Adiabatic limits and spectral sequences for Riemannian foliations

For general Riemannian foliations, spectral asymptotics of the Laplacian is studied when the metric on the ambient manifold is blown up in directions normal to the leaves (adiabatic limit). The number of "small" eigenvalues i s given in terms of the differentiable spectral sequence of the foliation. The asymptotics of the corresponding eigenforms also leads to a Hedge theor etic description of this spectral sequence. This is an extension of results of Mazzeo-Melrose and R. Forman.