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Van Est theorem for groups of diffeomorphisms

For M a closed, oriented, smooth manifold, we study the group Diff/sub o/ (M) of diffeomorphisms of M isotopic to the identity, viewed as an infinite-dimensional Lie group. A classical theorem of van Est relating the topology of a Lie group to its Lie algebra cohomology via a spectral sequence is generalized to this case. The van Est theorem in the classical case also interprets a crucial element of this spectral sequence, the smooth cohomology of the Lie group, in terms of the homogeneous space G/K, where k is a maximal compact subgroup of G. In the case at hand, we show that, although K rarely exists, the space of Riemannian metrics on M may serve as a substitute. Also, techniques are given for working with the space of Riemannian metrics.