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

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