We construct spaces of manifolds of various dimensions following
Vassiliev's approach to the theory of knots. These are infinite-dimensional
spaces with hypersurface, corresponding to manifolds with Morse singularities.
Connected components of the complement to this discriminant are homotopy
equivalent to the covering spaces of BDiff(M). These spaces appear to be a
natural base over which one can consider parametrised versions of Floer and
Seiberg-Witten theories.
