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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2024 MCQ for Journal of the American Mathematical Society is 4.83.

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Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence
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by Xianzhe Dai
J. Amer. Math. Soc. 4 (1991), 265-321
DOI: https://doi.org/10.1090/S0894-0347-1991-1088332-0

Abstract:

We first prove an adiabatic limit formula for the $\eta$-invariant of a Dirac operator, generalizing the recent work of J.-M. Bismut and J. Cheeger. An essential part of the proof is the study of the spectrum of the Dirac operator in the adiabatic limit. A new contribution arises in the adiabatic limit formula, in the form of a global term coming from the (asymptotically) very small eigenvalues. We then proceed to show that, for the signature operator, these very small eigenvalues have a purely topological significance. In fact, we show that the Leray spectral sequence can be recast in terms of these very small eigenvalues. This leads to a refined adiabatic limit formula for the signature operator where the global term is identified with a topological invariant, the signature of a certain bilinear form arising from the Leray spectral sequence. As an interesting application, we give intrinsic characterization of the non-multiplicativity of signature.
References
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Bibliographic Information
  • © Copyright 1991 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 4 (1991), 265-321
  • MSC: Primary 58G10; Secondary 55T10, 58G25
  • DOI: https://doi.org/10.1090/S0894-0347-1991-1088332-0
  • MathSciNet review: 1088332