Cohomology rings, Rochlin function, linking pairing and the Goussarov--Habiro theory of three--manifolds

We prove that two closed oriented 3-manifolds have isomorphic quintuplets (homology, space of spin structures, linking pairing, cohomology rings, Rochlin function) if, and only if, they belong to the same class of a certain surgery equivalence relation introduced by Goussarov and Habiro.