Preprints

Barriers to Topologically Minimal Surfaces

In earlier work we introduced topologically minimal surfaces as the analogue of geometrically minimal surfaces. Here we strengthen the analogy by showing that complicated amalgamations act as barriers to low genus, topologically minimal surfaces.


Heegaard splittings of sufficiently complicated 3-manifolds I: Stabilization

We construct families of pairs of Heegaard splittings that must be stabilized several times to become equivalent. The first such pair differs only by their orientation. These are genus $n$ splittings of a closed 3-manifold that must be stabilized at least $n-2$ times to become equivalent. The second is a pair of genus $n$ splittings of a manifold with toroidal boundary that must be stabilized at least $n-4$ times to become equivalent. The last example is a pair of genus $n$ splittings of a closed 3-manifold that must be stabilized at least $\frac{1}{2}n -3$ times to become equivalent, regardless of their orientations. All of these examples are splittings of manifolds that are obtained from simpler manifolds by gluing along incompressible surfaces via ``sufficiently complicated" maps. 


Heegaard splittings of sufficiently complicated 3-manifolds II: Amalgamation

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 amalgamation of low genus, unstabilized, boundary-unstabilized Heegaard splittings of $M_i$ is an unstabilized splitting of $M$, (2) every low genus, unstabilized Heegaard splitting of $M$ can be expressed as an amalgamation of unstabilized, boundary-unstabilized splittings of $M_i$, and possibly a Type II splitting of $F \times I$, and (3) if there is no Type II splitting in such an expression then it is unique.

NOTE: The preceeding trio of papers were combined and submitted as one. Please contact me if you would like a copy of this.


Heegaard structure respects complicated JSJ-decompositions (joint with R. Derby-Talbot and E. Sedgwick)

Let $M$ be a 3-manifold with torus boundary components $T_1$ and $T_2$. Let $\phi:T_1 \to T_2$ 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 $M_\phi$. We show that if $\phi$ is “sufficiently complicated” then any incompressible or strongly irreducible surface in $M_\phi$ can be isotoped to be disjoint from $T$. It follows that every Heegaard splitting of a 3-manifold admitting a “sufficiently complicated” JSJ decomposition is an amalgamation of Heegaard splittings of the components of the JSJ decomposition.

Surfaces that become isotopic after Dehn filling (joint with R. Derby-Talbot and E. Sedgwick)

We show that after generic filling along a torus boundary component of a 3-manifold, no two closed, 2-sided, essential surfaces become isotopic, and no closed, 2-sided, essential surface becomes inessential. That is, the set of essential surfaces (considered up to isotopy) survives unchanged in all suitably generic Dehn fillings. Furthermore, for all but finitely many non-generic fillings, we show that two essential surfaces can only become isotopic in a constrained way.