Stable loops and almost transverse surfaces
We show that the cone over a fibered face of a compact fibered hyperbolic 3-manifold is dual to the cone generated by the homology classes of finitely many curves called minimal stable loops living in the associated veering triangulation. We also present a new, more hands-on proof of Mosher’s Transverse Surface Theorem.
In this paper we use veering triangulations to study the suspension flows of pseudo-Anosov homeomorphisms on compact surfaces, which we call circular pseudo-Anosov flows.
Let be a compact hyperbolic 3-manifold with fibered face . There is a circular pseudo-Anosov flow , unique up to reparameterization and conjugation by homeomorphisms isotopic to the identity, which organizes the monodromies of all fibrations of corresponding to [Fri79]. We call this flow the suspension flow of the fibered face. The suspension flow has the following property: a cohomology class is Lefschetz dual to a class in if and only if is nonnegative on , the cone of homology directions of . Hence computing is equivalent to computing .
For a flow on , is the smallest closed cone containing the projective accumulation points of homology classes nearly closed orbits of [Fri79, Fri82]. For our circular pseudo-Anosov flow , has a more convenient characterization as the smallest closed cone containing the homology classes of the closed orbits of . Our main result gives a characterization of in terms of the veering triangulation associated to .
Theorem 4.7 (Stable loops).
Let be a compact hyperbolic 3-manifold with fibered face . Let and be the associated veering triangulation and circular pseudo-Anosov flow, respectively. Then is the smallest convex cone containing the homology classes of the minimal stable loops of .
The veering triangulation is a taut ideal triangulation of a cusped hyperbolic 3-manifold obtained from by deleting finitely many closed curves from . In Theorem 4.7 we are viewing it as embedded in as an ideal triangulation of an open submanifold. The stable loops of are a family of closed curves carried by the so-called stable train track of , which lies in the 2-skeleton of and is defined in Section 4.3. They correspond to nontrivial elements of the fundamental groups of leaves of the stable foliation of . A minimal stable loop is a stable loop traversing each switch of the stable train track at most once. Our proof uses the fact that the 2-skeleton of the veering triangulation is a branched surface, and (Proposition 4.4) a surface carried by this branched surface is a fiber of if and only if it is infinitely flippable in a sense defined in Section 4.4.
Flippability is a condition depending on the combinatorics of the veering triangulation. We begin the paper by developing some of these combinatorics. This allows us to give a hands-on, combinatorial proof of Mosher’s Transverse Surface Theorem111This is actually a slight generalization of Mosher’s theorem, as he proved Theorem 3.5 only in the case when . He later proved a version for pseudo-Anosov flows which are not necessarily circular [Mos92]. [Mos91]:
Theorem 3.5 (Almost transverse surfaces).
Let be a compact hyperbolic 3-manifold, with a fibered face of and associated suspension flow . Let be an integral homology class. Then if and only if is represented by a surface almost transverse to .
A surface is almost transverse to if it is transverse to a closely related flow , obtained from by a process called dynamically blowing up singular orbits. The precise definition of almost transversality is found in Section 3.2.
Loosely speaking, the strategy of our proof of Theorem 3.5 is to arrange a surface to lie in a regular neighborhood of the veering triangulation away from the singular orbits of , and then use our knowledge of how the veering triangulation sits in relation to in order to appropriately blow up the flow near the singular orbits. Mosher, without the machinery of veering triangulations available to him, performed a deep analysis of the dynamics of the lift of to the cyclic cover of associated to the surface. Including this dynamical analysis, his complete proof of the theorem spans [Mos89, Mos90, Mos91].
Taut ideal triangulations were introduced by Lackenby in [Lac00] as combinatorial analogues of taut foliations, where he uses them to give an alternative proof of Gabai’s theorem that the singular genus of a knot is equal to its genus. He states “One of the principal limitations of taut ideal triangulations is that they do not occur in closed 3âmanifolds,” and asks:
Is there a version of taut ideal triangulations for closed 3âmanifolds?
While we do not claim a comprehensive answer to Lackenby’s question, a a theme of this paper and [Lan18] is that for fibered hyperbolic 3-manifolds (possibly closed), a veering triangulation of a dense open subspace is a useful version of a taut ideal triangulation.
Veering triangulations are introduced by Agol in [Ago10]. There is a canonical veering triangulation associated to any fibered face of a hyperbolic 3-manifold, and Guéritaud showed [Gue15] that it can be built directly from the suspension flow. If taut ideal triangulations are combinatorial analogues of taut foliations, Guéritaud’s construction allows us to view veering triangulations as combinatorializations of pseudo-Anosov flows222The forthcoming paper [SchSeg19] will make this explicit..
We include two appendices. In Appendix A we prove that a result of Fried from [Fri79], which was stated and proved for closed hyperbolic 3-manifolds, holds for manifolds with boundary. The result, which we have been assuming so far in this introduction and which is necessary for the results of this paper, states that the suspension flow is canonically associated to in the sense described above. In Appendix B, we explain how to use Theorem 3.5 to show the results of [Lan18] hold for manifolds with boundary. In particular we get the following corollary.
I thank Yair Minsky, James Farre, Samuel Taylor, and Ian Agol for stimulating conversations. I additionally thank Yair Minsky, my PhD advisor, for generosity with his time and attention during this research and throughout my time as a graduate student.
I gratefully acknowledge the support of the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1122492, and of the National Science Foundation under Grant No. DMS-1610827 (PI Yair Minsky). Any opinions, findings, and conclusions or recommendations expressed in this material are my own and do not necessarily reflect the views of the National Science Foundation.
In this paper, all manifolds are orientable and all homology and cohomology groups have coefficients in .
2.1 The Thurston norm, fibered faces, relative Euler class
We review some facts about the Thurston norm, which can be found in [Thu86]. Let be a compact, irreducible, boundary irreducible, atoroidal, anannular 3-manifold ( may be empty). If is a connected surface embedded in , define
where denotes Euler characteristic. If is disconnected, let where the sum is taken over the connected components of . For any integral homology class , we can find an embedded surface representing . Define
Then extends by linearity and continuity from the integer lattice to a vector space norm on called the Thurston norm [Thu86]. We mention that Thurston defined more generally to be a seminorm on for any compact orientable . However, in this paper will always be a norm, since the manifolds we consider will not admit essential surfaces of nonnegative Euler characteristic.
The unit ball of is denoted by . As a consequence of taking integer values on the integer lattice, is a finite-sided polyhedron with rational vertices. Our convention in this paper is that a face of is a closed cell of the polyhedron.
We say an embedded surface is if it is incompressible and realizes the minimal in . If is the fiber of a fibration then is taut, any taut surface representing ] is isotopic to , and lies in for some top-dimensional face of . Moreover, any other integral class representing a class in is represented by the fiber of some fibration of over . Such a top-dimensional face is called a fibered face.
Let be an oriented plane field on which is transverse to . If we fix an outward pointing section of , this determines a relative Euler class . For a relative 2-cycle , is the first obstruction to finding a nonvanishing section of agreeing with the outward pointing section on . A reference on relative Euler class is [Sha73].
If is a flow on tangent to , Let be the oriented line field determined by the tangent vectors to orbits of . Let be the oriented plane field which is the quotient bundle of by . We can think of as a subbundle of by choosing a Riemannian metric and identifying with the orthogonal complement of . For notational simplicity we define , the relative Euler class of .
Fix a fibration , which allows us to express for some homeomorphism of . Let be the tangent plane field to the foliation of by ’s. Let be the suspension flow of , which moves points in along lines for fixed , gluing by at the boundary of . We have and hence . For some fibered face , we have . We have , i.e. and agree on . In fact, more is true: is exactly the subset of on which and agree.
It can be fruitful to think of a properly embedded surface in as representing both a homology class in and a cohomology class in mapping homology classes of closed curves to their intersection number with . As such we will sometimes think of as a norm on via Lefschetz duality. The image in of a face of will be denoted , and in general the subscript , when attached to an object, will denote the Lefschetz dual of that object.
2.2 Circular pseudo-Anosov flows
Let be a flow on which is tangent to . Recall that a cross section to is a fiber of a fibration whose fibers are transverse to . Flows which admit cross sections are called circular. We call the suspension flow of a pseudo-Anosov map on a compact surface a circular pseudo-Anosov flow.
Such a preserves two transverse measured foliations called the stable and unstable foliations of which are transverse everywhere except the boundary of the surface (see [Thu88]). These foliations give two transverse codimension-1 foliations in preserved by called the stable and unstable foliations of . The closed orbits corresponding to the singular points of lying in are called singular orbits. The closed orbits lying on , which correspond to the boundary components of certain stable and unstable leaves, are called -singular orbits. See Figure 3.
If is a cross section to a flow , the first return map of is the map sending to the first point in its forward orbit under lying in . If is a circular pseudo-Anosov flow, the first return map of any cross section to will be pseudo-Anosov (this was proved in [Fri79] for closed manifolds, but the proof in general is essentially the same).
2.3 Review of the veering triangulation
A taut tetrahedron is an ideal tetrahedron with the following edge and face decorations. Four edges are labeled 0, and two are labeled . Two faces are cooriented outwards and two are cooriented inwards, and faces of opposite coorientation meet only along edges labeled 0. The edge labels should be thought of as the interior angles of the corresponding edges. See Figure 4. We define the top (resp. bottom) of a taut tetrahedron to be the union of the two faces whose coorientations point out of (resp. into) . The top (resp. bottom) -edge of will be the edge labeled lying in the top (resp. bottom) of .
Up to orientation-preserving combinatorial equivalence, there are two types of taut tetrahedron with a distinguished 0-edge. We call these and and they are shown in Figure 5.
A taut ideal triangulation of a 3-manifold is an ideal triangulation by taut tetrahedra such that
when two faces are identified their coorientations agree,
the sum of interior angles around a single edge is , and
no two interior angles of are adjacent around an edge.
A consequence of the third condition above is that each edge of a taut ideal triangulaton has degree .
Let be an edge of a taut ideal triangulation . If has the property that all tetrahedra for which is a 0-edge are of type when is distinguished, we say is right veering. Symmetrically, if they are all of type we say is left veering. If every edge of is either right or left veering, is veering.
The 0 and labels tell us how to “pinch” a taut ideal triangulation along its edges so that the 2-skeleton has a well-defined tangent space at every point. We will always assume the 2-skeleton of a taut ideal triangulation is embedded in such a way.
2.4 The veering triangulation of a fibered face
In this section we move somewhat delicately between compact manifolds and their interiors. The reason for this is that we wish to work in a compact manifold and study its second homology rel boundary, but veering triangulations live most naturally in cusped manifolds.
Veering triangulations were introduced by Agol in [Ago10], where he canonically associated a veering triangulation to a pseudo-Anosov surface homeomorphism. Let be a pseudo-Anosov map on a compact surface with associated stable and unstable measured foliations and . Let and , and let and be the respective mapping tori of and . Agol constructs an ideal veering triangulation of from a sequence of Whitehead moves between ideal triangulations of which are dual to a periodic splitting sequence of measured train tracks carrying the stable lamination of . Each Whitehead move corresponds to gluing a taut tetrahedron to , and the resulting taut ideal triangulation of glues up to give a taut ideal triangulation of . We call a taut ideal triangulation of this type layered on . Agol shows that up to combinatorial equivalence there is only one veering triangulation of which is layered on , and we call this the veering triangulation of .
Let be the universal cover of , and be the space obtained by attaching a point to each lift of an end of to . The measured foliation lifts to and gives rise to a measured foliation on with singularities at points of . We call this the vertical foliation, and the analogous measured foliation coming from is called the horizontal foliation. In pictures, we will arrange the vertical and horizontal foliations so they are actually vertical and horizontal in the page.
The transverse measures on the vertical and horizontal foliations give a singular flat structure. A singularity-free rectangle is a subset of which can be identified with such that for all , (resp. ) is a leaf of the vertical (resp. horizontal) foliation.
We consider the family of singularity-free rectangles which are maximal with respect to inclusion. Any such maximal rectangle has one singularity in the interior of each edge.
Each maximal rectangle defines a map of a taut tetrahedron into which “flattens” and has the following properties: the pullback of the orientation on induces the correct coorientation on each triangle in , the top (resp. bottom) -edge of is mapped to a segment connecting the singularities on the horizontal (resp. vertical) edges of , and is a geodesic in the singular flat structure of . See Figure 7.
We can build a complex from , where the union is taken over all maximal rectangles, by making all the identifications of the following type: let and be maximal rectangles, and suppose and are faces of and such that . Then we identify and . The resulting complex is a taut ideal triangulation of , and one checks that it is veering. Guéritaud showed that it descends to a layered veering triangulation of , which must be the one constructed by Agol.
While the veering triangulation is canonically associated to , it is in fact “even more canonical” than that. To elaborate, we need the following result.
Theorem 2.3 (Fried).
Let be a circular pseudo-Anosov flow on a compact 3-manifold . Let be a cross section of and let be the fibered face of such that . An integral class lies in if and only if is represented by a cross section to . Up to reparameterization and conjugation by homeomorphisms of isotopic to the identity, is the only such circular pseudo-Anosov flow.
Thus we we will speak of the suspension flow of a fibered face. Theorem 2.3 says that all the monodromies of fibrations coming from are realized as first return maps of the suspension flow. Theorem 2.3 is proven by Fried in [Fri79] in the case where is closed. The proof of this more general result, when the fiber possibly has boundary, does not seem to exist in the literature. For the reader’s convenience we provide a proof in Appendix A (Theorem A.7).
Set . Let be the face of such that , let be the associated suspension flow, and let .
The lift of to the universal cover of is product covered, i.e. conjugate to the unit-speed flow in the direction on . Consequently the quotient of by the flowing action of , which we call the flowspace of and denote by , is homeomorphic to . In addition, has 2 transverse (unmeasured) foliations which are the quotients of the lifts of the (2-dimensional) stable and unstable foliations of to .
Let be another fiber of over such that , with monodromy . Let , , and be obtained from and in the same way that , , and were obtained from and .
By Theorem 2.3, both and and their vertical and horizontal foliations can be identified with and its foliations by forgetting measures. Hence we can identify and together with their vertical and horizontal foliations. The maximal rectangles from Guéritaud’s construction depend only on the vertical and horizontal foliations, and not on their transverse measures. The geodesics defining the edges of a tetrahedron do depend on the measures (and hence on and ), but for either pair of measures, the geodesics will be transverse to both the vertical and horizontal foliations. We see then that the triangles of the veering triangulation of are well-defined up to isotopy. It follows that the veering triangulations of and are the same up to isotopy in .
Synthesizing the above discussion, we have shown the following.
Theorem 2.4 (Agol).
Let be a compact hyperbolic 3-manifold, and suppose and are fibers of fibrations with monodromies and such that for some fibered face of . Then the veering triangulations of and are combinatorially equivalent.
It therefore makes sense to speak of the veering triangulation of a fibered face of .
2.5 Some notation
We now fix some notation which will hold for the remainder of the paper.
Let be a compact hyperbolic 3-manifold, and let be a circular pseudo-Anosov flow on . Let be the union of the singular orbits of and let be a small regular neighborhood of . Let be a small regular neighborhood of and put and . Let be the (closed) fibered face of determined by , with associated veering triangulation .
The homology long exact sequence associated to the triple contains the sequence . By excision, . Hence there is an injective map
At the level of chains, the map corresponds to sending a relative 2-chain to . We call the puncturing map, and if we often write to mean .
2.6 Veering triangulations, branched surfaces, and boundaries of fibered faces
Recall that a branched surface in is an embedded 2-complex transverse to which is locally modeled on the quotient of a stack of disks such that for , is glued to along the closure of a component of the complement of a smooth arc through . The quotient is given a smooth structure such that the inclusion of each is smooth (see Figure 8). Branched surfaces were introduced in [FO84] and experts will note that the above definition is different than the original; ours is the slightly more general one found in [Oer86].
Let be a branched surface. The union of points in with no neighborhood homeomorphic to a disk is called the branching locus of , and the components of the complement of the branching locus are called sectors. Let be the closure of a regular neighborhood of . Then can be foliated by closed intervals transverse to , and we call this the normal foliation of . If the normal foliation is oriented we say is an oriented branched surface. In this paper all branched surfaces will be oriented. If is a cooriented surface properly embedded in which is positively transverse to the normal foliation in the sense that its coorientation is compatible with orientation of the normal foliation, we say is carried by . A surface carried by gives a system of nonnegative integer weights on the sectors of which is compatible with natural linear equations along the branching locus of . Any system of nonnegative real weights satisfying these linear equations gives a 2-cycle and thus determines a homology class . We say is carried by . The collection of homology classes carried by is clearly a convex cone.
Put . Then the 2-skeleton of has the structure of a cooriented branched surface. We call this branched surface .
Corollary 2.5 (Agol).
Let be the closed face of determined by , and let be the face of such that . The cone of classes in carried by is equal to .
Let be a fiber of such that . By Theorem 2.4, can be built as a layered triangulation on . It follows that is carried by . Therefore any integral class in is carried, so every rational class in is carried.
We can find a closed oriented transversal through each point of , so is a homology branched surface in the sense of [Oer86]; it follows that the cone of classes it carries is closed (see [Lan18] for details). Since each class in is approximable by a sequence of rational classes in the interior of the cone, the proof is finished. ∎
2.7 Relating and
The complex is a decomposition of into truncated taut tetrahedra. Let be a taut tetrahedron of , and let be the truncation of obtained by deleting . The top, bottom, and top and bottom -edges of are defined to be the restrictions to of the corresponding parts of . Each of the four faces of corresponding to ideal vertices of is a triangle with two interior angles of 0 and one of , which we call a flat triangle. Because the faces of are cooriented, a flat triangle inherits a coorientation on its edges. We say a flat triangle which is cooriented outwards (respectively inwards) at its vertex is an upward (resp. downward) flat triangle, as in Figure 10. The upwards flat triangles of correspond to the ideal vertices connected by the top -edge of .
The union of all upward flat triangles is a collection of annuli. The triangulation restricted to each annulus component of has the property that each edge of a flat triangle either traverses the annulus or lies on the boundary of the annulus. The same holds for the union of all downward flat triangles. A component of is called an upward ladder and a component of is called a downward ladder. We call the boundary components of ladders ladderpoles.
The 1-skeleton of this triangulation by flat triangles of is a cooriented train track, which we call . We define a notion of left and right on each branch of : orient inwards, and map some neighborhood of homeomorphically to so that is identified with and the pushforward of ’s coorientation points in the positive direction. Define the left (resp. right) switch of to be the preimage of (resp. 1). We orient each branch of from right to left, and this consistently determines an orientaion on . If an oriented curve is carried by such that orientations agree, we say is positively carried by . By our choice of orientation if is a surface carried by then its boundary, given the orientation induced by an outward-pointing vector field on , is positively carried by .
We call branches of contained in ladderpoles ladderpole branches, and branches that traverse ladders rungs. We define the left (resp. right) ladderpole of a ladder to be the ladderpole containing the left (resp. right) boundary switches of its rungs. A closed oriented curve positively carried by and traversing only ladderpole branches is called a ladderpole curve.
Note that a switch of corresponds to an edge of . The combinatorics of the flat and veering triangulations are related by the following Lemma, the proof of which is elementary.
Let be a switch of corresponding to an edge of . If lies in the left ladderpole of an upward ladder then is right veering. If lies in the right ladderpole of an upward ladder then is left veering.
Consider a singular leaf of the stable foliation of meeting a singular orbit of . Topologically is the quotient of a half-closed annulus by a map which wraps around some finite number of times. The flow lines of converge in forward time to . Since is dense in , the intersection has many components; we will call the component containing a stable flow prong of . An unstable flow prong of is defined symmetrically with a singular leaf of the unstable foliation of .
A stable flow prong of intersects in the interior of an upward ladder. Symmetrically, an unstable 2-prong of intersects in the interior of a downward ladder.
Fix a fiber of with , and let and be as in Section 2.4. Let .
Pick a singular orbit of and consider the left ladderpole of an upward ladder of . The vertices of define a family of triangles of . Let , let be a component of the lift of to , the universal cover of , let be the union of triangles in the veering triangulation of intersecting , and let be the closure in of the projection of . This is a bi-infinite sequence of triangles, each sharing an edge with the next, and all sharing a single vertex at a singular point of .
The vertical and horizontal foliations define infinitely many quadrants each meeting in a corner of angle , which are each bounded by one vertical and one horizontal leaf. We claim lives in only one of these quadrants, a situation depicted in Figure 12.
In , the edges of positive slope are right veering while the edges of negative slope are left veering. Hence every edge in meeting lies in a quadrant with horizontal left boundary leaf and vertical right boundary leaf (where our notion of left and right is determined looking at from inside ).
Fix one of the triangles of , defined by edges and meeting . The two edges lie in the interior of a singularity-free rectangle with vertical and horizontal sides containing in its boundary. Such a rectangle lies in the union of two adjacent quadrants of , only one of which can have horizontal left boundary and vertical right boundary. Therefore and lie in the same quadrant, and all the edges of lie in the same quadrant by induction.
Applying this analysis to each ladderpole yields the result. ∎
3 Almost transverse surfaces
3.1 Dynamic blowups
We now describe the process of dynamically blowing up a singular periodic orbit of a pseudo-Anosov flow , which can be thought of as replacing a singular orbit by the suspension of a homeomorphism of a tree. For more details, the reader can consult [Mos92, Mos91, Mos90].
Let , . Define a pseudo-Anosov star to be a directed tree embedded in the plane with edges meeting at a central vertex , such that the orientations of edges around alternate between inward and outward with respect to . We say a directed tree is a dynamic blowup of if the closed neighborhood of each vertex of is a pseudo-Anosov star, and there exists a cellular map preserving edge orientations such that is injective on the complement of . See Figure 14 for two examples.
Let be a singular periodic orbit of meeting stable and unstable flow prongs, and suppose rotates the flow prongs by traveling once around , where .
The intersection of the flow prongs of with a local cross section of gives a pseudo-Anosov star with edges, with each edge oriented according to whether points in that flow prong spiral towards or away from in forward time. Let be a dynamic blowup of that is invariant under rotation by , and let be the preimage of the central vertex of under the collapsing map .
There is a flow on which replaces by the suspension of a homeomorphism with the following properties. Each edge of is mapped by to its image under rotation by , and fixes vertices and moves interior points in the direction inherits from .
The orbit of under is a complex of annuli invariant under . A point interior to an annulus of spirals away from one boundary circle and towards the other in forward time, and the boundaries of annuli are closed orbits of . The flows and are semiconjugate via a map collapsing to which is injective on the complement of .
We say is obtained from by dynamically blowing up . A flow obtained from by dynamically blowing up some collection of singular orbits is a dynamic blowup of .
The effect of dynamically blowing up a singular orbit is to pull apart its flow prongs, otherwise leaving the dynamics of the flow unchanged. See Figure 15.
3.2 Statement of Transverse Surface Theorem, and Mosher’s approach
Let be a circular pseudo-Anosov flow on a 3-manifold . We say an oriented surface embedded in is almost transverse to if there is a dynamic blowup of such that is transverse to and the orientation of agrees with that of the tangent bundle of at every point in .
Transverse Surface Theorem (Mosher).
Let be a closed hyperbolic 3-manifold with fibered face and associated suspension flow . An integral class lies in if and only if it is represented by a surface which is almost transverse to .
Let be a surface which is almost transverse to , and thus transverse to some dynamic blowup of . Then is taut, and . This is true even when has boundary, as we will now show.
Note that since is a topologically transitive flow, meaning it has a dense orbit, is as well. Consider a point for some component of . There is an open neighborhood of which is homeomorphic to such that the restricted flow lines of correspond to the vertical lines for , and corresponds to . Let be a dense orbit of . We can take a segment of with endpoints near each other in and attach endpoints with a short path to obtain a closed curve positively intersecting , so is homologically nontrivial. In particular it follows that has no sphere or disk components.
We record a lemma:
Let be a dynamic blowup of . Then the tangent vector fields of and are homotopic, and consequently .
Next, note that the restriction of to is homotopic to , so . Here denotes the pairing of with . Hence
The first inequality holds because has no sphere or disk components. The final inequality follows from the fact that is a supremum of finitely many linear functionals on which include . The fact that means that is taut, while means that and agree on so .
In light of the above discussion, to prove the Transverse Surface Theorem it suffices to produce an almost transverse representative of any integral class in , and since any such class in the interior of is represented by a cross section, it suffices to produce an almost transverse representative for any integral class in .
Mosher’s proof of the Transverse Surface Theorem spans [Mos89, Mos90, Mos91]; we give a brief summary here. Given an integral class lying in , we consider its Poincaré dual . Associated to is an infinite cyclic covering space . Mosher shows that there is a way to dynamically blow up a collection of singular orbits of to get a dynamic blowup that lifts to a flow on with nice dynamics. More specifically, he defines a natural partial order on the set of chain components of and shows that has finitely many chain components up to the deck action of . He constructs a strongly connected directed graph with vertices the deck orbits of chain components of , and edges determined by . He shows that flow isotopy classes of surfaces transverse to and compatible with are in bijection with positive cocycles on representing a cohomology class which is determined by . Finally he proves the existence of such a cocycle.
3.3 A veering proof of the Transverse Surface Theorem
The proof of the Transverse Surface Theorem which we present in this section depends on the combinatorial Lemma 3.4, the statement of which requires some definitions.
A pseudo-Anosov tree is a directed tree such that the closed neighborhood of each non-leaf vertex is a pseudo-Anosov star. We can embed in a disk such that its leaves lie in .
Let be the closure of a component of . Then is homeomorphic to a closed disk and is a union of edges in and one closed interval in . The boundary of is composed of two leaves of . Each leaf can be assigned a or depending on whether the corresponding edge of points into or away from the leaf, respectively. We endow with the orientation pointing from to .
Let be complementary regions of which are incident along a single vertex of , and the orientations on and are opposite. Then there is a unique dynamic blowup of with one more edge than such that and are incident along an edge of .
A picture makes this obvious; in lieu of a proof, see Figure 16 for a diagram of this dynamic blowup (for simplicity, we are abusing notation slightly by identifying and with the corresponding complementary regions of ).
If we choose an orientation for , then any point lying in a component of can be given a sign according to whether the orientation of agrees with the orientation of the component of containing .
An even family for a pseudo-Anosov tree is a finite subset of which represents 0 in when each is given a sign as in the previous paragraph and is viewed as a 0-chain. We say an even family can be filled in over if there exists a family of disjoint cooriented line segments with such that:
in the cooriented sense, i.e. the coorientations at each point in agree with the orientation of each segment of , and
for each segment of , intersects only transversely in the interior of edges such that the coorientation of agrees with the orientation of the intersected edge.
If and above are symmetric under rotation of by an angle of , and can be chosen to respect this symmetry, we say can be -symmetrically filled in over . Figure 17 shows an example.
Let and be a pseudo-Anosov star and an even family for respectively that are symmetric under rotation by . There exists a dynamic blowup of such that can be -symmetrically filled in over .
Note that the lemma statement includes the case .
Choose a pair of points in of opposite sign which are circularly adjacent and let and be all their images, without repeats, under rotation of by . Let , be the components of corresponding to respectively. If and are incident along an edge of then there is a family of cooriented line segments filling in over . Otherwise and are incident at the vertex of and determine a dynamic blowup of as in Lemma 3.3. Since the pairs are unlinked in , we may perform all of these dynamic blowups in concert to obtain a dynamic blowup of such that can be -symmetrically filled in over .
Let be the family of cooriented line segments filling in . Let . By construction, is contained in a single component of , and has a single vertex which is preserved under rotation by . There exists a closed disk centered around which is also preserved under rotation by . We can connect each point in by a cooriented line segment to its image under a retraction such that the union of these segments is invariant under rotation by . Let , and . A picture of this situation is shown in Figure 18.
We see that is an even family for which is smaller than . Iterating this procedure, we eventually -symmetrically fill in over a dynamic blowup of . ∎
Now we are equipped to prove the Transverse Surface Theorem. We actually will prove a generalization to compact manifolds which might have boundary.
Theorem 3.5 (Almost transverse surfaces).
Let be a compact hyperbolic 3-manifold, with a fibered face of and associated circular pseudo-Anosov suspension flow . Let be an integral homology class. Then if and only if is represented by a surface almost transverse to .
By the discussion following the statement of the Transverse Surface Theorem in Section 3.2, the homology class of any surface almost transverse to lies in . Hence we need only produce a representative of transverse to some dynamic blowup of . Since any integral class in is represented by a cross section, we assume .
Our strategy will be to take a nice representative of which is transverse to and complete it over by gluing in disks and annuli which are also transverse to . Where necessary we dynamically blow up some singular orbits of and glue in annuli which are transverse to the blown up flow.
By Corollary 2.5, has some representative which is carried by , so we can assume lies in transverse to the normal foliation. Since is transverse to , so is .
First, to each boundary component of lying in (see Section 2.5 for notation), we glue an annulus which extends that component of to maintaining transversality to .
Next, let be a singular orbit of whose flow prongs rotates by .
By [Lan18], is either
a collection of meridians of , or
a collection of ladderpole curves which is nulhomologous in .
In case (a) we have nothing to do. In case (b) the curves can be capped off by meridional disks of transverse to .
In case (c), we consider a meridional disk of which is transverse to . The intersection of the flow prongs of with gives a pseudo-Anosov star . Further, we claim is an even family for .
Because is a collection of ladderpole curves nulhomologous in , it consists of equal numbers of left and right ladderpole curves of upward ladders. It follows that is an even family for .
By Lemma 3.4 there exists a dynamic blowup of such that can be -symmetrically filled in by a collection of cooriented line segments over . The tree determines a dynamic blowup of . We can suspend to a family of annuli with boundary that are transverse to (see Figure 20 and caption).
By gluing these annuli to , we eliminate all boundary components of meeting . The coorientations agree along by construction.
By repeating this procedure at every singular orbit of , we obtain a surface which is transverse to a dynamic blowup of . The image of under the puncturing map is evidently . Since is injective, . ∎
4 Homology directions and
Our notation for this section is the same as defined at the beginning of Section 2.5: is a circular pseudo-Anosov flow on a compact 3-manifold , and is the associated fibered face with veering triangulation . By we mean .
4.1 Convex polyhedral cones
We recall some facts about convex polyhedral cones (for a reference see e.g. [Ful93] 1.2). Let be a subset of a finite-dimensional real vector space . Define the dual of to be
A convex polyhedral cone in is the collection of all linear combinations, with nonnegative coefficients, of finitely many vectors. If is a convex polyhedral cone in , then is a convex polyhedral cone in . We have the relation .
A face of is defined to be the intersection of with the kernel of an element in . The dimension of a face of a convex polyhedral cone is the dimension of the vector subspace generated by points in the face. A top-dimensional proper face of a convex polyhedral cone is called a facet of the cone. If is a face of , define
Then is a face of with , and defines a bijection between the faces of and the faces of . We indulge in some foreshadowing by remarking that, in particular, restricts to a bijection between the one-dimensional faces of and the facets of .
We can identify with via the universal coefficients theorem, so if is a convex polyhedral cone in we will view as living in and vice versa.
Recall that is the the 2-skeleton of viewed as a branched surface. Let be a truncated taut tetrahedron of . If is carried with positive weights on both of the sectors of corresponding to the bottom of , then may be isotoped upwards through to a new surface carried by such that the uppermost (with respect to the orientation of the normal foliation of ) portion of which was carried by the bottom of is now carried by the top of . Outside of a neighborhood of this isotopy is the identity. We call this isotopy an upward flip. If is the image of under a single upward flip of , we say is an upward flip of .
4.3 A train track on
Let be a traintrack embedded in the 2-skeleton of with a single trivalent switch lying in the interior of each ideal triangle such that each edge of intersects in a single point. Since each edge of has degree at least 4, this point is a switch of with valence . Note that is nonstandard in 2 ways: it is embedded in the 2-skeleton of rather than a surface, and it is not trivalent.
Let be a triangle of , and let be the switch of interior to . The interior of is divided into 3 disks by , one of which has a cusp at . The branch which is disjoint from the cusped region is called a large branch of . A branch which is not large is called a small branch of .
We now define a particular train track in the 2-skeleton of which we call the stable train track of and denote by ). For each triangle of we place a trivalent switch in the interior of , and connect the large branch to the unique edge of which is the bottom -edge of the taut tetrahedron which bounds below. We connect the two small branches to the other two edges of . Gluing so that intersects each edge of in a point yields our desired train track. This is precisely the train track we would get from Agol’s construction of by fixing a fibration, building as a layered triangulation on the fiber by looking at dual triangulations to a periodic maximal splitting sequence of the monodromy’s stable train track, and recording the switches of stable train tracks on the relevant ideal triangles. We will also view as living in . Figure 1 shows a portion of a veering triangulation with its stable train track.
Any surface carried by naturally inherits a trivalent train track from which we call . There is a natural cellular map , which allows us to identify each branch of with two branches of . This map need be neither surjective nor injective. A branch of composed of 2 large branches of is called a large branch of .
By construction, we have the following.
Let be a surface carried by . There exists an upward flip of if and only if contains a large branch.
Any curve carried by corresponds to a curve carried by under the map , and we will abuse terminology slightly by considering a curve carried by to also be carried by .
4.4 Infinite flippability
A finite or infinite sequence of surfaces carried by such that each successive element is an upward flip of the previous element is called a flipping sequence. Any element of an infinite flipping sequence is called infinitely flippable.
If is a surface carried by a branched surface with positive weights on each sector, we say is fully carried.
We use these new words in a mathematical sentence:
Let be a surface carried by which is a fiber of . Because can be built as a layered triangulation on the extension of to , is infinitely flippable and some positive integer multiple of is fully carried by .
The cone of homology directions of a flow , denoted , is the smallest closed cone containing the projective accumulation points of nearly closed orbits of . Since in our case is a circular pseudo-Anosov flow, there is a more convenient characterization of as the smallest closed, convex cone containing the homology classes of the closed orbits of (see the proof of Lemma A.3 in Appendix A). In fact, it suffices to take the smallest convex cone containing a certain finite collection of closed orbits. It follows that is a rational convex polyhedral cone.
Let be the veering triangulation of . Define a -transversal to be an oriented curve in which is positively transverse to the 2-skeleton of , i.e. intersects the 2-skeleton only transversely and agreeing with the coorientation of . Let be the smallest closed cone containing the homology class of each closed -transversal.
Let be a closed -transversal. Then . Moreover, .
Since both cones are closed, to show it suffices to show that the homology class of every closed orbit lies in , and that the homology class of every closed -transversal lies in .
Suppose is a closed orbit of . If lies interior to , then is already a closed -transversal. Otherwise is a singular or -singular orbit and can be isotoped onto a closed -transversal lying in the interior of a ladder in . Hence .
Now suppose is a closed -transversal. We can isotope into such that is positively transverse to . Let be an integral class, and let be a representative of carried by . Since is a fiber of , by Observation 4.2 there exists a surface (topologically is parallel copies of for some positive integer ) representing which is fully carried by . We can cap off the boundary components of to obtain a surface in representing whose intersection with is . Since is positively transverse to , it has positive intersection with . Letting denote the Lefschetz dual of , we see . Viewing as a linear functional on , we see is strictly positive on , so is nonnegative on , whence .
Note that by Theorem A.7 we have . Hence , so . ∎
A surface carried by is a fiber of if and only if it is infinitely flippable.
One direction of this is just Observation 4.2.
For the other direction, we will show that if a flipping sequence is such that there is a 3-cell of which is not swept across by a flip in the sequence, then the sequence is finite. Therefore if is infinitely flippable, some integer multiple of will be fully carried by . Any fully carried homology class has positive intersection with any closed transversal to and thus represents a fiber by Proposition 4.3 and Theorem A.7.
Let be a flipping sequence. We assume is connected, as otherwise we can apply the following reasoning to the flipping sequence associated to each component of . Let be the union of all 2-cells in every , as well as every 3-cell swept across by an upward flip in the sequence. Suppose . We claim there is some 3-cell of whose bottom -edge lies in and which is not swept across by an upward flip in our sequence.
There is some 3-cell incident to along one of its edges. This edge meets two torus boundary components of ; pick one and call it . Then is a nonempty proper subset of which is simplicial with respect to the triangulation of by flat triangles coming from . To find a 3-cell not swept across by an upward flip whose bottom -edge lies in , it suffices to find a flat triangle in such that and such that has an edge lying in whose coorientation points into . This is equivalent to finding an edge of lying in whose coorientation points into . We will call such an edge an outward pointing edge of .
To find an outward pointing edge of , we use our knowledge of the combinatorics of the triangulation of . If the intersection of with the interior of some ladder is a nonempty proper subset of the ladder, we can find a rung of that ladder which is an outward pointing edge of . Otherwise, contains the interior of at least one ladder. As was observed in [Lan18], any closed curve carried by which is not a ladderpole curve must traverse each ladder of . We conclude that must be a collection of ladderpole curves for all . Any two edges of a ladderpole which share a vertex cannot correspond to a pair of 2-cells of forming the bottom of a 3-cell, because each bottom of a 3-cell intersects in at least one rung. Therefore there are no flips of incident to for any and we conclude that is a collection of ladderpole curves, so every edge of is outward pointing.
Hence, as claimed, we can produce a 3-cell of whose bottom -edge lies in , and thus in for some . (We have actually produced with one of its bottom faces lying in ).
Any surface carried by inherits an ideal triangulation of its interior, and a flipping sequence gives a sequence of diagonal exchanges between ideal triangulations of a reference copy of the surface. Let be a reference copy of . Then