Continuous Cohomology of the Group of Volume-preserving and Symplectic Diffeomorphisms, Measurable Transfer and Higher Asymptotic Cycles

Topology of a manifold is reflected in its diffeomorphism group. It is challenging therefore to understand the diffeomorphism group Diff(M) both as a topological and discrete group. Twenty years ago, some work has been done, in connection with characteristic classes of foliations, in constructing continuous cohomology classes for Diff(M). For M closed oriented n-dimensional manifold, a class in H cont(Diff(M),R) has been explicitly written down by Bott [Bo] [Br]. This class is defined as follows. The group Diff(M) acts in the multiplicative group C + (M) of positive smooth functions, and on its torsor An(M) of volume forms. Hence one gets a cocycle in H cont(Diff(M), C ∞ + (M)), defined by λ(f) = f∗(v) v = Jacv(f), where ν ∈ An(M) and f ∈ Diff(M). The Bott class is ∫