Kazuo Habiro RIMS, Kyoto University In this talk, I plan to explain our recent results joint with T. Widmer on Kirby calculus for framed links in 3-manifolds [5, 6]. It is well known that every closed oriented 3-manifold can be obtained from S 3 by surgery along a framed link [8, 11]. Kirby [7] gave a criterion for two framed links in S 3 to give the same (i.e., orientation-preserving homeomorphic) results of surgery using two kinds of moves called stabilization and handle slides. One can use Kirby's theorem in order to define a 3-manifold invariant, by con- structing framed link invariant which is invariant under the Kirby moves. For example, the Reshetikhin–Turaev invariant is constructed in this way. Fenn and Rourke [3] gave a characterization of the equivalence relation on framed links in a closed oriented 3-manifold generated by Kirby's two types of moves. We generalized this result to 3-manifolds with boundary in [5]. As an application, we proved the following result for null-homotopic framed links, which are framed links whose components are null-homotopic. Theorem 1 ([5]). For null-homotopic framed links L and L' in a compact oriented 3-manifold M with connected boundary, the following conditions are equivalent. 1. L and L' are related by a sequence of stabilizations and handle-slides. 2. There exists an orientation-preserving homeomorphism h : ML → ML, relative to boundary satisfying the following commutative diagram Here the surjective homomorphisms e (resp. e') are defined using the 4- manifold W L (resp. W L ) constructed by adding 2-handles on the cylinder M × [0,1] along L × {1} (resp. L' × {1}). 1 For a more precise and general statement see [5, Theorem 3.1]. In [6], we considered null-homologous framed links in a 3-manifold. Here a framed link L is Q-null-homologous if every component of L is Q-null-homologous in the 3-manifold.  Similarly, Z -null-homologous framed links are defined.  Based on our generalization of Fenn and Rourke's theorem, we proved the following result. Theorem 2 ([6]). Let M be a compact, connected, oriented 3-manifold with non-empty boundary. Let P ⊂ ∂M be a subset containing exactly one point of each connected component of ∂M . Let L and L' be Q-null-homologous framed links in M. Then the following conditions are equivalent. 1. L and L' are related by a sequence of stabilization, handle-slides, "Q-null- homologous K3-moves", and “IHX-moves”. 2. There is an orientation-preserving homeomorphism h : ML ⇒ ML , re- stricting to the canonical identification ∂M L ⇒ ∂M L ′ such that the follow- ing diagram commutes.
Here er, and er,  are defined by using the 4-manifolds W L and W L , re- spectively. See [6, Theorem 1.1] for a more precise statement and the definition of Q- null-homologous K 3 -moves and IHX-moves. We note here that the IHX-moves are related to the IHX relations in the theory of finite type invariants of links and 3-manifolds. We also consider a more special class of framed links, called admissible framed links. Here a framed link L is admissible if L is Z -null-homologous and the link- ing matrix of L is diagonal with diagonal entries ±1. Surgery along admissible framed links is studied e.g. in [1, 2, 9]. We have the following result. Theorem 3 ([6]). Let M  and P be as in Theorem 2. Suppose that H 1 ( M ; Z ) is torsion-free. Let L and L' be admissible framed links in M. Then the following conditions are equivalent. 1. L and L' are related by a sequence of stabilizations, "band-slides", "pair- moves”, “admissible IHX-moves” and “lantern-moves”. 2. There is an orientation-preserving homeomorphism h : ML ⇒ ML , re- stricting to the identification map M L = M L , such that the following dia- gram commutes. See [6, Theorem 1.4] for a more precise statement. The case where M = S 3 has been proved in [4], where we do not need pair-moves, admissible IHX-moves and lantern-moves. In the talk, I plan to discuss also some applications and generalizations of the results. [1] T. D. Cochran, A. Gerges and K. Orr. Dehn surgery equivalence relations on 3-manifolds, Math. Proc. Camb. Phil. Soc. 131 (2001) 97–127. [2] T. D. Cochran and P. M. Melvin. Finite type invariants of 3-manifolds. Invent. Math. 140 (2000) 45–100. [3] R. Fenn and C. Rourke, On Kirby's calculus of links, Topology 18 (1979) 1–15. [4] K. Habiro. Refined Kirby calculus for integral homology spheres. Geom. Topol. 10 (2006) 1285–1317. [5] K. Habiro and T. Widmer. On Kirby calculus for null-homotopic framed links in 3-manifolds. Alg. Geom. Topol. 14 (2014) 115–134. [6] K. Habiro and T. Widmer. Kirby calculus for null-homologous framed links in 3-manifolds. preprint, 2014. [7] R. Kirby. A calculus for framed links in S 3 . Invent. Math. 45 (1978) 35–56. [8] W. B. R. Lickorish. A representation of orientable combinatorial 3- manifolds. Ann. of Math. (2) 76 (1962) 531–540. [9] T. Ohtsuki. Finite type invariants of integral homology 3-spheres. J. Knot Theory Ramif. 5 (1996) 101–115. [10] N. Yu. Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103 (1991) 547–597. [11] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math 12 (1960) 503–528. (2)