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

TL;DR

This paper demonstrates that any isomorphism between the first homology groups of closed 3-manifolds, which preserves linking pairings, can be realized through a sequence of borromean surgeries, refining previous classification results.

Contribution

It proves that all such homology isomorphisms are realizable via borromean surgeries, linking algebraic invariants to topological operations in 3-manifolds.

Findings

Borromean surgeries induce canonical isomorphisms preserving linking pairings.

Any algebraic presentation of the homology group can be realized through surgery.

Homology isomorphisms compatible with linking pairings are realizable by borromean surgeries.

Abstract

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