In this paper we develop the theory of finite-type invariants for homologic
ally nontrivial 3-manifolds. We construct an infinite-dimensional Euclidean
space with a hypersurface in it corresponding to manifolds with Morse sing
ularities. The connected components of the complement to the discriminant c
orrespond to homeomorphism classes of spin 3-manifolds. We give the definit
ion of the invariant of finite type, suggested by the geometry of the space
and give an example or the invariant of order 1. (C) 2000 Academie des sci
ences/Editions scientifiques et medicales Elsevier SAS.