Dehn surgery equivalence relations on 3-manifolds (Q2743678)

The authors examine the equivalence relation on closed oriented 3-manifolds generated by Dehn surgeries which preserve integral or rational homology groups, or which preserve some property of the fundamental group. For example, if the second lower central subgroup is preserved, then the two 3-manifolds are said to be surgery equivalent. Three of their theorems (corollaries) are:NEWLINENEWLINENEWLINECorollary 3.6. A closed, connected 3-manifold with \(H_1= Z^m\) is surgery equivalent to the connected sum of \(m\) copies of \(S^1\times S^2\) iff its integral triple cup product vanishes identically.NEWLINENEWLINENEWLINECorollary 3.11. If two closed, orientable 3-manifolds are orientation-preserving homotopy equivalent, then they are surgery equivalent.NEWLINENEWLINENEWLINECorollary 3.12. If two closed orientable 3-manifolds have isomorphic abelian fundamental groups then they are surgery equivalent iff they are orientation-preserving homotopy equivalent.