Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence

Dai, Xianzhe

doi:10.1090/S0894-0347-1991-1088332-0

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

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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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