Dirac operators on cobordisms: Degenerations and surgery

Cibotaru, Daniel; Nicolaescu, Liviu

doi:10.1090/S0002-9947-2012-05476-6

Dirac operators on cobordisms: Degenerations and surgery
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by Daniel F. Cibotaru and Liviu I. Nicolaescu

Trans. Amer. Math. Soc. 364 (2012), 3185-3216

DOI: https://doi.org/10.1090/S0002-9947-2012-05476-6

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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