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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Dirac operators on cobordisms: Degenerations and surgery


Authors: Daniel F. Cibotaru and Liviu I. Nicolaescu
Journal: Trans. Amer. Math. Soc. 364 (2012), 3185-3216
MSC (2010): Primary 58J20, 58J28, 58J30, 58J32, 53B20, 35B25
Published electronically: February 8, 2012
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Abstract: We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator $ D_t$ on each regular level set $ C_t$ of a fixed Morse function defining this cobordism. We show that as we approach the critical level set $ C_0$ from above and from below these operators converge in the gap topology to (different) selfadjoint operators $ D_\pm $ that we describe explicitly. We also relate the Atiyah-Patodi-Singer index of the Dolbeault operator on the cobordism to the spectral flows of the operators $ D_t$ on the complement of $ C_0$ and the Kashiwara-Wall index of a triplet of finite dimensional Lagrangian spaces canonically determined by $ C_0$.


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Additional Information

Daniel F. Cibotaru
Affiliation: Instituto de Matemática, Universidade Federal Fluminense, Rua Maria Santos Braga, 24020-140 Niterói, RJ-Brasil
Email: daniel@mat.uff.br

Liviu I. Nicolaescu
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556-4618
Email: nicolaescu.1@nd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05476-6
PII: S 0002-9947(2012)05476-6
Received by editor(s): February 9, 2010
Received by editor(s) in revised form: September 22, 2010
Published electronically: February 8, 2012
Additional Notes: The second author was partially supported by NSF grant DMS-1005745.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.