We outline a rigorous algorithm, first suggested by Casson, for determining whether a closed orientable $3$-manifold $M$ is hyperbolic, and to compute the hyperbolic structure, if one exists. The algorithm requires that a procedure has been given to solve the word problem in ${\pi }_1\left(M\right)$.

We prove a conjecture of Hutchings and Lee relating the Seiberg–Witten invariants of a closed 3–manifold $X$ with $b_1\ge 1$ to an invariant that “counts” gradient flow lines—including closed orbits—of a circle-valued Morse function on the manifold. The proof is based on a method described by Donaldson for computing the Seiberg–Witten invariants of 3–manifolds by making use of a “topological quantum field theory,” which makes the calculation completely explicit. We also realize a version of the Seiberg–Witten invariant of $X$ as the intersection number of a pair of totally real submanifolds of a product of vortex moduli spaces on a Riemann surface constructed from geometric data on $X$. The analogy with recent work of Ozsváth and Szabó suggests a generalization of a conjecture of Salamon, who has proposed a model for the Seiberg–Witten–Floer homology of $X$ in the case that $X$ is a mapping torus.

We construct examples of non-isotrivial algebraic families of smooth complex projective curves over a curve of genus 2. This solves a problem from Kirby’s list of problems in low-dimensional topology. Namely, we show that 2 is the smallest possible base genus that can occur in a 4–manifold of non-zero signature which is an oriented fiber bundle over a Riemann surface. A refined version of the problem asks for the minimal base genus for fixed signature and fiber genus. Our constructions also provide new (asymptotic) upper bounds for these numbers.

We show that every subgroup of the mapping class group $MCG\left(S\right)$ of a compact surface $S$ is either virtually abelian or it has infinite dimensional second bounded cohomology. As an application, we give another proof of the Farb–Kaimanovich–Masur rigidity theorem that states that $MCG\left(S\right)$ does not contain a higher rank lattice as a subgroup.

We develop a theory of convex cocompact subgroups of the mapping class group $MCG$ of a closed, oriented surface $S$ of genus at least 2, in terms of the action on Teichmüller space. Given a subgroup $G$ of $MCG$ defining an extension $1\to {\pi }_1\left(S\right)\to {\Gamma }_G\to G\to 1$, we prove that if ${\Gamma }_G$ is a word hyperbolic group then $G$ is a convex cocompact subgroup of $MCG$. When $G$ is free and convex cocompact, it is called a Schottky subgroup.

We define a laminar branched surface to be a branched surface satisfying the following conditions: (1) Its horizontal boundary is incompressible; (2) there is no monogon; (3) there is no Reeb component; (4) there is no sink disk (after eliminating trivial bubbles in the branched surface). The first three conditions are standard in the theory of branched surfaces, and a sink disk is a disk branch of the branched surface with all branch directions of its boundary arcs pointing inwards. We will show in this paper that every laminar branched surface carries an essential lamination, and any essential lamination that is not a lamination by planes is carried by a laminar branched surface. This implies that a 3–manifold contains an essential lamination if and only if it contains a laminar branched surface.

In this paper we discuss the topology of the symplectomorphism group of a product of two 2–dimensional spheres when the ratio of their areas lies in the interval $\left(1,2\right]$. More precisely we compute the homotopy type of this symplectomorphism group and we also show that the group contains two finite dimensional Lie groups generating the homotopy. A key step in this work is to calculate the mod 2 homology of the group of symplectomorphisms. Although this homology has a finite number of generators with respect to the Pontryagin product, it is unexpected large containing in particular a free noncommutative ring with 3 generators.

We study a notion of deformation for simplicial trees with group actions ($G$–trees). Here $G$ is a fixed, arbitrary group. Two $G$–trees are related by a deformation if there is a finite sequence of collapse and expansion moves joining them. We show that this relation on the set of $G$–trees has several characterizations, in terms of dynamics, coarse geometry, and length functions. Next we study the deformation space of a fixed $G$–tree $X$. We show that if $X$ is “strongly slide-free” then it is the unique reduced tree in its deformation space. These methods allow us to extend the rigidity theorem of Bass and Lubotzky to trees that are not locally finite. This yields a unique factorization theorem for certain graphs of groups. We apply the theory to generalized Baumslag–Solitar groups and show that many have canonical decompositions. We also prove a quasi-isometric rigidity theorem for strongly slide-free $G$–trees.

We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg–Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we establish its validity for a large class of singularities: some rational and minimally elliptic (including the cyclic quotient and “polygonal”) singularities, and Brieskorn–Hamm complete intersections. Some of the verifications are based on a result which describes (in terms of the plumbing graph) the Reidemeister–Turaev sign refined torsion (or, equivalently, the Seiberg–Witten invariant) of a rational homology 3–manifold $M$, provided that $M$ is given by a negative definite plumbing. These results extend previous work of Artin, Laufer and S S-T Yau, respectively of Fintushel–Stern and Neumann–Wahl.

We continue the study of the linear programming bounds for sphere packing introduced by Cohn and Elkies. We use theta series to give another proof of the principal theorem, and present some related results and conjectures.

Yasutaka Nakanishi asked in 1981 whether a 3–move is an unknotting operation. In Kirby’s problem list, this question is called the Montesinos–Nakanishi 3–move conjecture. We define the $n$th Burnside group of a link and use the 3rd Burnside group to answer Nakanishi’s question; ie, we show that some links cannot be reduced to trivial links by 3–moves.

Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the “empty disks” circumscribing the regions of the decomposition. The main result of this paper is a characterization of when a given topological decomposition and angle assignment can be realized as the data of an actual Delaunay decomposition of a hyperbolic surface.

We prove the long-standing Montesinos conjecture that any closed oriented PL 4–manifold $M$ is a simple covering of $S^4$ branched over a locally flat surface (cf [Trans. Amer. Math. Soc. 245 (1978) 453–467]). In fact, we show how to eliminate all the node singularities of the branching set of any simple 4–fold branched covering $M\to S^4$ arising from the representation theorem given in [Topology 34 (1995) 497–508]. Namely, we construct a suitable cobordism between the 5–fold stabilization of such a covering (obtained by adding a fifth trivial sheet) and a new 5–fold covering $M\to S^4$ whose branching set is locally flat. It is still an open question whether the fifth sheet is really needed or not.

If a knot $K$ has Seifert matrix $V_K$ and has a prime power cyclic branched cover that is not a homology sphere, then there is an infinite family of non–concordant knots having Seifert matrix $V_K$.

The question was raised as to whether the cut number of a 3–manifold $X$ is bounded from below by $\frac{1}{3}{\beta }_1\left(X\right)$. We show that the answer to this question is “no.” For each $m\ge 1$, we construct explicit examples of closed 3–manifolds $X$ with ${\beta }_1\left(X\right)=m$ and cut number 1. That is, ${\pi }_1\left(X\right)$ cannot map onto any non-abelian free group. Moreover, we show that these examples can be assumed to be hyperbolic.

In 1987 S Cappell and J Shaneson constructed an $s$–cobordism $H$ from the quaternionic 3–manifold $Q$ to itself, and asked whether $H$ or any of its covers are trivial product cobordism? In this paper we study $H$, and in particular show that its 8–fold cover is the product cobordism from $S^3$ to itself. We reduce the triviality of $H$ to a question about the 3–twist spun trefoil knot in $S^4$, and also relate this to a question about a Fintushel–Stern knot surgery.