroianu,
 -Theory of suspended pseudodifferential operators,  -Theory 28 (2003), 167-181. MR 1995875 (2005a:58041) - 18.
- S. Moroianu, Cusp geometry and the cobordism invariance of the index, Adv. Math. 194 (2005), 504-519. MR 2139923
- 19.
- L. I. Nicolaescu, On the cobordism invariance of the index of Dirac operators, Proc. Amer. Math. Soc. 125 (1997), 2797-2801. MR 1402879 (97j:58148)
- 20.
- B. Vaillant, Index- and spectral theory for manifolds with generalized fibered cusps, Dissertation, Bonner Mathematische Schriften 344 (2001), Rheinische Friedrich-Wilhelms-Universität Bonn. MR 1933455 (2003h:58034)
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Additional Information:
Sergiu
Moroianu
Affiliation:
Institutul de Matematica al Academiei Române P.O. Box 1-764, RO-014700 Bucharest, Romania
Email:
moroianu@alum.mit.edu
DOI:
10.1090/S0002-9939-06-08347-X
PII:
S 0002-9939(06)08347-X
Keywords:
Cusp pseudodifferential operators,
noncommutative residues
Received by editor(s):
November 19, 2004
Received by editor(s) in revised form:
May 11, 2005
Posted:
May 11, 2006
Additional Notes:
This research was partially supported by RTN HPRN-CT-2002-00280 ``Quantum Spaces -- Noncommutative Geometry'' and Marie Curie MERG 006375 funded by the European Commission, and by a CERES contract (2004)
Communicated by:
Mikhail Shubin
Copyright of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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