New proof of the cobordism invariance of the index

Author:
Maxim Braverman

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1095-1101

MSC (1991):
Primary 32L20; Secondary 58G10, 14F17

Published electronically:
October 3, 2001

MathSciNet review:
1873784

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Abstract | References | Similar Articles | Additional Information

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.

**1.**Maxim Braverman and Michael Farber,*Novikov type inequalities for differential forms with non-isolated zeros*, Math. Proc. Cambridge Philos. Soc.**122**(1997), no. 2, 357–375. MR**1458239**, 10.1017/S0305004197001734**2.**H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon,*Schrödinger operators with application to quantum mechanics and global geometry*, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR**883643****3.**Mikhael Gromov and H. Blaine Lawson Jr.,*Positive scalar curvature and the Dirac operator on complete Riemannian manifolds*, Inst. Hautes Études Sci. Publ. Math.**58**(1983), 83–196 (1984). MR**720933****4.**Nigel Higson,*A note on the cobordism invariance of the index*, Topology**30**(1991), no. 3, 439–443. MR**1113688**, 10.1016/0040-9383(91)90024-X**5.**Matthias Lesch,*Deficiency indices for symmetric Dirac operators on manifolds with conic singularities*, Topology**32**(1993), no. 3, 611–623. MR**1231967**, 10.1016/0040-9383(93)90012-K**6.**Liviu I. Nicolaescu,*On the cobordism invariance of the index of Dirac operators*, Proc. Amer. Math. Soc.**125**(1997), no. 9, 2797–2801. MR**1402879**, 10.1090/S0002-9939-97-03975-0**7.**Michael Reed and Barry Simon,*Methods of modern mathematical physics. IV. Analysis of operators*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**0493421****8.**M. A. Shubin,*Pseudodifferential operators and spectral theory*, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. Translated from the Russian by Stig I. Andersson. MR**883081****9.**M. A. Shubin,*Semiclassical asymptotics on covering manifolds and Morse inequalities*, Geom. Funct. Anal.**6**(1996), no. 2, 370–409. MR**1384616**, 10.1007/BF02247891**10.**Mikhail Shubin,*Spectral theory of the Schrödinger operators on non-compact manifolds: qualitative results*, Spectral theory and geometry (Edinburgh, 1998) London Math. Soc. Lecture Note Ser., vol. 273, Cambridge Univ. Press, Cambridge, 1999, pp. 226–283. MR**1736869**, 10.1017/CBO9780511566165.009**11.**Edward Witten,*Supersymmetry and Morse theory*, J. Differential Geom.**17**(1982), no. 4, 661–692 (1983). MR**683171**

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Additional Information

**Maxim Braverman**

Affiliation:
Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

Email:
maxim@neu.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-01-06250-5

Keywords:
Vanishing theorem,
Clifford bundle,
Dirac operator,
Andreotti-Grauert theorem,
Melin inequality

Received by editor(s):
October 11, 2000

Published electronically:
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

Article copyright:
© Copyright 2001
American Mathematical Society