Hyperbolic Manifolds Containing High Topological Index Surfaces
If a graph is in bridge position in a 3-manifold so that the graph complement is irreducible and boundary irreducible, we generalize a result of Bachman and Schleimer to prove that the complexity of a surface properly embedded in the complement of the graph bounds the graph distance of the bridge surface. We use this result to construct, for any natural number , a hyperbolic manifold containing a surface of topological index .
It has become increasingly common and useful to measure distances in complexes associated to surfaces between certain important sub-complexes associated with the surface embedded in a 3-manifold. These techniques provide a means to indicate the inherent complexity of links in a manifold, decomposing surfaces, or the manifold itself. In  Bachman defined the topological index of a surface as a topological analogue of the index of an unstable minimal surface. When the distance is small, the notion of topological index refines this distance, by looking at the homotopy type of a certain sub-complex.
In the same way that incompressible surfaces share important properties with strongly irreducible surfaces (distance ) despite being compressible, the topological index provides a degree of measurement of how similar irreducible, but weakly reducible (distance ) surfaces are to incompressible surfaces. In a series of papers [1, 2, 3], Bachman has shown that surfaces with a well-defined topological index in a 3-manifold can be put into a sort of normal form with respect to a trianglulation of the manifold, generalizing the ideas of normal form introduced by Kneser  and almost normal form introduced by Rubinstein , and mirroring results about geometrically minimal surfaces due to Colding and Minicozzi [10, 11, 12, 13, 14].
Lee  has shown that an irredubible manifold containing an incompressible surface contains topologically minimal surfaces of arbitrarily high genus, but has only shown that the topological index of such surfaces is at least two. In  Johnson and Bachman showed that surfaces of arbitrarily high index exist. These surfaces are the lifts of Heegaard surfaces in an n-fold cover of a manifold obtained by gluing together boundary components of the complement of a link in . A by-product of their construction is that the resulting manifolds are toroidal.
This leaves open the question of whether the much more ubiquitous class of hyperbolic manifolds can also contain high topological index surfaces. Here we construct certain hyperbolic manifolds containing such surfaces. We generalize the construction in  by gluing along the boundary components of the complement of a graph in to show:
There is a closed 3-manifold , with an index 1 Heegaard surface , such that for each , the lift of to some -fold cover of has topological index . Moreover, is hyperbolic for all .
In order to guarantee the hyperbolicity of we must rule out the existence of high Euler characteristic surfaces in the graph complement. To that end, we define the graph distance, , of graphs in , an analogue of bridge distance of links. In the spirit of Hartshorn  and Bachman-Schleimer  we show that the complexity of an essential surface is bounded below by the graph bridge distance:
Let be a graph in a closed, orientable 3-manifold which is in bridge position with respect to a Heegaard surface , so that is irreducible and boundary irreducible. Let be a properly embedded, orientable, incompressible, boundary-incompressible, non-boundary parallel surface in . Then is bounded above by .
Given a link , a bridge sphere for is a sphere, , embedded in , intersecting the link transversely, and dividing into two -balls, and , so that there exist disks and properly embedded in and , respectively, so that and are each a collection of arcs.
In , Goda introduced the notion of a bridge sphere for a spatial -graph, and this was extended by Ozawa in . A bridge sphere for a (spatial) graph is a sphere, , embedded in , instersecting transversely in the interior of edges, and dividing into two -balls, and , so that there exist disks and properly embedded in and , respectively, so that and are each a collection of trees and/or arcs.
If is a bridge sphere for a link , then a bridge disk is a disk properly embedded in one of the components of , whose boundary consists of exactly two arcs, meeting at their endpoints, with one arc essential in , and the other essential in . We refer to the arc in the boundary of the disk that is contained in as a bridge arc. Similarly, if is a bridge sphere for a graph , then a graph-bridge disk is a disk properly embedded in one of the components of , whose boundary consists of exactly two arcs, meeting at their endpoints, with one arc essential in , and the other essential in . We refer to the arc in the boundary of the disk that is contained in as a graph-bridge arc.
The curve complex for a surface with (possibly empty) boundary is the complex with vertices corresponding to the isotopy classes of essential simple closed curves in , so that a collection of vertices defines a simplex if representatives of the corresponding isotopy classes can be chosen to be pairwise disjoint. We will denote the curve complex for a surface by .
The arc and curve complex for a surface with boundary is the complex with vertices corresponding to the (free) isotopy classes of essential simple closed curves and properly embedded arcs in . A collection of vertices defines a simplex if representatives of the corresponding isotopy classes can be chosen to be pairwise disjoint. We will denote the arc and curve complex for a surface by .
If is a surface embedded in a manifold, and a 1-dimensional complex intersects transversely, we will refer to the surface obtained by removing a neighborhood of the 1-complex by . We will often refer to simply by , and simply by .
Let be a surface with at least two distinct, essential curves. Given two collections and of vertices in the complex (resp., ), the distance between and , denoted (resp., ), is the minimal number of edges in any path in (resp., ) from a vertex in to a vertex in . When the surface is understood, we often just write (resp., ).
We will be working with four subtly different but closely related sub-complexes, and some associated notions of distance.
Let be a properly embedded surface separating a manifold into two components, and . Define the disk set of (resp., ), denoted , (resp. ), as the set of all vertices corresponding to essential simple closed curves in that bound embedded disks in (resp., ). Define the disk set of , denoted , as the set of all vertices corresponding to essential simple closed curves in that bound embedded disks in .
Let be a bridge sphere for a link , bounding -balls and , with at least marked points corresponding to the transverse intersections of with . The distance of the bridge surface, denoted , is , the distance in the curve complex of between and .
The fundamental building block in our construction will be the exterior of a graph that is highly complex as viewed from the arc and curve complex. The existence of such a block will follow from a result of Blair, Tomova, and Yoshizawa. It is a special case of Corollary 5.3 from .
Theorem 2.6 ().
Given non-negative integers and , with , there exists a -component link in , and a bridge sphere for so that is -bridge with respect to and .
Let be a bridge sphere for a link , bounding -balls and . Define the bridge disk set of (resp., ), denoted (resp., ), as the set of all vertices either corresponding to essential simple closed curves in that bound embedded disks in (resp., ), or corresponding to bridge arcs in .
Let be a bridge sphere for a link , bounding -balls and . The bridge distance of the bridge surface , denoted is , the distance in the arc and curve complex of between and .
Lemma 2.9 (, Lemma 2).
If is a bridge surface which is not a sphere with four or fewer punctures, then .
Let be a bridge sphere for graph , bounding 3-balls and . The graph disk set of (resp., ) denoted (resp., ), is the set of all vertices either corresponding to essential simple closed curves in that bound embedded disks in (resp., ), or corresponding to graph-bridge arcs in .
Let be a bridge sphere for graph . The graph distance of the bridge surface, denoted is , the distance in the arc and curve complex of between and .
Let be a link in bridge position with respect to bridge sphere , bounding 3-balls and , and let be a graph in bridge position with respect to formed by adding edges to in that are simultaneously parallel into in the complement of , and so that has at least two components.
If is a graph-bridge disk for , then there is a bridge disk for in which is disjoint from .
Let , …, be the connected components , and let be the component of to which is incident.
Over all bridge disks for disjoint from , choose one which minimizes . Suppose the intersection is non-empty. Any loops of intersection can be removed because is a handlebody and therefore irreducible. Any points of intersection between and are contained in and . Choose an arc of . The arc cuts into two disks and . For one of or , is contained in . Call that disk . Consider an arc of outermost in . If the interior of is disjoint from then take to be . The arc cuts off a disk from and cuts into two disks and only one of whose (say ) boundary is incident to . The disk is a bridge disk for and intersects fewer times than , contradicting the minimality of . ∎
The above implies that the distance in the arc and curve complex of between and is less than or equal to one.
Let and be as above. Then .
Since contains no graph-bridge disks, . Thus . Lemma 2.12 shows that , and so by the triangle inequality we have that . ∎
In , Hartshorn proved that an essential closed surface in a 3-manifold creates an upper bound on the possible distances of Heegaard splittings of that manifold in terms of the genus of the essential surface.
Theorem 2.14 (Hartshorn, Theorem 1.2 of ).
Let be a Haken 3-manifold containing an incompressible surface of genus . Then any Heegaard splitting of has distance at most .
This idea has been generalized in numerous ways, including by Bachman and Schleimer, who show in  that the distance of a bridge Heegaard surface in a knot complement is bounded by twice the genus plus the number of boundary components of an essential properly embedded surface.
Theorem 2.15 (Bachman-Schleimer, Theorem 5.1 of ).
Let be a knot in a closed, orientable 3-manifold which is in bridge position with respect to a Heegaard surface . Let be a properly embedded, orientable, essential surface in . Then the distance of with respect to is bounded above by twice the genus of plus .
We will need a yet more general version, since we will be concerned with surfaces properly embedded in graph complements.
The essence of both results is that the distance of a bridge or Heegaard surface is bounded above in terms of the complexity of an essential properly embedded surface. We will generalize this result to link and graph complements, with the additional benefit of avoiding many of the technical details of  necessary to treat the boundary components. Unfortunately, our bound will be worse than that obtained by Bachman and Schleimer, though it will be sufficient for many applications of this type of bound (e.g., , , , , and ). We note also that our proof requires a minimal starting position similar to that used by Hartshorn, an assumption the Bachman-Schleimer method was able to avoid.
We now prove the following.
Proof of Theorem 1.
In the case that is closed, we note that the proofs of both Theorem 2.15 and Theorem 2.14 apply to closed surfaces in manifolds with boundary as long as the manifold is irreducible. In the case that we will double along to obtain a closed surface and show that the surface can be made to fulfill all the hypotheses necessary to use the machinery in the proof of Theorem 2.14 to obtain the bound on distance.
First, isotope to intersect minimally, among all isotopy representatives of . Let and be the handlebodies on either side of . Double along , and call the resulting manifold . Let the doubles of , , and be , , and , respectively, and let be in , with respective copies , , , and for .
Note that is a Heegaard surface for . (The proof of this is very similar to the proof of Proposition 3.3 below.) Also, note that since is incompressible and -incompressible in , is an incompressible closed surface in and since was incompressible in , is incompressible in .
Each of and are incompressible.
If, say, had a compressing disk , then since is incompressible in , there would have to be a disk in with , and . We may choose to be a compressing disk which intersects minimally. Further, since is incompressible, we may choose to intersect only in arcs, if at all. But is irreducible, so bounds a ball and we may isotope across this ball from to , lowering the number of intersections between and .
If , then this can be viewed as an isotopy of in which reduces the number of intersections between and , a contradiction.
If we still arrive at a contradiction. Consider a loop, , of intersection in , innermost in . Since only contains arcs, consists of two arcs, and in and respectively. Thus bounds a disk in , cuts off a subdisk of and cuts off a subdisk of , both of which are in either or , say . Now we have an isotopy of from to
Independent of whether intersected , we could have chosen to have fewer intersections with , contradicting our choice of to minimize intersections. ∎
Every intersection of with is essential in .
Curves of intersection in which are inessential in both surfaces would either give rise to a reduction in or could have come from the doubling of arcs in which would give rise to a reduction in in a fashion similar to the previous claim. ∎
There are no -parallel annular components of or .
Any such component disjoint from would have been eliminated when was minimized. The intersection of any such component intersecting with would be a -parallel disk which also would have been eliminated when was minimized. ∎
Now we have satisfied all the hypotheses to obtain the sequence of isotopic copies of described in Lemmas 4.4 and 4.5 of . Depending on whether either of or contain disk components or not, we apply either Lemma 4.4 or 4.5, respectively, of  to obtain a sequence of compressions of which give rise to a path in . A priori, this path would not restrict to a path in , but the following Claim shows that we can choose the compressions to be symmetric across , and so each compression will correspond to an edge in .
If there exists an elementary -compression of in (resp. ), then there exists an elementary compression of in (resp. ) which is symmetric across in the sense that either
the -compressing disk is disjoint from in , and there is a corresponding -compressing disk in , or
the -compression is along a disk that is symmetric across .
Let be an elementary -compression disk for, say, chosen to minimize . We may restrict attention to such disks with .
First, we observe that cannot contain any loops of intersection, for a loop of innermost in bounds a sub-disk of which would either give rise to a compression for or would provide a means of isotoping so as to lower . Thus, consists only of arcs. These arcs are either
vertical arcs: with one endpoint on each of and ,
-arcs: with both endpoints on , or
-arcs: with both endpoints on .
Consider an -arc of , outermost in , cutting off sub-disk from , with boundary consisting of in and in . Without loss of generality, assume . If is essential in , then is a boundary compression disk for in , which is impossible. If is inessential in , then it must co-bound a disk in together with an arc . The curve cannot be essential in , else would be a compressing disk for . Thus, bounds a disk, . Now is a sphere bounding a ball in , so is isotopic to , and replacing with results in an elementary boundary compressing disk for with fewer intersections with than . Thus we may assume that contains no -arcs.
Now consider a sub-disk of which is cut off by all the arcs of and whose boundary consists of no more than one vertical arc. With out loss of generality, assume . Suppose has -arcs, . Then all the are disjoint arcs on . If any of them are inessential in then they bound disks . If any of the are essential in , then they bound disks that are bridges disks for in . In either case, results in a boundary compressing disk for with fewer intersections with than . This boundary compressing disk is still elementary as the arc in remains unchanged. Thus, we may assume that consists solely of vertical arcs.
Let be an arc of outermost in , cutting off a sub-disk from . Without loss of generality, . The boundary of consists of three arcs; , and . By symmetry, there exists disk in , so that is a disk in with boundary consisting of arcs and , intersecting in exactly one arc, . Finally, we must show that is a “strongly essential” arc in .
If is not strongly essential then it is either the meridian of a boundary parallel annulus of which is
not possible since was a sub-arc of the original elementary compression disk , or is inessential in . If is inessential then it would co-bound a disk in together with an arc . This disk provides an isotopy in of to .
If the disk only intersects in then is a compressing disk for with fewer arcs of intersection with , as the disk can be isotoped away from . This disk is still an elementary compressing disk because is isotopic to , and so contradicts our original choice of .
Thus, is strongly essential in , and is a new compressing disk for that is symmetric across . ∎
We may, thus, proceed exactly as in Theorem 2.14. Each elementary boundary compression of towards either of or can be performed in a symmetric way, demonstrating a path from to in of length no greater than twice the genus of , which is .
Each time a boundary compression for corresponds to a pair of curves and in that contribute an edge in a path in from to , there is immediately a pair of curves and in also contributing an edge in a path from to , and this pair of paths corresponds to a single pair of curves and in contributing a single edge in . Each time a boundary compression for corresponds to a pair of curves intersecting that contributes an edge in a path in from to , the restriction of these curves to is a pair of arcs contributing an edge in .
Further, since the boundary compressions (and elimination of boundary parallel annuli) are all being performed symmetrically, the resulting disks from and from are symmetric. That is, either (resp., ) is disjoint from , so that we may assume that it sits in (resp., ), or it is symmetric across so that (resp., is a graph bridge disk for in . In either case, this demonstrates a path in from to of length no greater than . ∎
3. Theorem 1
In  Bachman defined the topological index of a surface. In contrast to the distances between sub-complexes each corresponding to some disks discussed in Section 2, he exploits the homotopy type of the complex of all disks.
The surface is said to be topologically minimal if either is empty, or if there exists an so that . If a surface is topologically minimal, then the topological index is defined to be the smallest so that , or if is empty.
In  Johnson and Bachman showed that surfaces of arbitrarily high index exist, but the manifolds they construct all contain essential tori. We prove an analogue of this.
3.1. The construction
Let be a positive integer. We will construct a hyperbolic manifold containing a Heegaard surface of topological index .
Using the machinery in Theorem 2.6, let be a -link in with two components, and , with bridge sphere of distance at least . Let and be the two 3-balls bounded by . Since is in bridge position, there exist disks and properly embedded in and , respectively, with , and . By modifying if necessary, we can find two arcs and in such that
each of and have endpoints on different components of .
Let , let , let , let , and let . Observe that is a graph in bridge position with respect to . Let , let , and let , and .
For each , let be homeomorphic to , along with homeomorphic copies of , of , of , and of .
Then, for each , identify with and identify with , all via the same homeomorphism. Call the resulting closed -manifold . Observe that the union of the is a closed surface that we will call . We will show that is a Heegaard surface for , that has high topological index, and that is hyperbolic.
For each , the surface is a genus Heegaard surface.
That the genus of is can be verified by an Euler characteristic count. It suffices, then, to verify that the complement of is two handlebodies, and .
Since was in bridge position with respect to , there are disks and properly embedded in and , respectively, so that and . Then and cut along is a collection of sub-disks.
The result of cutting along all these sub-disks of is a pair of 3-balls, each with two sub-disks, and , of contained in the boundary. Each identification of with (indices mod ) glues pairs of these sub-disks along arcs, resulting in disks in , and further cutting along copies of each of and results in a collection of -balls, showing that is a handlebody.
Similarly, the result of cutting along all of the sub-disks of is a pair of 3-balls, each with four sub-disks of contained in the boundary, , and . Each identification of with (indices mod ) glues pairs of these sub-disks along arcs, resulting in disks in , and further cutting along copies of each of , and results in a collection of -balls, showing that is a handlebody. ∎
3.2. Bounding from above
The surface has topological index at most .
Our proof will follow almost exactly as the proof of Proposition 5 from . In each copy of the manifold , we have the surface , a copy of , dividing the manifold into and , copies of and . Observe that in each , there is exactly one essential disk, with boundary contained in , just as in . However, in each , there are several essential disks with boundary contained in . We will call this collection of disks . From each , choose a single representative .
Define the sub-complex, , of spanned by the vertices corresponding to , which is homeomorphic to an -sphere. Then, define a map by the identity on , and by sending a vertex corresponding to a disk to the vertex corresponding to or , where either , or is the smallest index for which an essential outermost sub-disk of is contained in or , respectively.
Just as in , we claim that this map is a simplicial map that fixes each vertex of . To see this, consider any two disks and connected by an edge in (so that the disks are realized disjointly in ). Observe that by our construction of and Corollary 2.13, any disk contained in must intersect any disk contained in (whether either disk is a bridge disk, a graph-bridge disk, or the boundary is contained in ). So, if , then , and is joined to in . Thus, is a retraction onto the -sphere, , showing that is non-trivial, so the topological index of is at most . ∎
The topological index of is well-defined, and is topologically minimal.
3.3. Bounding from below
We make use of an important theorem in the development of topological index by Bachman:
Theorem 3.6 (Theorem 3.7 of ).
Let be a properly embedded, incompressible surface in an irreducible 3-manifold . Let be a properly embedded surface in with topological index . Then may be isotoped so that
meets in saddles, for some , and
the sum of the topological indices of the components of , plus is at most .
The surface has topological index no smaller than .
Suppose had topological index . By Theorem 3.6, can be isotoped to a surface, , so that meets in saddles, the sum of the topological indices of each component of is , and . Further, we may isotope any annular components of that are boundary parallel into completely into . Observe that this will have no effect on the Euler characteristic of , nor any effect on the topological index, since such a component will have topological index zero. We consider two different cases.
First, suppose that there is some component of with Euler characteristic less than . In this case, because the Euler characteristic of is , the sum of the Euler characteristics of the remaining components of must be greater than . This implies that there are at least components of with non-negative Euler characteristic. Again, as the sum of the topological indices of each component of is , there must be at least one component of with non-negative Euler characteristic and topological index zero. This is impossible by Theorem 1.
Second, suppose that the Euler characteristic of each component of is bounded below by . As the sum of the topological indices of each component of is , there must be at least one index so that every component of has topological index zero. Thus, there is a component, , of which is incompressible and has Euler characteristic bounded below by .
While may be boundary compressible, we may boundary compress maximally, if necessary, to obtain a surface that is incompressible, boundary incompressible, and not boundary parallel. Since boundary compressions only increase Euler characteristic, the resulting essential surface has Euler characteristic bounded below by .
We have now shown that contains a surface of topological index . To prove Theorem 1 it remains to show that is hyperbolic.
For , is hyperbolic.
Consider an essential surface in with Euler characteristic bounded below by zero, chosen to intersect minimally. If , we arrive at a contradiction to Theorem 1 as would lie in one of the copies of . If , the incompressibility and boundary incompressibility of guarantees that the curves of are essential in . Thus is a collection of one or more planar surfaces for some . This again contradicts Theorem 1. Thus, in particular, is prime and atoridal for all . Then, as is an incompressible surface in , we conclude that is hyperbolic. ∎
-  D. Bachman. Normalizing Topologically Minimal Surfaces I: Global to Local Index. ArXiv e-prints, math.GT 1210.4573, October 2012.
-  D. Bachman. Normalizing Topologically Minimal Surfaces II: Disks. ArXiv e-prints, math.GT 1210.4574, October 2012.
-  D. Bachman. Normalizing Topologically Minimal Surfaces III: Bounded Combinatorics. ArXiv e-prints, math.GT 1303.6643, March 2013.
-  David Bachman. Topological index theory for surfaces in 3-manifolds. Geom. Topol., 14(1):585–609, 2010.
-  David Bachman. Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds. Math. Ann., 355(2):697–728, 2013.
-  David Bachman and Jesse Johnson. On the existence of high index topologically minimal surfaces. Math. Res. Lett., 17(3):389–394, 2010.
-  David Bachman and Saul Schleimer. Distance and bridge position. Pacific J. Math., 219(2):221–235, 2005.
-  Ryan Blair, Marion Campisi, Jesse Johnson, Scott A. Taylor, and Maggy Tomova. Exceptional and cosmetic surgeries on knots. Mathematische Annalen, 367(1):581–622, 2017.
-  Ryan Blair, Maggy Tomova, and Michael Yoshizawa. High distance bridge surfaces. Algebr. Geom. Topol., 13(5):2925–2946, 2013.
-  Tobias H. Colding and William P. Minicozzi, II. The space of embedded minimal surfaces of fixed genus in a 3-manifold. I. Estimates off the axis for disks. Ann. of Math. (2), 160(1):27–68, 2004.
-  Tobias H. Colding and William P. Minicozzi, II. The space of embedded minimal surfaces of fixed genus in a 3-manifold. II. Multi-valued graphs in disks. Ann. of Math. (2), 160(1):69–92, 2004.
-  Tobias H. Colding and William P. Minicozzi, II. The space of embedded minimal surfaces of fixed genus in a 3-manifold. III. Planar domains. Ann. of Math. (2), 160(2):523–572, 2004.
-  Tobias H. Colding and William P. Minicozzi, II. The space of embedded minimal surfaces of fixed genus in a 3-manifold. IV. Locally simply connected. Ann. of Math. (2), 160(2):573–615, 2004.
-  Tobias H. Colding and William P. Minicozzi, II. The space of embedded minimal surfaces of fixed genus in a 3-manifold V; fixed genus. Ann. of Math. (2), 181(1):1–153, 2015.
-  Kun Du and Ruifeng Qiu. A note on the uniqueness of unstabilized Heegaard splittings of amalgamated 3-manifolds. Topology Appl., 204:135–148, 2016.
-  Hiroshi Goda. Bridge index for theta curves in the -sphere. Topology Appl., 79(3):177–196, 1997.
-  Kevin Hartshorn. Heegaard splittings of Haken manifolds have bounded distance. Pacific J. Math., 204(1):61–75, 2002.
-  H. Kneser. Geschlossene flächen in driedimensionalen mannigfaltigkeiten. Jahresbericht der Dent. Math. Verein., 28:248–260, 1929.
-  Jung Hoon Lee. On topologically minimal surfaces of high genus. Proc. Amer. Math. Soc., 143(6):2725–2730, 2015.
-  George Mossessian. Stabilizing Heegaard splittings of high-distance knots. Algebr. Geom. Topol., 16(6):3419–3443, 2016.
-  Hossein Namazi. Big Heegaard distance implies finite mapping class group. Topology Appl., 154(16):2939–2949, 2007.
-  Ken’ichi Ohshika and Makoto Sakuma. Subgroups of mapping class groups related to Heegaard splittings and bridge decompositions. Geom. Dedicata, 180:117–134, 2016.
-  Makoto Ozawa. Bridge position and the representativity of spatial graphs. Topology Appl., 159(4):936–947, 2012.
-  Joachim H. Rubinstein. An algorithm to recognize the -sphere. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 601–611. Birkhäuser, Basel, 1995.