Darryl McCullough

The topology of deformation spaces of Kleinian groups

Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and let AH(…1(M)) denote the space of (conjugacy classes of) discrete faithful representations of …1(M )i nto PSL 2(C). The components of the interior MP(…1(M)) of AH(…1(M)) (as a subset of the appropriate representation variety) are enumerated by the spaceA(M) of marked homeomorphism types of oriented, compact, irreducible 3-manifolds homotopy equivalent to M. In this paper, we give a topological enumeration of the components of the closure of MP(…1(M)) and hence a conjectural topological enumeration of the components of AH(…1(M)). We do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a sequence of isomorphic freely indecomposable Kleinian groups. We use this enumeration to exhibit manifolds M for which AH(…1(M)) has inflnitely many components. In this paper, we begin a study of the global topology of deformation spaces of Kleinian groups. The basic object of study is the space AH(…1(M)) of marked hyperbolic 3-manifolds homotopy equivalent to a flxed compact 3manifold M. The interior MP(…1(M)) of AH(…1(M)) is very well understood due to work of Ahlfors, Bers, Kra, Marden, Maskit, Sullivan and Thurston. In particular, the components ofMP(…1(M)) are enumerated by topological data, namely the set A(M) of marked, compact, oriented, irreducible 3-manifolds homotopy equivalent to M, while each component is parametrized by analytic data coming from the conformal boundaries of the hyperbolic 3-manifolds. Thurston’s Ending Lamination Conjecture provides a conjectural classiflcation for elements of AH(…1(M)) by data which are partially topologi

The genus 2 Torelli group is not finitely generated

Abstract Let F g be a closed orientable 2-manifold of genus g . The Torelli group is the kernel of the natural homomorphism from the mapping class group of F 1 to Aut( H 1 ( F g )). For g ⩾3 the Torelli group has been shown to be finitely generated by Dennis Johnson. We show that it is not finitely generated when g =2.

ISOMETRIES OF ELLIPTIC 3-MANIFOLDS

The closed 3-manifolds of constant positive curvature were classified long ago by Seifert and Threlfall. Using well-known information about the orthogonal group O(4), we calculate their full isometry groups Isom(M), determine which elliptic 3-manifolds admit Seifert fiberings that are invariant under all isometries, and verify that the inclusion of Isom(M) to Diff(M) is a bijection on path components.

The tree of knot tunnels

We present a new theory which describes the collection of all tunnels of tunnel number 1 knots in the 3-sphere (up to orientation-preserving equivalence in the sense of Heegaard splittings) using the disk complex of the genus-2 handlebody and associated structures. It shows that each knot tunnel is obtained from the tunnel of the trivial knot by a uniquely determined sequence of simple cabling constructions. A cabling construction is determined by a single rational parameter, so there is a corresponding numerical parameterization of all tunnels by sequences of such parameters and some additional data. Up to superficial differences in definition, the final parameter of this sequence is the Scharlemann-Thompson invariant of the tunnel, and the other parameters are the Scharlemann-Thompson invariants of the intermediate tunnels produced by the constructions. We calculate the parameter sequences for tunnels of 2-bridge knots. The theory extends easily to links, and to allow equivalence of tunnels by homeomorphisms that may be orientation-reversing.

Manifold covers of 3-orbifolds with geometric pieces

Abstract It is conjectured that any compact 3-orbifold containing no bad 2-suborbifolds is built from pieces whose interiors admit geometric structures. We prove that any 3-orbifold having such geometric pieces has a finite orbifold covering space which is a manifold. As applications, we extend some results of Hempel and Waldhausen for 3-manifolds to the orbifold setting.

Finiteness of Classifying Spaces of Relative Dieomorphism Groups of 3{Manifolds

The main theorem shows that if M is an irreducible compact connected ori- entable 3{manifold with non-empty boundary, then the classifying space BDi (M rel @M) of the space of dieomorphisms of M which restrict to the identity map on @M has the homotopy type of a nite aspherical CW{complex. This answers, for this class of manifolds, a question posed by M Kontsevich. The main theorem follows from a more precise result, which asserts that for these manifolds the mapping class group H(M rel @M) is built up as a se- quence of extensions of free abelian groups and subgroups of nite index in relative mapping class groups of compact connected surfaces.

Homeomorphisms which are Dehn twists on the boundary

A homeomorphism of a 3-manifold M is said to be Dehn twists on the boundary when its restriction to the boundary of M is isotopic to the identity on the complement of a collection of disjoint simple closed curves in the boundary of M. In this paper, we give various results about such collections of curves and the associated homeomorphisms. In particular, if M is compact, orientable, irreducible and the boundary of M is a single torus, and M admits a homeomorphism which is a nontrivial Dehn twist on the boundary of M, then M must be a solid torus.

Isotopies of 3-manifolds

Abstract An isotopy of a manifold M that starts and ends at the identity diffeomorphism determines an element of π 1 (Diff( M )). For compact orientable 3-manifolds with at least three nonsimply connected prime summands, or with one S 2 × S 1 summand and one other prime summand with infinite fundamental group, infinitely many integrally linearly independent isotopies are constructed, showing that π 1 (Diff( M )) is not finitely generated. The proof requires the assumption that the fundamental group of each prime summand with finite fundamental group imbeds as a subgroup of SO(4) that acts freely on S 3 (conjecturally, all 3-manifolds with finite fundamental group satisfy this assumption). On the other hand, if M is the connected sum of two irreducible summands, and for each irreducible summand P of M , π 1 (Diff( P )) is finitely generated, then results of Jahren and Hatcher imply that π 1 (Diff( M )) is finitely generated. The isotopies are constructed on submanifolds of M which are homotopy equivalent to a 1-point union of two 2-spheres and some finite number of circles. The integral linear independence is proven by obstruction-theoretic methods.

Group actions on nonclosed 2-manifolds

When a group G acts properly discontinuously on a surface S, there is an extension E of the fundamental group of S by G. When S is not closed, E is the fundamental group of a graph of groups with finite vertex groups. The groups E and the graphs of groups arising from actions in this way are characterized, providing a unified theory of such actions. Several applications are given.

Free actions on handlebodies

Abstract The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable three-dimensional handlebody of genus g ⩾1 can be enumerated in terms of sets of generators of G . They correspond to the equivalence classes of generating n -vectors of elements of G , where n =1+( g −1)/| G |, under Nielsen equivalence (or weak Nielsen equivalence). For Abelian and dihedral G , this allows a complete determination of the equivalence and weak equivalence classes of actions for all genera. Additional information is obtained for other classes of groups. For all G , there is only one equivalence class of actions on the genus g handlebody if g is at least 1+l(G) |G| , where l( G ) is the maximal length of a chain of subgroups of G . There is a stabilization process that sends an equivalence class of actions to an equivalence class of actions on a higher genus, and some results about its effects are obtained.