The moduli space of parallelizable 4-manifolds

In this paper we construct the space of smooth 4-manifolds and find the homotopy model for the connected components of the complement to the discriminant. The discriminant of this space is a singular hypersurface and its generic points correspond to manifolds with isolated Morse singularities. These spaces can be considered as a natural base for the recent theories studying invariants for families. We show that the theory of Bauer and Furuta can be raised to parametrized families on our configurational space and their invariant is the step-function on chambers. We also introduce the definition of the invariant of finite type and give a simple example of an invariant of order one.