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David Bachman

Publications and source records attributed to David Bachman.

At least 19 recordsLinked to original sources
arXiv2020

Cohomology fractals, Cannon-Thurston maps, and the geodesic flow

Cohomology fractals are images naturally associated to cohomology classes in hyperbolic three-manifolds. We generate these images for cusped, incomplete, and closed hyperbolic three-manifolds in real-time by ray-tracing to a fixed visual radius. We discovered cohomology fractals while attempting to illustrate Cannon-Thurston maps without using vector graphics; we prove a correspondence between these two, when the cohomology class is dual to a fibration. This allows us to verify our implementations by comparing our images of cohomology fractals to existing pictures of Cannon-Thurston maps. In a sequence of experiments, we explore the limiting behaviour of cohomology fractals as the visual radius increases. Motivated by these experiments, we prove that the values of the cohomology fractals are normally distributed, but with diverging standard deviations. In fact, the cohomology fractals do not converge to a function in the limit. Instead, we show that the limit is a distribution on the sphere at infinity, only depending on the manifold and cohomology class.

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arXiv2020

Cohomology fractals

We introduce cohomology fractals; these are certain images associated to a cohomology class on a hyperbolic three-manifold. They include images made entirely from circles, and also images with no geometrically simple features. They are closely related to limit sets of kleinian groups, but have some key differences. As a consequence, we can zoom in almost any direction to arbitrary depth in real time. We present an implementation in the setting of ideal triangulations using ray-casting.

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arXiv2016

Computing Heegaard genus is NP-hard

We show that {\sc Heegaard Genus $\leq g$}, the problem of deciding whether a triangulated 3-manifold admits a Heegaard splitting of genus less than or equal to $g$, is NP-hard. The result follows from a quadratic time reduction of the NP-complete problem {\sc CNF-SAT} to {\sc Heegaard Genus $\leq g$}.

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arXiv2015

Locally Helical Surfaces have bounded twisting

A topologically minimal surface may be isotoped into a normal form with respect to a fixed triangulation. If the intersection with each tetrahedron is simply connected, then the pieces of this normal form are triangles, quadrilaterals, and helicoids. Helical pieces can have any number of positive or negative twists. We show here that the net twisting of the helical pieces of any such surface in a given triangulated 3-manifold is bounded.

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arXiv2013

Normalizing Topologically Minimal Surfaces III: Bounded Combinatorics

We show that there are a finite number of possible pictures for a surface in a tetrahedron with local index $n$. Combined with previous results, this establishes that any topologically minimal surface can be transformed into one with a particular normal form with respect to any triangulation.

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arXiv2013

Heegaard structure respects complicated JSJ decompositions

Let $M$ be a 3-manifold with torus boundary components $T_1$ and $T_2$. Let $ϕ\colon T_1 \to T_2$ be a homeomorphism, $M_ϕ$ the manifold obtained from $M$ by gluing $T_1$ to $T_2$ via the map $ϕ$, and $T$ the image of $T_1$ in $M_ϕ$. We show that if $ϕ$ is "sufficiently complicated" then any incompressible or strongly irreducible surface in $M_ϕ$ 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.

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arXiv2013

Surfaces that become isotopic after Dehn filling

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.

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arXiv2013

Almost normal surfaces with boundary

We show that a strongly irreducible and boundary-strongly irreducible surface can be isotoped to be almost normal in a triangulated 3-manifold.

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arXiv2012

Normalizing Topologically Minimal Surfaces I: Global to Local Index

We show that in any triangulated 3-manifold, every index n topologically minimal surface can be transformed to a surface which has local indices (as computed in each tetrahedron) that sum to at most n. This generalizes classical theorems of Kneser and Haken, and more recent theorems of Rubinstein and Stocking, and is the first step in a program to show that every topologically minimal surface has a normal form with respect to any triangulation.

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arXiv2012

Normalizing Topologically Minimal Surfaces II: Disks

We show that a topologically minimal disk in a tetrahedron with index $n$ is either a normal triangle, a normal quadrilateral, or a normal helicoid with boundary length 4(n+1). This mirrors geometric results of Colding and Minicozzi.

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arXiv2012

Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds

Let M_1 and M_2 be compact, orientable 3-manifolds with incompressible boundary, and M the manifold obtained by gluing with a homeomorphism $ϕ:\bdy M_1 \to \bdy M_2$. We analyze the relationship between the sets of low genus Heegaard splittings of M_1, M_2, and M, assuming the map ϕis "sufficiently complicated." This analysis yields counter-examples to the Stabilization Conjecture, a resolution of the higher genus analogue of a conjecture of Gordon, and a result about the uniqueness of expressions of Heegaard splittings as amalgamations.

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arXiv2012

A Geometric Approach to Differential Forms

This is a draft of a textbook on differential forms. The primary target audience is sophmore level undergraduates enrolled in what would traditionally be a course in vector calculus. Later chapters will be of interest to advaced undergraduate and beginning graduate students. Applications include brief introductions to Maxwell's equations, foliations and contact structures, and DeRham cohomology.

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arXiv2009

Topological Index Theory for Surfaces in 3-Manifolds

The disk complex of a surface in a 3-manifold is used to define its {\it topological index}. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical surfaces. The main result is that one may always isotope a surface $H$ with topological index $n$ to meet an incompressible surface $F$ so that the sum of the indices of the components of $H \setminus N(F)$ is at most $n$. This theorem and its corollaries generalize many known results about surfaces in 3-manifolds, and often provides more efficient proofs. The paper concludes with a list of questions and conjectures, including a natural generalization of Hempel's {\it distance} to surfaces with topological index $\ge 2$.

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arXiv2009

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 ϕ. We show that when ϕis "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.

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arXiv2009

Degeneration of Heegaard genus, a survey

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 together the boundary components of surface times I, and (3) gluing a handlebody to the boundary of a 3-manifold. We detail those cases in which it is known when the Heegaard genus is less than what is expected after gluing.

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arXiv2009

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.

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arXiv2009

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 ${1/2}n -3$ 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.

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