Related papers: Lifted Heegaard Surfaces and Virtually Haken Manif…

We present a new and shorter proof of Stocking's result that any strongly irreducible Heegaard surface of a closed orientable triangulated 3-manifold is isotopic to an almost normal surface. We also re-prove a result of Jaco and Rubinstein…

We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two…

In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3-manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed,…

This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that \pi_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at…

Haken showed that the Heegaard splittings of reducible 3-manifolds are reducible, that is, a reducing 2-sphere can be found which intersects the Heegaard surface in a single simple closed curve. When the genus of the "interesting" surface…

In this paper, we find infinite hyperbolic 3-manifolds that admit no weakly symplectically fillable contact structures, using tools in Heegaard Floer theory. We also remark that part of these manifolds do admit tight contact structures.

Let M be a 3-manifold admitting a strongly irreducible Heegaard surface S and f:M \to M an involution. We construct an invariant Heegaard surface for M of genus at most 8 g(S) - 7. As a consequence, given a (possibly branched) double cover…

The existence of embedded minimal surfaces in non-compact 3-manifolds remains a largely unresolved and challenging problem in geometry. In this paper, we address several open cases regarding the existence of finite-area, embedded, complete,…

An L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic L-spaces with no symmetries. In particular, unlike all previously known L-spaces, these manifolds are not double…

We examine three key conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b_1 conjecture and the virtually fibred conjecture. We explore the interaction of these conjectures with the following seemingly…

If M is a closed simple 3-manifold whose fundamental group contains a genus-g surface group for some g>1, and if the dimension of H_1(M;Z_2) is at least max(3g-1,6), we show that M contains a closed, incompressible surface of genus at most…

Given a 3-manifold M containing an incompressible surface Q, we obtain an inequality relating the Heegaard genus of M and the Heegaard genera of the components of M - Q. Here the sum of the genera of the components of M - Q is bounded above…

We present a new construction of embedded minimal surfaces in hyperbolic space with $3$ asymptotically totally geodesic ends and arbitrary finite genus.

Understanding non-Haken 3-manifolds is central to many current endeavors in 3-manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere to…

The paper contains a new proof that a complete, non-compact hyperbolic $3$-manifold $M$ with finite volume contains an immersed, closed, quasi-Fuchsian surface.

Let $G$ be a finite group acting on a connected compact surface $\Sigma$, and $M$ be an integer homology 3-sphere. We show that if each element of $G$ is extendable over $M$ with respect to a fixed embedding $\Sigma\rightarrow M$, then $G$…

Non-isotopic Heegaard splittings of non-minimal genus were known previously only for very special 3-manifolds. We show in this paper that they are in fact a wide spread phenomenon in 3-manifold theory: We exhibit a large class of knots and…

We show that if two 3-manifolds with toroidal boundary are glued via a `sufficiently complicated' map then every Heegaard splitting of the resulting 3-manifold is weakly reducible. Additionally, if Z is a manifold obtained by gluing X and…

We show that there are at most finitely many one cusped orientable hyperbolic 3-manifolds which have more than eight non-hyperbolic Dehn fillings. Moreover, we show that determining these finitely many manifolds is decidable.

We show that cusped finite-volume hyperbolic 3-manifolds contain infinitely many simple closed geodesics.