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.
"Cohomology rings, Rochlin function, linking pairing and the Goussarov–Habiro theory of three-manifolds." Algebr. Geom. Topol. 3 (2) 1139 - 1166, 2003. https://doi.org/10.2140/agt.2003.3.1139