L3

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.