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New proof of the cobordism invariance of the index
Author(s):
Maxim
Braverman
Journal:
Proc. Amer. Math. Soc.
130
(2002),
1095-1101.
MSC (1991):
Primary 32L20;
Secondary 58G10, 14F17
Posted:
October 3, 2001
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Abstract:
We give a simple proof of the cobordism invariance of the index of an elliptic operator. The proof is based on a study of a Witten-type deformation of an extension of the operator to a complete Riemannian manifold. One of the advantages of our approach is that it allows us to treat directly general elliptic operators which are not of Dirac type.
References:
-
- 1.
- M. Braverman and M. Farber, Novikov type inequalities for differential forms with non-isolated zeros, Math. Proc. Cambridge Philos. Soc. 122 (1997), 357-375. MR 99b:58220
- 2.
- H.L. Cycon, R.G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with applications to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, 1987. MR 88g:35003
- 3.
- M. Gromov and B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. (1983), no. 58, 295-408. MR 85g:58082
- 4.
- N. Higson, A note on the cobordism invariance of the index, Topology 30 (1991), no. 3, 439-443. MR 92f:58171
- 5.
- M. Lesch, Deficiency indices for symmetric Dirac operators on manifolds with conic singularities, Topology 32 (1993), no. 3. MR 94e:58133
- 6.
- L. I. Nicolaescu, On the cobordism invariance of the index of Dirac operators, Proc. Amer. Math. Soc. 125 (1997). MR 97j:58148
- 7.
- M. Reed and B. Simon, Methods of modern mathematical physics IV: Analysis of operators, Academic Press, London, 1978. MR 58:12429c
- 8.
- M. Shubin, Pseudodifferential operators and spectral theory, Springer-Verlag, Berlin, New York, 1980. MR 88c:47105
- 9.
- -, Semiclassical asymptotics on covering manifolds and Morse inequalities, Geom. Funct. Anal. 6 (1996), 370-409. MR 97i:58171
- 10.
- -, Spectral theory of the Schrödinger operators on non-compact manifolds: qualitative results, Spectral theory and geometry (Edinburgh, 1998), Cambridge Univ. Press, Cambridge, 1999, pp. 226-283. MR 2001d:58037
- 11.
- E. Witten, Supersymmetry and Morse theory, J. of Diff. Geom. 17 (1982), 661-692. MR 84b:58111
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Additional Information:
Maxim
Braverman
Affiliation:
Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email:
maxim@neu.edu
DOI:
10.1090/S0002-9939-01-06250-5
PII:
S 0002-9939(01)06250-5
Keywords:
Vanishing theorem,
Clifford bundle,
Dirac operator,
Andreotti-Grauert theorem,
Melin inequality
Received by editor(s):
October 11, 2000
Posted:
October 3, 2001
Additional Notes:
This research was partially supported by grant No. 96-00210/1 from the United States-Israel Binational Science Foundation (BSF)
Communicated by:
Jozef Dodziuk
Copyright of article:
Copyright
2001,
American Mathematical Society
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