Related papers: Genus two 3-manifolds are built from handle number…

We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed.

We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into (torus with one boundary component times [0,1]. We use this condition to decide whether a…

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…

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…

Let $M_1$ and $M_2$ be orientable irreducible 3--manifolds with connected boundary and suppose $\partial M_1\cong\partial M_2$. Let $M$ be a closed 3--manifold obtained by gluing $M_1$ to $M_2$ along the boundary. We show that if the gluing…

We consider a Heegaard splitting M=H_1 \cup_S H_2 of a 3-manifold M having an essential disk D in H_1 and an essential surface F in H_2 with |D \cap F|=1. (We require that boundary of F is in S when H_2 is a compressionbody with non-empty…

We survey known (and unknown) results about the behavior of Heegaard genus of 3-manifolds constructed via various gluings. The constructions we consider are (1) gluing together two 3-manifolds with incompressible boundary, (2) gluing…

Let M_1 and M_2 be compact, orientable 3-manifolds, and M the manifold obtained by gluing some component F of \bdy M_1 to some component of \bdy M_2 by a homeomorphism \phi. We show that when \phi is "sufficiently complicated" then (1) the…

Let $M$ be a closed orientable 3-manifold with a genus two Heegaard splitting $(V_1, V_2; F)$ and a non-trivial JSJ-decomposition, where all components of the intersection of the JSJ-tori and $V_i$ are not $\partial$-parallel in $V_i$ for…

We construct a small, hyperbolic 3-manifold $M$ such that, for any integer $g\geq 2$, there are infinitely many separating slopes $r$ in $\partial M$ so that $M(r)$, the 3-manifold obtained by attaching a 2-handle to $M$ along $r$, is…

Extending Haken's Theorem to product annuli and disks for Heegaard splittings of sutured manifolds, we show that the handle number of an irreducible sutured manifold equals the handle number of its guts. We further show that reduced sutured…

Let $M$ be a 3-manifold with torus boundary components $T_1$ and $T_2$. Let $\phi \colon T_1 \to T_2$ be a homeomorphism, $M_\phi$ the manifold obtained from $M$ by gluing $T_1$ to $T_2$ via the map $\phi$, and $T$ the image of $T_1$ in…

We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite…

Let $M$ be a simple manifold, and $F$ be a component of $\partial M$ of genus two. For a slope $\gamma$ on $F$, we denote by $M(\gamma)$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood of $\gamma$ on $F$.…

The Heegaard genus g of an irreducible closed orientable 3-manifold puts a limit on the number and complexity of the pieces that arise in the Jaco-Shalen-Johannson decomposition of the manifold by its canonical tori. For example, if p of…

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…

Let $M$ be a closed orientable 3-manifold with a genus two Heegaard splitting $(V_1, V_2; F)$ and a non-trivial JSJ-decomposition, where all components of the intersection of the JSJ-tori and $V_i$ are not $\partial$-parallel in $V_i$ for…

We construct a sequence of pairs of 3-manifolds each with torus boundary and with the following two properties: 1) For the result of a carefully chosen glueing of the nth pair of 3-manifolds along their boundary tori, the ratio of the genus…

Let M and M' be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that M and M' are glued via a homeomorphism between their boundaries. Then we show that, provided the gluing homeomorphism is `sufficiently…

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…