RESEARCH SUMMARY
DAVID BACHMAN
1. Introduction
My research program has revolved around finding and exploiting topological analogues
to geometric structures. More specifically, I have developed a purely topological theory of
surfaces in 3-manifolds that entirely mimics the geometric theory of minimal surfaces. Such
surfaces behave in important ways like a minimal surface, leading to properties that I have
been able to use to resolve several long-standing questions in low-dimensional topology.
This document is organized as follows. In the next section I will introduce the geometric
idea of a minimal surface, and describe how the existence of such objects can inform the study
of the topology of the spaces that contain them. Then I will describe various topological
classes of surfaces in 3-manifolds and describe how they are related to minimal surfaces.
Finally, I will describe how my recent research uses the “minimal surface-like” properties
of these (topological) surfaces to establish theorems. I conclude with the open questions
that are currently driving my research. This work is currently supported by NSF grant
DMS–0906151.
2. Soap Films, Minimal Surfaces and Topology.
An idealization of a soap bubble film is a surface which has the least area among all
surfaces that are subject to a given constraint. Such constraints might include, for example,
having a prescribed boundary.
At the center of differential geometry is the idea of curvature. There are various kinds of
curvatures that geometers typically study, including sectional curvature, scalar curvature,
mean curvature, Gaussian curvature, Ricci curvature, etc. It is a standard result that leastarea surfaces have zero mean curvature. However, this constraint is only necessary, not
sufficient. Surfaces with zero mean curvature are critical points for area. The least area
surfaces represent local minima, and as such represent only a limited collection of critical
points. An arbitrary surface that represents such a critical point is called a minimal surface.
The index of a critical point is a numerical measure of how unlike a local minimum it is.
Suppose, for example, that f : Rn → R1 has a critical point at the origin. Then f can be
approximated near the origin by


x1
f (x1 , ..., xn ) ≈ (x1 ... xn )M  ...  + c
xn
where M is an n × n matrix and c is a constant. We say the index of the critical point at the
origin is the number of (not necessarily distinct) negative eigenvalues of M. In particular, if
1
the origin is a local minimum for f , then its index is zero. We say the index of a minimal
surface is its index, when viewed as a critical point for the area function.
It is easier to “see” these ideas one dimension lower. For example, consider a torus,
represented by the unit square {(x, y) ∈ R2 |0 ≤ x, y ≤ 1} with opposite boundary edges
identified. The loop {(x, 12 ) ∈ R2 |0 ≤ x ≤ 1} is least length in its isotopy class, and as
such represents an index 0 critical point for the length function. In contrast, depending on
the metric that one uses, the sphere {p ∈ R3 ||p| = 1} may have no such least-length loops.
However, it is a deep result of [LŠ47] that given any generic metric on the sphere, there will
always be loops representing critical points of index 1, 2, and 3 for the length function. Since
the existence of such loops is independent of the metric, it is a topological property of the
sphere, rather than a geometric one. By complete analogy, the existence of certain types of
minimal surfaces in a 3-manifold M can give invaluable information about the topology of
M.
3. Incompressible Surfaces
An important topological class of surfaces in a 3-manifold are the so-called incompressible
ones. Such surfaces have the property that any loop on the surface that cannot be shrunk to
a point on the surface also cannot be shrunk to a point if it is allowed to leave the surface.
Incompressible surfaces became the key to answering a host of important questions about
3-manifolds. This body of work, initiated by Haken [Hak68] and Waldhausen [Wal68], culminated with Thurston [Thu82], who showed that any 3-manifold containing an incompressible
surface admits an extremely nice metric (usually hyperbolic). However, not all 3-manifolds
contain such surfaces. The question of whether all 3-manifolds admit nice metrics is precisely Thurston’s famous Geometrization conjecture, which was only recently established by
G. Perelmann.
Another important result about incompressible surfaces was established by Freedman,
Hass, and Scott, who showed that they are always isotopic to embedded least area (i.e.
index 0 minimal) surfaces [FHS83]. However, as stated in the previous paragraph, not all
3-manifolds contain such a surface. This is precisely like the situation one dimension lower,
where we saw in the previous section that not all surfaces contain loops that represent index
0 critical points for the length function. Thus, sometimes to find minimal surfaces we must
expand our search past the incompressible ones.
4. Heegaard surfaces
Heegaard surfaces, introduced by Poul Heegaard in [Hee98], have become a classical way
to study the topology of 3-manifolds. A 3-manifold that is homeomorphic to the regular
neighborhood of a (finite) connected graph in R3 is called a handlebody. Let M be a closed
3-manifold and F ⊂ M an embedded surface. We say F is a Heegaard surface in M if it
separates M into two handlebodies.
The utility of Heegaard surfaces is due to their intimate relationship with other techniques
used to study 3-manifolds:
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(1) Given a Heegaard surface of genus g one can write down a rank g presentation of
π1 (M). In most known examples a minimal genus Heegaard surface gives one a
presentation of the fundamental group of minimal rank.
(2) A Heegaard surface can be viewed as a level surface of a (Morse) height function on
M.
(3) Given a realization of M as a CW-complex, the boundary of a regular neighborhood
of the 1-skeleton is a Heegaard surface.
(4) Any “strongly irreducible” Heegaard surface can be isotoped into a normal form with
respect to a fixed triangulation of M [Rub95], [Sto00].
(5) Given a bumpy Riemannian metric on M, a “strongly irreducible” Heegaard surface
can be realized as an index 1 minimal surface [PR87].
It is the last result mentioned above that we focus on here. What was shown in [PR87]
is that a certain kind of Heegaard surface can always be isotoped to an index one minimal
surface. In the next section we discuss which surfaces are the most likely candidates for
those that can always be isotoped to be minimal, with index two.
5. Stabilization and Critical surfaces
Given a Heegaard surface we can construct a new one by taking the connected sum with
an unknotted torus. We call this operation stabilization. The new Heegaard surface has
genus one higher than that of the original. It is a theorem of Reidemeister [Rei36] and
Singer [Sin33] that any two Heegaard surfaces are stably equivalent. That is, given two
Heegaard surfaces, one may always stabilize the one of higher genus some number of times
to obtain a stabilization of the lower genus one. Up until recently, very little was known
about stabilization. In fact, in the 75 years following the work of Reidemeister and Singer
we did not know of a single example of a pair of Heegaard surfaces of the same genus that
require more than one stabilization to be equivalent. This was one of the many questions
that my research led to a resolution of.
Since one may always stabilize to obtain Heegaard surfaces of arbitrarily high genus,
given a pair of Heegaard surfaces, what is of real interest is their minimal genus common
stabilization. What I showed in [Bac02] is if the manifold did not contain an incompressible
surface, then any minimal genus common stabilization possesses a list of useful topological
properties which are conjecturally enough to guarantee that it is isotopic to a minimal surface
of index 2. Any surface satisfying these properties was dubbed critical.
Exploiting the topology of critical surfaces was the key to my resolution of Gordon’s
conjecture [Bac08]. This conjecture, made (completely coincidentally!) by my advisor about
10 years before I solved it, asserted that the connected sum of unstabilized Heegaaard surfaces
must also be an unstabilized Heegaard surface. The conjecture was listed as Problem 3.91
in Kirby’s problem list [Kir97], and several attempts had been made at a solution (see, for
example, [Edw01]). The conjecture was also solved by very different techniques by R. Qiu,
and published jointly with M. Scharlemann [SQ09].
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6. Topologically Minimal surfaces of arbitrary index
To proceed further, I developed a definition of topologically minimal surfaces in 3-manifolds,
and their associated topological index [Bacc]. This is a non-negative, integer-valued, topological invariant that mimics in every way the index of a (geometrically) minimal surface.
For example, suppose the boundaries of 3-manifolds M1 and M2 are homeomorphic to a
surface F . By gluing ∂M1 to ∂M2 by a “complicated” gluing map, the usual geometric model
shows the creation of a long product region in the resulting manifold which is homeomorphic
to F × I. Any surface S which crosses from M1 to M2 passes through this region, and thus
must have large area. But if S is a minimal surface then it will have negative Gaussian
curvature, and thus the fact that its area is large will imply (by the Gauss-Bonnet theorem)
that it has large genus. The conclusion is that any minimal surface that crosses from M1 to
M2 must have large genus.
In [Baca] I showed that topologically minimal surfaces behave in exactly the same way.
That is, if manifolds M1 and M2 are glued along their boundary by a “complicated” map,
then any topologically minimal surface in the resulting 3-manifold that crosses between them
must have large genus. This fact was the key technical tool used in several sequels.
In the first such sequel [Bacb] I presented the first counter-examples to the Stabilization
Conjecture, mentioned above. In this paper I was able to construct the first examples of
pairs of Heegaard surfaces that required many stabilizations to become equivalent. (Other
examples were concurrently announced by Hass, Thompson, and Thurston [HTT09], and
Johnson [Johb], [Joha].)
7. Current and Future work
My current research projects span several areas of low-dimensional topology, but they are
all related to the study of topologically minimal surfaces. Presently I will describe each area
where I am active, and discuss its relationship to my general research program.
7.1. Relationships between minimal surfaces in the smooth, topological, and PL
category. The pioneering work of Rubinstein has shown that in the Piecewise-Linear (PL)
category there are surfaces that behave in many ways like minimal surfaces of various indices.
Let M now denote a triangulated 3-manifold, i.e. a collection of tetrahedra which are
glued along their triangular faces to form a manifold. A surface in M that intersects each
tetrahedron is said to be normal if its intersection with each tetrahedron is a collection of
triangles and quadrilaterals.
A crude measure of the area of a surface in M is the number of times it meets the 1skeleton of the triangulation. It is a classical result of Haken that any incompressible surface
can be isotoped to meet the 1-skeleton a minimal number of times, and such a representative
will be normal [Hak61]. In this way, the normal surfaces act precisely like index 0 minimal
surfaces.
Recall that Pitts and Rubinstein showed that the so-called “strongly irreducible” surfaces
were isotopic to index 1 minimal surfaces [PR87]. Rubinstein also showed that any such
surface can be isotoped to meet a triangulation in a collection of triangles and quadrilaterals,
together with one exceptional piece in a single tetrahedron [Rub95] (see also [Sto00]). This
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exception looks like either a saddle or a small unknotted tube. Any surface which meets the
triangulation in this way was dubbed “almost normal.” Thus, there is a complete dichotomy
between index 1 minimal surfaces and almost normal surfaces. Hence, it is fruitful to think
of the almost normal surfaces as “PL minimal surfaces of index 1.” (Note that saddles and
unknotted tubes are typical pictures of a smooth index 1 minimal surface, strengthening the
analogy.)
In some of my early work I was able to extend Rubinstein’s results to strongly irreducible
surfaces with non-empty boundary [Bac01], [Bac04]. This was instrumental in several later
results, e.g. [BSS06]. Now the challenge is to find PL analogues of topologically minimal
surfaces of arbitrary index, not just index 0 and 1. Preliminary results have been extremely
fruitful. I now have a proof that there is a normal-like form for topologically minimal surfaces
that have index 2 (i.e. critical surfaces), and am close to a general classification for arbitrary
index.
7.2. Dehn Filling. When the relationships between topologically and PL minimal surfaces
are complete, it will shed light on many other areas of 3-manifold topology. But even before
then, just understanding this relationship in the topological index 2 case is proving to be
quite fruitful. As just one example of this, I will describe how it relates to another area of
low-dimensional topology.
A common way to construct a 3-manifold is to begin with a second manifold with torus
boundary, and glue on a solid torus. This process is known as Dehn filling, and has been
studied extensively. In fact, it has been shown that every 3-manifold can be obtained from
S 3 by removing a collection of solid tori, and Dehn filling the resulting boundary components
in some way [Wal60], [Lic62]. See [Gor03] for a survey of recent results about Dehn filling.
From the perspective of Dehn filling, a natural question is how the set of Heegaard surfaces
can change when you glue on a solid torus. The above described work will yield a complete
answer to this question: For suitably generic filling, the set of Heegaard surfaces does not
change at all. That is, there is a one-to-one correspondence between the Heegaard surfaces
before and after filling; no two have become isotopic after filling, and no new Heegaard
surfaces were created when we filled.
7.3. The Virtually Haken Conjecture. One of the biggest open questions in the theory
of 3-manifolds is Thurston’s Virtually Haken Conjecture. This conjecture asserts that every
(closed, aspherical) 3-manifold has a finite cover that contains an (embedded) incompressible
surface. It has been shown that the Hyperbolization Conjecture (an important part of the
Geometrization conjecture) is implied by the Virtually Haken Conjecture, independent of
Perelmann’s proof of the Geometrization Conjecture. The theory of topologically minimal
surfaces provides a unique approach to the Virtually Haken Conjecture.
Suppose a manifold M contains a (geometrically) minimal surface H of index n. Then it
is immediate that H lifts to a minimal surface in any d-fold cover of M whose index is dn.
It is an open question as to whether topologically minimal surfaces behave at all the same
way. On the one hand, the only construction of high index topologically minimal surfaces was
given by myself and Jesse Johnson in [BJ], where such surfaces arise as covers of a topological
index 1 surface. On the other hand, there are example known of covers of topological index 1
surfaces that are also index 1. So the best we can hope for is that perhaps every topological
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index 1 surface lifts in some finite cover to a high index surface. Given the currently known
examples, this seems likely.
Now, recall that the goal is to produce a cover with an incompressible surface. But manifolds with no incompressible surfaces have extremely simple Heegaard structures. In such
manifolds the set of Heegaard surfaces forms a tree, whose leaves correspond to topological index 1 surfaces, and whose branch points correspond to topological index 2 surfaces.
From this viewpoint, it seems highly unlikely that manifolds with no incompressible surfaces
contain topologically minimal surfaces with index ≥ 3. If one can prove this, and prove
that every topological index 1 surface lifts in some cover to a high index surface, then the
virtually Haken conjecture would follow.
Preliminary explorations of these issues have already yielded significant results. As mentioned above, I now have a jointly-authored paper with a construction of high index topologically minimal surfaces. I am currently working on a result that asserts that S 3 contains
no topologically minimal surfaces, which would be an important step in this program. I also
have many partial results dealing with covers of topologically minimal surfaces, and minimal
surfaces in manifolds that do not contain incompressible surfaces.
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