Steven Boyer

Finite Dehn surgery on knots

Let K be a knot with a closed tubular neighbourhood N(K) in a connected orientable closed 3-manifold W , such that the exterior of K, M = W − intN(K), is irreducible. We consider the problem of which Dehn surgeries on K, or equivalently, which Dehn fillings on M , can produce 3-manifolds with finite fundamental group. For convenience, a surgery is called a G-surgery if the resultant 3-manifold has fundamental group G. If G is cyclic or finite, the surgery is also called a cyclic surgery or a finite surgery. Similar terminology will be used for Dehn fillings. The manifold obtained by the Dehn filling on M along ∂M with slope r is denoted by M(r). Let ∆(r1, r2) denote the minimal geometric intersection number (the distance) between two slopes r1 and r2 on ∂M . According to [T1], M belongs to one of the following three mutually exclusive categories: (I) M is a Seifert fibred space admitting no essential torus. (II) M is a hyperbolic manifold (i.e. intM admits a complete hyperbolic metric of finite volume). (III) M contains an essential torus. It is a remarkable result, the so-called cyclic surgery theorem [CGLS], that if M is not Seifert fibred, then all cyclic surgery slopes of K have mutual distance no larger than 1, and consequently, there are at most 3 cyclic surgeries on K. In this paper, we consider finite Dehn surgery on K and we prove, for instance, that if M is not a manifold of type (I) and is not a union along a torus of the twisted I-bundle over the Klein bottle and a cabled space, then there are at most 6 finite and cyclic surgeries on K of maximal mutual distance 5. Henceforth, we shall use finite/cyclic to mean either finite or (infinite) cyclic. It turns out to be convenient to consider the three cases mentioned above separately. In case (I) it is well-known that one can completely classify the finite/cyclic surgeries on K. Considering the torus knots for instance, one sees that there exist infinitely many knots whose exteriors are of type (I), each of which admit an infinity of distinct finite (cyclic or non-cyclic) surgery slopes. Our contributions deal with the cases (II) and (III). For the former we obtain

Varieties of group representations and Casson’s invariant for rational homology 3-spheres

Andrew Cassons Z-valued invariant for Z-homology 3-spheres is shown to extend to a Q-valued invariant for Q-homology 3-spheres which is additive with respect to connected sums. We analyze conditions under which the set of abelian SL2(C) and SU(2) representations of a finitely generated group is isolated. A formula for the dimension of the Zariski tangent space to an abelian SL2(C) or SU(2) representation is obtained. We also derive a sum theorem for Cassons invariant with respect to toroidal splittings of a Z-homology 3-sphere. Andrew Cassons lectures at MSRI in the spring of 1985 introduced an integer valued invariant of oriented integral homology 3-spheres. This invariant, constructed by means of representation spaces, yields interesting new results in low dimensional topology. In this paper we examine the extent to which Cassons procedure for defining his invariant can be used to obtain a rational valued invariant for oriented rational homology 3-spheres. Let 7r be a finitely generated group and G a Lie group. It is well known that the set R(7, G) of all homomorphisms of 7r into G can be given the structure of an analytic set in a natural manner. If G is an algebraic group, R(7r, G) becomes an algebraic set. The closed subspace of R(7r, G) consisting of representations 7r -* G with abelian image will be denoted by A(7r, G) . Let Rn(7,, G) be the union of those components of R(7r, G) which do not meet A(7r, G). When G is understood from the context, it will be dropped from the notation. If R is a commutative ring, an R-homology 3-sphere is a closed, orientable (over Z) 3-manifold with homology isomorphic to H*(S3; R). Let H(R) be the set of oriented homeomorphism types of oriented R-homology 3-spheres. For M E 11(Z) Casson defined an integer valued invariant A(M). We briefly recall his definition (see [AM] for a comprehensive exposition of Cassons MSRI lectures). Let M = WI UF W2 be a Heegard decomposition of M, where F = & W is of genus g and let F* be F punctured once. The diagram of I Received by the editors May 19, 1987 and, in revised form, December 9, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M99, 57M05, 20F99.

Reducing Dehn filling and toroidal Dehn filling

It is shown that if M is a compact, connected, orientable hyperbolic 3-manifold whose boundary is a torus, and 7‘1, FZ are two slopes on i7M whose associated fillings are respectively a reducible manifold and one containing an essential torus, then the distance between these slopes is bounded above by 4. Under additional hypotheses this bound is improved Consequently the cabling conjecture is shown to hold for genus 1 knots in the 3-sphere. ~e~~~~: Dehn filling; Reducible slope; Essential torus slope; Cabling conjecture AMS classijicatian: 57M25; 57R65

On character varieties, sets of discrete characters, and nonzero degree maps

A {\it knot manifold} is a compact, connected, irreducible, orientable

Every nontrivial knot in ³ has nontrivial A-polynomial

3

Graph manifold ℤ-homology 3-spheres and taut foliations

-manifold whose boundary is an incompressible torus. We first investigate virtual epimorphisms between the fundamental groups of small knot manifolds and prove minimality results for small knot manifolds with respect to nonzero degree maps. These results are applied later in the paper where we fix a small knot manifold

Dehn Surgery on Knots

M

Characteristic subsurfaces and Dehn filling

and investigate various sets of characters of representations

On the local structure of SL(2,C)-character varieties at reducible characters

\rho: \pi_1(M) \to {\rm PSL}_2(\Bbb{C})

On proper powers in free products and Dehn surgery

whose images are discrete. We show that the topology of these sets is intimately related to the algebraic structure of the