Adiabatic Limits, Nonmultiplicativity of Signature, and Leray Spectral Sequence

We first prove an adiabatic limit formula for the rf-invariant of aDirac 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 op-erator in the adiabatic limit. A new contribution arises in the adiabatic limitformula, in the form of a global term coming from the (asymptotically) verysmall eigenvalues.We then proceed to show that, for the signature operator, these very smalleigenvalues have a purely topological significance. In fact, we show that theLeray spectral sequence can be recast in terms of these very small eigenvalues.This leads to a refined adiabatic limit formula for the signature operator wherethe global term is identified with a topological invariant, the signature of acertain bilinear form arising from the Leray spectral sequence.As an interesting application, we give intrinsic characterization of the non-multiplicativity of signature. DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE,MASSACHUSETTS 02139 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use