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We refine Matveev's result asserting that any two closed oriented 3-manifoldscan be related by a sequence of borromean surgeries if and only if they haveisomorphic first homology groups and linking pairings. Indeed, a borromeansurgery induces a canonical isomorphism between the first homology groups ofthe involved 3-manifolds, which preserves the linking pairing. We prove thatany such isomorphism is induced by a sequence of borromean surgeries. As anintermediate result, we prove that a given algebraic square finite presentationof the first homology group of a 3-manifold, which encodes the linking pairing,can always be obtained from a surgery presentation of the manifold.

Realizing isomorphisms between first homology groups of closed 3-manifolds by borromean surgeries