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Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence


Author: Xianzhe Dai
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
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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.


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DOI: https://doi.org/10.1090/S0894-0347-1991-1088332-0
Article copyright: © Copyright 1991 American Mathematical Society

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