Hyperbolic manifolds containing high topological index surfaces
aa r X i v : . [ m a t h . G T ] J un HYPERBOLIC MANIFOLDS CONTAINING HIGH TOPOLOGICALINDEX SURFACES
MARION CAMPISI AND MATT RATHBUN
Abstract.
If a graph is in bridge position in a 3-manifold so that the graph com-plement is irreducible and boundary irreducible, we generalize a result of Bachmanand Schleimer to prove that the complexity of a surface properly embedded in thecomplement of the graph bounds the graph distance of the bridge surface. We usethis result to construct, for any natural number n , a hyperbolic manifold containinga surface of topological index n . Introduction
It has become increasingly common and useful to measure distances in complexesassociated to surfaces between certain important sub-complexes associated with thesurface embedded in a 3-manifold. These techniques provide a means to indicate theinherent complexity of links in a manifold, decomposing surfaces, or the manifold itself.In [4] Bachman defined the topological index of a surface as a topological analogue ofthe index of an unstable minimal surface. When the distance is small, the notion oftopological index refines this distance, by looking at the homotopy type of a certainsub-complex.In the same way that incompressible surfaces share important properties with stronglyirreducible surfaces (distance >
2) despite being compressible, the topological indexprovides a degree of measurement of how similar irreducible, but weakly reducible(distance = 1) 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-manifoldcan 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 [18] and almost normalform introduced by Rubinstein [24], and mirroring results about geometrically mini-mal surfaces due to Colding and Minicozzi [10, 11, 12, 13, 14].Lee [19] has shown that an irredubible manifold containing an incompressible surfacecontains topologically minimal surfaces of arbitrarily high genus, but has only shownthat the topological index of such surfaces is at least two. In [6] Johnson and Bachmanshowed that surfaces of arbitrarily high index exist. These surfaces are the lifts ofHeegaard surfaces in an n-fold cover of a manifold obtained by gluing together boundarycomponents of the complement of a link in S . A by-product of their construction isthat the resulting manifolds are toroidal.This leaves open the question of whether the much more ubiquitous class of hyper-bolic manifolds can also contain high topological index surfaces. Here we constructcertain hyperbolic manifolds containing such surfaces. We generalize the construction in [6] by gluing along the boundary components of the complement of a graph in S to show: Theorem 1.1.
There is a closed 3-manifold M , with an index 1 Heegaard surface S ,such that for each n , the lift of S to some n -fold cover M n of M has topological index n . Moreover, M n is hyperbolic for all n . In order to guarantee the hyperbolicity of M n we must rule out the existence ofhigh Euler characteristic surfaces in the graph complement. To that end, we definethe graph distance , d G , of graphs in S , an analogue of bridge distance of links. In thespirit of Hartshorn [17] and Bachman-Schleimer [7] we show that the complexity of anessential surface is bounded below by the graph bridge distance: Theorem 1.2.
Let Γ be a graph in a closed, orientable 3-manifold M which is inbridge position with respect to a Heegaard surface B , so that M r n (Γ) is irreducibleand boundary irreducible. Let S be a properly embedded, orientable, incompressible,boundary-incompressible, non-boundary parallel surface in M r n (Γ) . Then d G ( B, Γ) is bounded above by g ( S ) + | ∂S | − . In Section 2 we lay out the definitions of the various complexes and distances wewill use, and prove Theorem 1.2. In Section 3, we prove Theorem 1.1.2.
Definitions
Given a link
L ⊂ S , a bridge sphere for L is a sphere, B , embedded in S , inter-secting the link L transversely, and dividing S into two 3-balls, V and W , so thatthere exist disks D V and D W properly embedded in V and W , respectively, so that L ∩ V ⊂ D V and L ∩ W ⊂ D W are each a collection of arcs.In [16], Goda introduced the notion of a bridge sphere for a spatial θ -graph, and thiswas extended by Ozawa in [23]. A bridge sphere for a (spatial) graph Γ is a sphere, B ,embedded in S , instersecting Γ transversely in the interior of edges, and dividing S into two 3-balls, V and W , so that there exist disks D V and D W properly embeddedin V and W , respectively, so that Γ ∩ V ⊂ D V and Γ ∩ W ⊂ D W are each a collectionof trees and/or arcs.If B is a bridge sphere for a link L , then a bridge disk is a disk properly embeddedin one of the components of ( S r n ( L )) r B ), whose boundary consists of exactly twoarcs, meeting at their endpoints, with one arc essential in B r n ( L ), and the otheressential in ∂n ( L ). We refer to the arc in the boundary of the disk that is containedin B as a bridge arc . Similarly, if B is a bridge sphere for a graph Γ, then a graph-bridge disk is a disk properly embedded in one of the components of ( S r n (Γ)) r B ),whose boundary consists of exactly two arcs, meeting at their endpoints, with one arcessential in B r n (Γ), and the other essential in ∂n (Γ). We refer to the arc in theboundary of the disk that is contained in B as a graph-bridge arc . Definition 2.1.
The curve complex for a surface B with (possibly empty) boundaryis the complex with vertices corresponding to the isotopy classes of essential simpleclosed curves in B , so that a collection of vertices defines a simplex if representatives ofthe corresponding isotopy classes can be chosen to be pairwise disjoint. We will denotethe curve complex for a surface B by C ( B ). YPERBOLIC MANIFOLDS CONTAINING HIGH TOPOLOGICAL INDEX SURFACES 3
Definition 2.2.
The arc and curve complex for a surface B ′ with boundary is thecomplex with vertices corresponding to the (free) isotopy classes of essential simpleclosed curves and properly embedded arcs in B ′ . A collection of vertices defines asimplex if representatives of the corresponding isotopy classes can be chosen to bepairwise disjoint. We will denote the arc and curve complex for a surface B ′ by AC ( B ′ ).If B is a surface embedded in a manifold, and a 1-dimensional complex intersects B transversely, we will refer to the surface obtained by removing a neighborhood of the1-complex by B ′ . We will often refer to C ( B ′ ) simply by C ( B ), and AC ( B ′ ) simply by AC ( B ). Definition 2.3.
Let B be a surface with at least two distinct, essential curves. Giventwo collections X and Y of vertices in the complex C ( B ) (resp., AC ( B )), the distancebetween X and Y , denoted d C ( B ) ( X, Y ) (resp., d AC ( B ) ( X, Y )), is the minimal numberof edges in any path in C ( B ) (resp., AC ( B )) from a vertex in X to a vertex in Y . Whenthe surface is understood, we often just write d C (resp., d AC ).We will be working with four subtly different but closely related sub-complexes, andsome associated notions of distance. Definition 2.4.
Let B be a properly embedded surface separating a manifold M intotwo components, V and W . Define the disk set of V (resp., W ), denoted D V ⊂ C ( B ),(resp. D W ⊂ C ( B )), as the set of all vertices corresponding to essential simple closedcurves in B that bound embedded disks in V (resp., W ). Define the disk set of B ,denoted D B , as the set of all vertices corresponding to essential simple closed curvesin B that bound embedded disks in M . Definition 2.5.
Let B be a bridge sphere for a link L , bounding 3-balls V and W ,with at least 6 marked points corresponding to the transverse intersections of L with B .The distance of the bridge surface , denoted d C ( B, L ), is d C ( B ′ ) ( D V , D W ), the distancein the curve complex of B ′ between D V and D W .The fundamental building block in our construction will be the exterior of a graphthat is highly complex as viewed from the arc and curve complex. The existence ofsuch a block will follow from a result of Blair, Tomova, and Yoshizawa. It is a specialcase of Corollary 5.3 from [9]. Theorem 2.6 ([9]) . Given non-negative integers b and d , with b ≥ , there exists a -component link L in S , and a bridge sphere B for L so that L is b -bridge with respectto B and d C ( B, L ) ≥ d . Definition 2.7.
Let B be a bridge sphere for a link L , bounding 3-balls V and W .Define the bridge disk set of V (resp., W ), denoted BD V ⊂ AC ( B ) (resp., BD W ), asthe set of all vertices either corresponding to essential simple closed curves in B ′ thatbound embedded disks in V r L (resp., W r L ), or corresponding to bridge arcs in B ′ . Definition 2.8.
Let B be a bridge sphere for a link L , bounding 3-balls V and W .The bridge distance of the bridge surface B , denoted d BD ( B, L ) is d AC ( B ′ ) ( BD V , BD W ),the distance in the arc and curve complex of B ′ between BD V and BD W . MARION CAMPISI AND MATT RATHBUN
Lemma 2.9 ([8], Lemma 2) . If B is a bridge surface which is not a sphere with fouror fewer punctures, then d BD ( B, L ) ≤ d C ( B, L ) ≤ d BD ( B, L ) . Definition 2.10.
Let B be a bridge sphere for graph Γ, bounding 3-balls V and W .The graph disk set of V (resp., W ) denoted GD V ⊂ AC ( B ) (resp., GD W ⊂ AC ( B )), isthe set of all vertices either corresponding to essential simple closed curves in B r n (Γ)that bound embedded disks in V r n (Γ) (resp., W r n (Γ)), or corresponding to graph-bridge arcs in B r n (Γ). Definition 2.11.
Let B be a bridge sphere for graph Γ. The graph distance of thebridge surface , denoted d G ( B, Γ) is d AC ( B ′ ) ( GD V , GD W ), the distance in the arc andcurve complex of B ′ = B r n (Γ) between GD V and GD W . Lemma 2.12.
Let L be a link in bridge position with respect to bridge sphere B ,bounding 3-balls V and W , and let Γ L be a graph in bridge position with respect to B formed by adding edges to L in V that are simultaneously parallel into B in thecomplement of L , and so that Γ L ∩ V has at least two components.If D ⊂ ( V r n (Γ L )) is a graph-bridge disk for Γ L , then there is a bridge disk D ′ for L in ( V r n ( L )) which is disjoint from D .Proof. Let Γ , . . . , Γ ℓ be the connected components Γ L ∩ V , and let Γ i be the componentof Γ L ∩ V to which D is incident.Over all bridge disks E ⊂ V for L disjoint from Γ i , choose one which minimizes | D ∩ E | . Suppose the intersection is non-empty. Any loops of intersection can beremoved because ( V r n (Γ)) is a handlebody and therefore irreducible. Any points ofintersection between ∂D and ∂E are contained in ∂D ∩ B and ∂E ∩ B . Choose anarc γ of | D ∩ E | . The arc γ cuts D into two disks D γ and D γ . For one of i = 1or 2, ∂D γ i ∩ ∂D is contained in B . Call that disk D γ . Consider an arc α of | D ∩ E | outermost in D γ . If the interior of D γ is disjoint from E then take α to be γ . The arc α cuts off a disk D α from D γ and cuts E into two disks E and E only one of whose(say E ) boundary is incident to L . The disk E ∪ D α = E ′ is a bridge disk for L andintersects D fewer times than E , contradicting the minimality of | D ∩ E | . (cid:3) The above implies that the distance in the arc and curve complex of B r n (Γ) between GD V and BD V is less than or equal to one. Corollary 2.13.
Let L and Γ L be as above. Then d BD ( B, L ) ≤ d G ( B, Γ L ). Proof.
Since W r n (Γ) contains no graph-bridge disks, GD W = BD W . Thus d G ( B, Γ L ) = d AC ( GD V , BD W ). Lemma 2.12 shows that d AC ( GD V , BD V ) ≤
1, and so by the triangleinequality we have that d BD ( B, L ) ≤ d G ( B, Γ L ). (cid:3) In [17], Hartshorn proved that an essential closed surface in a 3-manifold creates anupper bound on the possible distances of Heegaard splittings of that manifold in termsof the genus of the essential surface.
Theorem 2.14 (Hartshorn, Theorem 1.2 of [17]) . Let M be a Haken 3-manifold con-taining an incompressible surface of genus g . Then any Heegaard splitting of M hasdistance at most g . YPERBOLIC MANIFOLDS CONTAINING HIGH TOPOLOGICAL INDEX SURFACES 5
This idea has been generalized in numerous ways, including by Bachman and Schleimer,who show in [7] that the distance of a bridge Heegaard surface in a knot complement isbounded by twice the genus plus the number of boundary components of an essentialproperly embedded surface.
Theorem 2.15 (Bachman-Schleimer, Theorem 5.1 of [7]) . Let K be a knot in a closed,orientable 3-manifold M which is in bridge position with respect to a Heegaard surface B . Let S be a properly embedded, orientable, essential surface in M r n ( K ) . Then thedistance of K with respect to B is bounded above by twice the genus of S plus | ∂S | . We will need a yet more general version, since we will be concerned with surfacesproperly embedded in graph complements.The essence of both results is that the distance of a bridge or Heegaard surface isbounded above in terms of the complexity of an essential properly embedded surface.We will generalize this result to link and graph complements, with the additionalbenefit of avoiding many of the technical details of [7] necessary to treat the boundarycomponents. Unfortunately, our bound will be worse than that obtained by Bachmanand Schleimer, though it will be sufficient for many applications of this type of bound( e.g. , [20], [15], [22], [5], and [21]). We note also that our proof requires a minimalstarting position similar to that used by Hartshorn, an assumption the Bachman-Schleimer method was able to avoid.We now prove the following.
Theorem 1.2.
Let Γ be a graph in a closed, orientable 3-manifold M which is inbridge position with respect to a Heegaard surface B , so that M r n (Γ) is irreducibleand boundary irreducible. Let S be a properly embedded, orientable, incompressible,boundary-incompressible, non-boundary parallel surface in M r n (Γ) . Then d G ( B, Γ) is bounded above by g ( S ) + | ∂S | − .Proof of Theorem 1.2. In the case that S is closed, we note that the proofs of bothTheorem 2.15 and Theorem 2.14 apply to closed surfaces in manifolds with boundaryas long as the manifold is irreducible. In the case that ∂S = ∅ we will double M r n (Γ)along ∂n (Γ) to obtain a closed surface and show that the surface can be made to fulfillall the hypotheses necessary to use the machinery in the proof of Theorem 2.14 toobtain the bound on distance.First, isotope S to intersect B minimally, among all isotopy representatives of S .Let V and W be the handlebodies on either side of B . Double M r n (Γ) along ∂n (Γ),and call the resulting manifold c M . Let the doubles of S , B , V and W be b S , b B , b V and c W , respectively, and let G be ∂n (Γ) in c M , with respective copies M i , S i , B i , V i and W i for i = 1 , b B is a Heegaard surface for c M . (The proof of this is very similar tothe proof of Proposition 3.2 below.) Also, note that since S is incompressible and ∂ -incompressible in M r n (Γ), b S is an incompressible closed surface in c M and since ∂n (Γ) was incompressible in M r n (Γ), G is incompressible in c M . Claim 1.
Each of b S ∩ b V and b S ∩ c W are incompressible. MARION CAMPISI AND MATT RATHBUN
Proof.
If, say, b S ∩ b V had a compressing disk D , then since b S is incompressible in c M ,there would have to be a disk D ′ in b S with ∂D ′ = ∂D , and D ′ ∩ b B = ∅ . We maychoose D to be a compressing disk which intersects G minimally. Further, since G is incompressible, we may choose D to intersect G only in arcs, if at all. But c M isirreducible, so D ∪ D ′ bounds a ball and we may isotope b S across this ball from D ′ to D , lowering the number of intersections between b S and b B .If D ′ ∩ G = ∅ , then this can be viewed as an isotopy of S in M r n (Γ) which reducesthe number of intersections between S and B , a contradiction.If D ′ ∩ G = ∅ we still arrive at a contradiction. Consider a loop, ℓ , of intersection in( D ∪ D ′ ) ∩ G , innermost in D ∪ D ′ . Since D ∩ G only contains arcs, ℓ consists of twoarcs, α and α ′ in D and D ′ respectively. Thus ℓ bounds a disk D ℓ in G , α cuts off asubdisk D α of D and α ′ cuts off a subdisk D α ′ of D ′ , both of which are in either M or M , say M . Now we have an isotopy of S from D α ∪ D α ′ to D ℓ Independent of whether D α ′ intersected B , we could have chosen D to have fewerintersections with G , contradicting our choice of D to minimize intersections. (cid:3) Claim 2.
Every intersection of b S with b B is essential in b B . Proof.
Curves of intersection in b S ∩ b B which are inessential in both surfaces wouldeither give rise to a reduction in | S ∩ B | or could have come from the doubling of arcsin S ∩ B which would give rise to a reduction in | S ∩ B | in a fashion similar to theprevious claim. (cid:3) Claim 3.
There are no ∂ -parallel annular components of b S ∩ c W or b S ∩ b V . Proof.
Any such component disjoint from G would have been eliminated when | S ∩ B | was minimized. The intersection of any such component intersecting G with M would be a ∂ -parallel disk which also would have been eliminated when | S ∩ B | wasminimized. (cid:3) Now we have satisfied all the hypotheses to obtain the sequence of isotopic copies of b S described in Lemmas 4.4 and 4.5 of [17]. Depending on whether either of b S ∩ b V or b S ∩ c W contain disk components or not, we apply either Lemma 4.4 or 4.5, respectively,of [17] to obtain a sequence of compressions of b S which give rise to a path in AC ( b S ).A priori, this path would not restrict to a path in AC ( S ), but the following Claimshows that we can choose the compressions to be symmetric across G , and so eachcompression will correspond to an edge in AC ( S ). Claim 4.
If there exists an elementary ∂ -compression of b S in b V (resp. c W ), then thereexists an elementary compression of b S in b V (resp. c W ) which is symmetric across G inthe sense that either(1) the ∂ -compressing disk D is disjoint from G in M , and there is a corresponding ∂ -compressing disk D in M , or(2) the ∂ -compression is along a disk that is symmetric across G . Proof.
Let D be an elementary ∂ -compression disk for, say, b S ∩ b V chosen to minimize | D ∩ G | . We may restrict attention to such disks with | D ∩ G | > YPERBOLIC MANIFOLDS CONTAINING HIGH TOPOLOGICAL INDEX SURFACES 7
First, we observe that D ∩ G cannot contain any loops of intersection, for a loopof D ∩ G innermost in D bounds a sub-disk of D which would either give rise to acompression for G or would provide a means of isotoping D so as to lower | D ∩ G | .Thus, D ∩ G consists only of arcs. These arcs are either • vertical arcs: with one endpoint on each of b S and b B , • b S -arcs: with both endpoints on b S , or • b B -arcs: with both endpoints on b B .Consider an b S -arc of D ∩ G , outermost in D , cutting off sub-disk D ′ from D , withboundary consisting of σ in b S and γ in G . Without loss of generality, assume D ′ ⊂ M .If σ is essential in b S ∩ M , then D ′ is a boundary compression disk for S in M , whichis impossible. If σ is inessential in b S ∩ M , then it must co-bound a disk E in b S ∩ M together with an arc σ ′ ⊆ ∂ ( b S ∩ M ). The curve γ ∪ σ ′ cannot be essential in G , else D ′ ∪ E would be a compressing disk for G . Thus, γ ∪ σ ′ bounds a disk, F ⊆ G . Now F ∪ D ′ ∪ E is a sphere bounding a ball in M , so D ∪ E is isotopic to F , and replacing D ′ with F results in an elementary boundary compressing disk for b S ∩ V with fewerintersections with G than D . Thus we may assume that D ∩ G contains no b S -arcs.Now consider a sub-disk D ′ of D which is cut off by all the arcs of D ∩ G and whoseboundary consists of no more than one vertical arc. With out loss of generality, assume D ′ ⊆ M . Suppose ∂D ′ has b B -arcs, β , β , . . . , β k . Then all the β i are disjoint arcs on G . If any of them are inessential in G ∩ b V then they bound disks B i ⊆ G ∩ V . If anyof the β i are essential in G ∩ b V , then they bound disks B i ⊆ V that are bridges disksfor n (Γ) in V . In either case, D ′ ∪ (cid:16)S ki =1 B i (cid:17) results in a boundary compressing diskfor S ∩ b V with fewer intersections with G than D . This boundary compressing disk isstill elementary as the arc in b S remains unchanged. Thus, we may assume that D ∩ G consists solely of vertical arcs.Let γ be an arc of D ∩ G outermost in D , cutting off a sub-disk D from D . Withoutloss of generality, D ⊆ M . The boundary of D consists of three arcs; γ ⊆ G , σ ⊆ S and β ⊆ B . By symmetry, there exists disk D ⊆ M in M , so that D ∪ D is a diskin b V with boundary consisting of arcs σ = σ ∪ σ ⊆ b S and β = β ∪ β ⊆ b B , intersecting G in exactly one arc, γ . Finally, we must show that σ is a “strongly essential” arc in b S ∩ b V .If σ is not strongly essential then it is either the meridian of a boundary parallelannulus of b S ∩ b V which isnot possible since σ was a sub-arc of the original elementary compression disk D ,or σ is inessential in b S ∩ b V . If σ is inessential then it would co-bound a disk E in b S together with an arc σ ′ ⊆ b S ∩ b B . This disk provides an isotopy in b S of σ to σ .If the disk D ′ = D r D only intersects D in γ then D ′ ∪ D is a compressing diskfor b S ∩ b V with fewer arcs of intersection with G , as the disk can be isotoped away from γ . This disk is still an elementary compressing disk because σ is isotopic to σ , andso contradicts our original choice of D .Thus, σ is strongly essential in b S ∩ b V , and D ∪ D is a new compressing disk for b S ∩ b V that is symmetric across G . (cid:3) MARION CAMPISI AND MATT RATHBUN
We may, thus, proceed exactly as in Theorem 2.14. Each elementary boundarycompression of b S towards either of b V or c W can be performed in a symmetric way,demonstrating a path from D b V to D c W in C ( b S ) of length no greater than twice thegenus of b S , which is 2( g ( S ) + | ∂S | − b S corresponds to a pair of curves b c i and d c i +1 in S that contribute an edge in a path in C ( b S ) from D b V to D c W , there is immediately apair of curves d c i +2 and d c i +3 in S also contributing an edge in a path from D V to D W ,and this pair of paths corresponds to a single pair of curves c i and c i +1 in S contributinga single edge in AC ( S ). Each time a boundary compression for b S corresponds to a pairof curves intersecting G that contributes an edge in a path in C ( b S ) from D b V to D c W ,the restriction of these curves to S is a pair of arcs contributing an edge in AC ( S ).Further, since the boundary compressions (and elimination of boundary parallelannuli) are all being performed symmetrically, the resulting disks D b V ∈ D b V from b S ∩ b V and D c W ∈ D c W from b S ∩ c W are symmetric. That is, either D b V (resp., D c W ) is disjointfrom G , so that we may assume that it sits in V (resp., W ), or it is symmetric across G so that D b V ∩ M (resp., D c W ∩ M ) is a graph bridge disk for Γ in M . In eithercase, this demonstrates a path in AC ( S ) from DG V to DG W of length no greater than2( g ( S ) + | ∂S | − (cid:3) Theorem 1.1
In [4] Bachman defined the topological index of a surface. In contrast to the distancesbetween sub-complexes each corresponding to some disks discussed in Section 2, heexploits the homotopy type of the complex of all disks.
Definition 3.1.
The surface B is said to be topologically minimal if either D B is empty,or if there exists an n ∈ N so that π n ( D B ) = 0. If a surface B is topologically minimal,then the topological index is defined to be the smallest n ∈ N so that π n − ( D B ) = 0,or 0 if D B is empty.In [6] Johnson and Bachman showed that surfaces of arbitrarily high index exist, butthe manifolds they construct all contain essential tori. We prove an analogue of this. Theorem 1.1.
There is a closed 3-manifold M , with an index 1 Heegaard surface S ,such that for each n , the lift of S to some n -fold cover M n of M has topological index n . Moreover, M n is hyperbolic for all n . The construction.
Let n be a positive integer. We will construct a hyperbolicmanifold containing a Heegaard surface of topological index n .Using the machinery in Theorem 2.6, let L be a (0 , S with two components, L and K , with bridge sphere B of distance at least 32 n + 7. Let V and W be the two3-balls bounded by B . Since L is in bridge position, there exist disks D V and D W properly embedded in V and W , respectively, with ( L ∩ V ) ⊂ D V , and ( L ∩ W ) ⊂ D W .By modifying D V if necessary, we can find two arcs τ L and τ K in B such that(1) τ L ∪ τ K ⊂ D V ,(2) τ L ∩ τ K = ∅ ,(3) τ L ∩ L = ∂τ L ⊂ L and τ K ∩ L = ∂τ K ⊂ K , YPERBOLIC MANIFOLDS CONTAINING HIGH TOPOLOGICAL INDEX SURFACES 9 (4) each of τ K and τ L have endpoints on different components of L ∩ V .Let L ′ = L ∪ τ L , let G L = ∂n ( L ′ ), let K ′ = K ∪ τ K , let G K = ∂n ( K ′ ), and letΓ = L ∪ τ L ∪ τ K = L ′ ∪ K ′ . Observe that Γ is a graph in bridge position with respectto B . Let M ′ = S r n (Γ), let V ′ = V r n (Γ), and let W ′ = W r n (Γ) = W r n ( L ),and B ′ = B r n (Γ) = B r n ( L ).For each i = 1 , , . . . , n , let M ′ i be homeomorphic to M ′ , along with homeomorphiccopies L i of L , ( G L ) i of G L , ( G K ) i of G K , and B ′ i of B ′ .Then, for each i = 1 , , . . . , ( n − G K ) i with ( G L ) i +1 and identify ( G K ) n with ( G L ) , all via the same homeomorphism. Call the resulting closed 3-manifold M n .Observe that the union of the B ′ i is a closed surface that we will call B n . We will showthat B n is a Heegaard surface for M n , that B n has high topological index, and that M n is hyperbolic. Proposition 3.2.
For each n , the surface B n ⊂ M n is a genus 3 n +1 Heegaard surface. Proof.
That the genus of B n is 3 n + 1 can be verified by an Euler characteristic count.It suffices, then, to verify that the complement of B n is two handlebodies, V n and W n .Since Γ was in bridge position with respect to B , there are disks D V and D W properlyembedded in V and W , respectively, so that Γ ∩ V ⊂ D V and Γ ∩ W ⊂ D W . Then D V and D W cut along Γ is a collection of sub-disks.The result of cutting V r n (Γ) along all these sub-disks of D V is a pair of 3-balls, eachwith two sub-disks, D +1 and D +2 , of n (Γ) contained in the boundary. Each identificationof ( G K ) i with ( G L ) i +1 (indices mod n ) glues pairs of these sub-disks along arcs, resultingin disks in V n , and further cutting along ( n −
1) copies of each of D +1 and D +2 resultsin a collection of 3-balls, showing that V n is a handlebody.Similarly, the result of cutting W r n (Γ) along all of the sub-disks of D W is a pairof 3-balls, each with four sub-disks of n (Γ) contained in the boundary, D − , D − , D − ,and D − . Each identification of ( G K ) i with ( G L ) i +1 (indices mod n ) glues pairs of thesesub-disks along arcs, resulting in disks in W n , and further cutting along ( n −
1) copiesof each of D − , D − , D − , and D − results in a collection of 3-balls, showing that W n is ahandlebody. (cid:3) Bounding from above.Proposition 3.3.
The surface S n has topological index at most n . Proof.
Our proof will follow almost exactly as the proof of Proposition 5 from [6]. Ineach copy M ′ i of the manifold M ′ , we have the surface B ′ i , a copy of B ′ , dividing themanifold into V ′ i and W ′ i , copies of V ′ and W ′ . Observe that in each V ′ i , there is exactlyone essential disk, D + i with boundary contained in B ′ i , just as in [6]. However, in each W ′ i , there are several essential disks with boundary contained in B ′ i . We will call thiscollection of disks D − i . From each D − i , choose a single representative D − i .Define the sub-complex, P , of D M spanned by the vertices corresponding to S i { D + i , D − i } ,which is homeomorphic to an ( n − F : D M → P bythe identity on P , and by sending a vertex corresponding to a disk D S i { D + i , D − i } to the vertex corresponding to D + j or D − j , where either D ∈ D − j , or j is the smallestindex for which an essential outermost sub-disk of D r ( S i G i ) is contained in V ′ j or W ′ j , respectively. Just as in [6], we claim that this map F is a simplicial map that fixes each vertexof P . To see this, consider any two disks D and D connected by an edge in D M (sothat the disks are realized disjointly in M ). Observe that by our construction of M ′ and Corollary 2.13, any disk contained in V ′ j must intersect any disk contained in W ′ j (whether either disk is a bridge disk, a graph-bridge disk, or the boundary is containedin B ′ j ). So, if D ± i = F ( D ) = F ( D ) = D ± j , then i = j , and F ( D ) is joined to F ( D )in P . Thus, F is a retraction onto the ( n − P , showing that π n − ( D M ) isnon-trivial, so the topological index of B n is at most n . (cid:3) Corollary 3.4.
The topological index of B n is well-defined, and B n is topologicallyminimal.3.3. Bounding from below.
We make use of an important theorem in the develop-ment of topological index by Bachman:
Theorem 3.5 (Theorem 3.7 of [4]) . Let G be a properly embedded, incompressiblesurface in an irreducible 3-manifold M . Let B be a properly embedded surface in M with topological index n . Then B may be isotoped so that (1) B meets G in p saddles, for some p ≤ n , and (2) the sum of the topological indices of the components of B r n ( G ) , plus p is atmost n . Proposition 3.6.
The surface B n has topological index no smaller than n . Proof.
Suppose S n had topological index ι < n . By Theorem 3.5, B n can be isotopedto a surface, B n , so that B n meets H = n ( G ) in σ saddles, the sum of the topologicalindices of each component of B n r n ( H ) is k , and k + σ ≤ ι . Further, we may isotopeany annular components of B n r H that are boundary parallel into ∂H completely into H . Observe that this will have no effect on the Euler characteristic of B n r H , norany effect on the topological index, since such a component will have topological indexzero. We consider two different cases.First, suppose that there is some component of B n r H with Euler characteristicless than − n . In this case, because the Euler characteristic of B n is − n , the sum ofthe Euler characteristics of the remaining components of B n r H must be greater than2 n . This implies that there are at least n + 1 components of B n r H with non-negativeEuler characteristic. Again, as the sum of the topological indices of each component of B n r H is k < n , there must be at least one component of B n r H with non-negativeEuler characteristic and topological index zero. This is impossible by Theorem 1.2.Second, suppose that the Euler characteristic of each component of B n r H isbounded below by − n . As the sum of the topological indices of each componentof B n r H is k < n , there must be at least one index j so that every component of B n ∩ M j has topological index zero. Thus, there is a component, B ′′ , of B n ∩ M j whichis incompressible and has Euler characteristic bounded below by − n .While B ′′ may be boundary compressible, we may boundary compress B ′′ maximally,if necessary, to obtain a surface that is incompressible, boundary incompressible, andnot boundary parallel. Since boundary compressions only increase Euler characteristic,the resulting essential surface has Euler characteristic bounded below by − n . YPERBOLIC MANIFOLDS CONTAINING HIGH TOPOLOGICAL INDEX SURFACES 11
By Lemma 2.9 and Corollary 2.13, in M j with B j a copy of B ′ , we have d C ( B j , L ) ≤ d BD ( B j , L ) ≤ d G ( B j , Γ)). By Theorem 1.2, d G ( B j , Γ) ≤ g ( B ′′ ) + | ∂B ′′ | − L and the fact that χ ( S ) = 2 − g ( S ) − | ∂S | , we have 32 n + 7 ≤ d C ( B j , L ) ≤ d G ( B j , Γ) ≤ g ( B ′′ ) + 4 | ∂B ′′ | − − χ ( B ′′ ) + 6. On the other handwe have just shown that − n ≤ χ ( B ′′ ), a contradiction. In either case, we find thatthe topological index of B n cannot be less than n . (cid:3) Hyperbolicity.
We have now shown that M n contains a surface of topologicalindex n . To prove Theorem 1.1 it remains to show that M n is hyperbolic. Proposition 3.7.
For n > M n is hyperbolic. Proof.
Consider an essential surface S in M n with Euler characteristic bounded belowby zero, chosen to intersect G minimally. If S ∪ G = ∅ , we arrive at a contradiction toTheorem 1.2 as S would lie in one of the copies of M ′ . If S ∪ G = ∅ , the incompressibilityand boundary incompressibility of G guarantees that the curves of S ∪ G are essentialin S . Thus S ∩ M ′ i is a collection of one or more planar surfaces for some i . Thisagain contradicts Theorem 1.2. Thus, in particular, M n is prime and atoridal for all n .Then, as G is an incompressible surface in M n , we conclude that M n is hyperbolic. (cid:3) References [1] D. Bachman. Normalizing Topologically Minimal Surfaces I: Global to Local Index.
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