W. B. R. Lickorish [Ann. Math., II. Ser. 76, 531-540 (1962; Zbl 0106.37102)] and A . H. Wallace [Can. J. Math. 12, 503-528 (1960; Zbl 0108.36101)] showed that any closed, orientable \(3\)-manifold can be obtained from \(S^3\) by surgery along a link in \(S^3\). Thus there is a bijection from the set of ‘surgery-links’ in the \(3\)-sphere \(S^3\) (or a \(3\)-manifold) to the set of orientable \(3\)-manifolds. In [Invent. Math. 45, 35-56 (1978; Zbl 0377.55001)] R. Kirby defined an equivalence relation on the set of surgery-links in \(S^3\) by using certain moves, Kirby’s moves, and showed that homeomorphism classes of closed orientable \(3\)-manifolds correspond bijectively to the equivalence classes of surgery-link in \(S^3\). Furthermore, in [Topology 18, 1-15 (1979; Zbl 0413.57006)] R. Fenn and C. Rourke define an equivalence relation on the set of surgery-links in a \(3\)-manifold by using extended Kirby’s moves and an additional move and showed that homeomorphism classes of closed orientable \(3\)-manifolds correspond bijectively to the equivalence classes of surgery-links in a \(3\)-manifold. In this paper, by using the same equivalence relation on links as in [Fenn and Rourke, loc. cit.], the authors showed that homeomorphism classes of compact, orientable \(3\)-manifolds (possibly with boundaries) correspond bijectively to the equivalence classes of surgery-links.
For non-orientable \(3\)-manifolds, similar results hold as follows. By [W. B. R. Lickorish, Proc. Camb. Philos. Soc. 59, 307-317 (1963; Zbl 0115.40801)], any non-orientable \(3\)-manifold can be obtained from the non-orientable \(S^2\) bundle over \(S^1\), \(S^1\widetilde{\times}S^2\), by surgery along a link in \(S^1\widetilde{\times}S^2\). Fenn and Rourke [loc. cit.] gave a classification of closed, non-orientable \(3\)-manifolds by using an equivalence relation on links in \(S^1\widetilde{\times}S^2\). For a compact, non-orientable \(3\)-manifold \(M\) (possibly with boundary), the authors also gave a classification theorem by using an equivalence relation, similar to but different from that defined by Fenn and Rourke [loc. cit.].