2.1 Veering triangulations

A veering triangulation of a $3$ -manifold M is a taut ideal triangulation together with a coherent assignment of veers to its edges. We begin by explaining each of these terms.

A taut ideal tetrahedron is an ideal tetrahedron (that is, a tetrahedron without vertices) along with a coorientation on each face so that it has two inward pointing faces, called its bottom faces, and two outward pointing faces, called its top faces. Each of its edges is then assigned either angle $\pi $ or $0$ depending on whether the coorientations on the adjacent faces agree or disagree, respectively.

Following Lackenby [Reference LackenbyLac00], an ideal triangulation of M is taut if each of its faces has been cooriented so that each ideal tetrahedron is taut and the angle sum around each edge is $2\pi $ . The local structure around each edge $\textit {e}$ is as follows: $\textit {e}$ includes as a $\pi $ -edge into two tetrahedra. For the other tetrahedra meeting $\textit {e}$ , $\textit {e}$ includes as an $0$ -edge and these tetrahedra are divided into the two sides, called fans of $\textit {e}$ , each of which is linearly ordered by the coorientation on faces. The length of each fan is one less than the degree of e on that side. See Figure 1.

Figure 1 The edge e has one fan of length 1 and one fan of length 3.

A veering triangulation $\tau $ of M is a taut ideal triangulation of M in which each edge has a consistent veer. This means that each edge is labeled to be either right or left veering such that each tetrahedron of $\tau $ admits an orientation-preserving isomorphism to the model veering tetrahedron pictured in Figure 2, in which the veers of the 0-edges are specified: right veering edges have positive slope and left veering edges have negative slope. The $\pi $ -edges can veer either way, as long as adjacent tetrahedra satisfy the same rule. If the $\pi $ -edges of a tetrahedron have opposite veer, the tetrahedron is said to be hinge; otherwise it is non-hinge.

Figure 2 A model veering tetrahedron and its cusps with their coorientations (in colour online).

2.2 The dual graph, flow graph, and stable branched surface

The stable branched surface $B^s$ in M associated to the veering triangulation $\tau $ , introduced in [Reference Schleimer and SegermanSS19] as the upper branched surface in dual position and in [Reference Landry, Minsky and TaylorLMT20, §4], plays a central role throughout this paper. We refer the reader to [Reference Floyd and OertelFO84, Reference OertelOer84] for general facts about branched surfaces.

Topologically, the stable branched surface $B^s$ is the dual complex of $\tau $ in M, and as such, it is a deformation retract of M. Note that $B^s$ is two-dimensional since $\tau $ has no vertices. For each tetrahedron t, we define a smooth structure on $B^s_t = B^s \cap t$ as follows: if the top edge of t is left veering, then we smooth according to the left-hand side of Figure 3 and otherwise we smooth according the the right-hand side. It is proven in [Reference Landry, Minsky and TaylorLMT20, Lemma 4.3] that this produces a well-defined global smooth structure making $B^s$ into a branched surface.

Figure 3 The stable branched surface $B^s$ within a single tetrahedron. Its intersection with the flow graph, where edges are directed upward, is shown in green (note: the number of incoming edges at a vertex may vary) (in colour online).

The stable branched surface contains two directed graphs related to $\tau $ that are also of central importance. The first, is the dual graph $\Gamma $ of $\tau $ which is defined to be the $1$ -skeleton of $B^s$ whose edges are directed by the coorientation on the faces of $\tau $ . Alternatively, $\Gamma $ is the graph with a vertex interior to each tetrahedron and a directed edge crossing each cooriented face from the vertex in the tetrahedron below the face to the vertex in the tetrahedron above the face. See Figure 3. The directed cycles of $\Gamma $ are called dual cycles or $\Gamma $ -cycles. Here and throughout, a directed cycle of a directed graph is an oriented loop determined by a cyclic concatenation of directed edges.

As the $1$ -skeleton of the branched surface $B^s$ , each turn in the graph $\Gamma $ is either branching, that is, realized by a smooth arc in $B^s$ , or else what we call anti-branching (or AB). In greater detail, a turn of $\Gamma $ is an ordered pair $(e_1,e_2)$ of directed $\Gamma $ -edges so that the terminal vertex v of $e_1$ equals the initial vertex of $e_2$ . The turn is branching if the arc $e_1 \cup e_2$ is smooth as an arc in the singular locus of $B^s$ and is anti-branching (or AB) otherwise. A directed path, ray, or cycle in $\Gamma $ that makes only branching turns is called a branch path, ray, or cycle, respectively. Similarly, a directed path, ray, or cycle in $\Gamma $ that makes only AB turns is called an AB path, ray, or cycle. We note that since for each vertex of $\Gamma $ each incoming edge is part of exactly one branching turn and one AB turn, there are only finitely many branch and AB cycles in $\Gamma $ .

Branching and anti-branching turns of $\Gamma $ can be characterized solely in terms of the veering combinatorics [Reference Landry, Minsky and TaylorLMT20, Lemma 4.5] and from this we can deduce a few important properties of the sectors of $B^s$ .

Each sector $\sigma $ of $B^s$ is a topological disk pierced by a single $\tau $ -edge, as in Figure 4. The $\Gamma $ -edges bounding $\sigma $ are oriented so that exactly one vertex is a source, which we call the bottom of $\sigma $ , and one is a sink, which we call the top of $\sigma $ . The top and bottom divide the boundary of $\sigma $ into two oriented $\Gamma $ -paths called sides. Each side has at least two $\Gamma $ -edges because the $\tau $ -edge piercing $\sigma $ has a non-empty fan on each side. According to the following lemma, which appears as [Reference Landry, Minsky and TaylorLMT20, Lemma 4.6], if you remove the last edge in any side of any sector of $B^s$ , the resulting path is a branch segment, and the entire side is never a branch segment. See Figure 4, where the AB turns appear as corners of the sector. We call these vertices the corner vertices of the sector.

Figure 4 A sector of $B^s$ and its intersection with $\Phi $ (green) and $\tau ^{(2)}$ (gray). The triple points of $B^s$ , which are the vertices of $\Gamma $ and $\Phi $ , are in black. All edges are directed upward; the top is the northmost vertex, the bottom is the southmost, and the corners are westmost/eastmost (in colour online).

The second directed graph embedded in $B^s$ is the flow graph $\Phi $ of $\tau $ , which was introduced in [Reference Landry, Minsky and TaylorLMT20, §4.3]. The vertices of $\Phi $ are in correspondence with $\tau $ -edges, and for each tetrahedron t of $\tau $ , there are $\Phi $ -edges from the bottom $\tau $ -edge of each tetrahedron to its top $\tau $ -edge and the two side $\tau $ -edges whose veer is opposite that of the top $\tau $ -edge.

This defines $\Phi $ as an abstract directed graph, but it also comes equipped with an embedding $\iota : \Phi \to B^s$ , which was called dual position in [Reference Landry, Minsky and TaylorLMT20]. Each $\tau $ -edge e is at the bottom of a unique tetrahedron $t_e$ and $\iota $ maps the vertex of $\Phi $ corresponding to e to the vertex of $\Gamma $ contained in $t_e$ . Each directed edge of $\Phi $ is then mapped into a single sector of $B^s$ so that it is positively transverse to $\tau ^{(2)}$ . See Figure 3. According to [Reference Landry, Minsky and TaylorLMT20, Lemma 4.7], for each sector $\sigma $ of $B^s$ , there is a directed edge of $\iota (\Phi ) \cap \sigma $ coming into the top vertex of $\sigma $ from each vertex of $\sigma $ other than its two corner vertices. See Figure 4. This characterizes the flow graph in dual position according to its intersection with each sector of $B^s$ .

The directed cycles of $\Phi $ , along with their images under $\iota $ , are called flow cycles or $\Phi $ -cycles. When convenient, we sometimes identify $\Phi $ with its image under $\iota $ .

2.3 The veering polynomial

Fix a finitely generated, free abelian group G and denote its group ring with integer coefficients by $\mathbb {Z}[G]$ . Let $P \in \mathbb {Z}[G]$ and write $P= \sum _{g\in G} a_g \cdot g$ . The support of P is

$$ \begin{align*} \mathrm{supp}(P) = \{g \in G : a_g \neq 0 \}. \end{align*} $$

For $P \in \mathbb {Z}[G]$ with $P = \sum _{g\in G} a_g \cdot g$ and $\alpha \in \mathrm {Hom}(G,\mathbb {R})$ , the specialization of P at $\alpha $ is the single variable expression $P^{\alpha }$ in $\mathbb {Z}[u^r : r \in \mathbb {R}]$ given by

$$ \begin{align*} P^{\alpha}(u) = \sum_{g\in G} a_g \cdot u^{\alpha(g)}. \end{align*} $$

These generalities will be used in the specific setting of veering polynomials. For this, let M be a $3$ -manifold with veering triangulation $\tau $ , and set $G = H_1(M;\mathbb {Z})/\mathrm {torsion}$ . In [Reference Landry, Minsky and TaylorLMT20, §2], we defined a polynomial invariant $V_{\tau } \in \mathbb {Z}[G]$ , called the veering polynomial of $\tau $ . Here, we recall an alternative characterization of $V_{\tau }$ in terms of the Perron polynomial of the flow graph $\Phi $ . We refer the reader to [Reference Landry, Minsky and TaylorLMT20, §4] for additional details.

For a directed graph D, let A denote the matrix with entries

(2.1) $$ \begin{align} A_{ab} = \sum_{\partial e=b-a} e, \end{align} $$

where the sum is over all edges e from the vertex a to the vertex b. We call A the adjacency matrix for D. The Perron polynomial of D is defined to be $P_D = \det (I - A)$ . By definition, this is an element of $\mathbb {Z}[C_1(D)]$ , where $C_1(D)$ is the group of simplicial 1-chains in D.

Following McMullen [Reference McMullenMcM15], we define the cycle complex $\mathcal {C}(D)$ of D to be the graph whose vertices are directed simple cycles of D and whose edges correspond to disjoint cycles. We recall that $P_D$ equals the clique polynomial of $\mathcal {C}(D)$ , which in particular shows that $P_D$ is an element of the subring $\mathbb {Z}[H_1(D)]$ (see [Reference McMullenMcM15, Theorem 1.4 and §3]). Here, the clique polynomial associated to $\mathcal {C}(D)$ is

(2.2) $$ \begin{align} P_D = 1 + \sum_{C} (-1)^{|C|} C \in \mathbb{Z}[H_1(D)], \end{align} $$

where the sum is over non-empty cliques C of the graph $\mathcal {C}(D)$ , that is, over simple multicycles of D, and $|C|$ is the number of vertices of C, that is, the number of components of the multicycle. Note that the support of P is the set $\mathrm {supp}(P_D) = \{ C\} \subset H_1(D)$ of directed simple multicycles appearing in equation (2.2).

Now let $\iota : \Phi \to M$ be the flow graph with its embedding into M. This induces a ring homomorphism $\iota _*: \mathbb {Z}[H_1(\Phi )] \to \mathbb {Z}[G]$ and we set

$$ \begin{align*} V_{\tau} = \iota_*(P_{\Phi}), \end{align*} $$

where $P_{\Phi }$ is the Perron polynomial of $\Phi $ . According to [Reference Landry, Minsky and TaylorLMT20, Theorem 4.8], this agrees with the original definition of the veering polynomial.

2.4 Surfaces carried by $\tau $ and cones in (co)homology

As noted by Lackenby [Reference LackenbyLac00], tautness of $\tau $ naturally gives its $2$ -skeleton $\tau ^{(2)}$ the structure of a transversely oriented branched surface in M. The smooth structure on $\tau ^{(2)}$ can be obtained by, within each tetrahedron, smoothing along the $\pi $ -edges and pinching along the $0$ -edges, thus giving $\tau ^{(2)}$ a well-defined tangent plane field at each of its points.

As a transversely oriented branched surface, $\tau ^{(2)}$ can carry surfaces similarly to the way a train track on a surface can carry curves. We let $\mathrm {cone}_2(\tau )$ be the closed cone in $H_2(M,\partial M)$ positively generated by classes that are represented by the surfaces that $\tau $ carries. We call $\mathrm {cone}_2(\tau )$ the cone of carried classes.

In a bit more detail, the branched surface $\tau ^{(2)}$ has a branched surface fibered neighborhood $N= N(\tau ^{(2)})$ foliated by intervals such that collapsing N along its I-fibers recovers $\tau ^{(2)}$ . The transverse orientation on the faces of $\tau $ orients the fibers of N, and a properly embedded oriented surface S in M is carried by $\tau ^{(2)}$ if it is contained in N where it is positively transverse to its I-fibers. We also say that S is carried by $\tau $ .

A carried surface S embedded in N transverse to the fibers defines a non-negative integral weight on each face of $\tau $ given by the number of times the I-fibers over that face intersect S. These weights satisfy the matching (or switch) conditions stating that the sum of weights on one side of an edge match the sum of weights on the other side. Conversely, a collection of non-negative integral weights satisfying the matching conditions gives rise to a surface embedded in N transverse to the fibers in the usual way. More generally, any collection of non-negative weights on faces of $\tau $ satisfying the matching conditions defines a non-negative relative cycle giving an element of $H_2(M, \partial M; \mathbb {R})$ and we say that a class is carried by $\tau ^{(2)}$ if it can be realized by such a non-negative cycle. Hence, $\mathrm {cone}_2(\tau )$ is precisely the subset of $H_2(M,\partial M)$ consisting of carried classes.

The following theorem is a summary of results in [Reference Landry, Minsky and TaylorLMT20, Theorems 5.1 and 5.12]. For its statement, we let $\mathrm {cone}_1(\Gamma ) \subset H_1(M; \mathbb {R})$ denote the cone positively spanned by the direct cycles of the dual graph $\Gamma $ . We call $\mathrm {cone}_1(\Gamma )$ the cone of homology directions of $\tau $ and note that it is equal to the cone positively generated by all closed curves which are positively transverse to $\tau ^{(2)}$ at each point of intersection. We write $\mathrm {cone}_1^{\vee }(\Gamma )$ for its dual cone in $H^1(M; \mathbb {R})$ , which consists of classes that are non-negative on all dual cycles.

So, for example, a class $\alpha \in H_2(M,\partial M)$ is carried by $\tau $ if and only if $\langle \alpha , \iota (c) \rangle \ge 0$ for each simple directed cycle c of $\Phi $ .

3.1 Descending sets and dynamic planes

For any branched surface B, let $N(B)$ denote a regular neighborhood of B foliated in the standard way by intervals. Let

$$ \begin{align*} \operatorname{\mathrm{coll}}: N(B)\twoheadrightarrow B \end{align*} $$

be the map which collapses all the intervals.

If B is a branched surface with generic branching, then we denote its branch locus, that is, its collection of non-manifold points, by $\operatorname {\mathrm {brloc}} (B)$ . The maw vector field is a vector field tangent to B defined on $\operatorname {\mathrm {brloc}} (B)$ that always points from the 2-sheeted side to the 1-sheeted side. Note that the maw vector field is defined even at triple points; see Figure 5.

Figure 5 The maw vector field (in colour online).

A descending path in B is an oriented immersed curve in B whose tangent vector at each point of intersection with $\operatorname {\mathrm {brloc}}(B)$ is equal to the maw vector field at that point.

We next consider the stable branched surface $B^s$ . Note that, up to homotopy, any closed descending path in $B^s$ is negatively transverse to $\tau ^{(2)}$ (see Figure 4), and is therefore homotopically non-trivial in M [Reference Schleimer and SegermanSS20, Theorem 3.2]. Let $\widetilde B^s$ , $\widetilde \Gamma $ , and $\widetilde \Phi $ be the preimages of $B^s$ , $\Gamma $ , and $\Phi $ , respectively, in the universal cover $\widetilde M$ of M.

Let $\sigma $ be a sector of the branched surface $\widetilde B^s$ . The descending set of $\sigma $ , denoted $\Delta (\sigma )$ , is defined to be the union of all sectors $\sigma '$ of $\widetilde B^s$ such that there exists a descending path from $\sigma $ to $\sigma '$ . Before describing $\Delta (\sigma )$ in detail, recall that by a path, ray, or line in $\Gamma $ or $\Phi $ , we always mean a directed path, ray, or line. If $\ell $ is a branch line in $\widetilde \Gamma $ through a vertex v, then the negative subray of $\ell $ at v is the portion of the branch line $\ell $ that lies below v.

Before the proof, let us establish a few facts that we will need. By [Reference Schleimer and SegermanSS19, Theorem 8.1], the branched surface $B^s$ fully carries a unique two-dimensional lamination $\mathcal L^s$ without parallel leaves such that $\mathcal L$ is essential and each leaf of $\mathcal L^s$ is either a plane, an $\pi _1$ -injective annulus, or a $\pi _1$ -injective Möbius band. Denote by $\widetilde {\mathcal L}^s$ the lamination lifted to the universal cover $\widetilde M$ whose leaves are planes. Note that since $\widetilde {\mathcal L}^s$ is carried by $\widetilde B^s$ , each leaf inherits a tessellation corresponding to the sectors of $\widetilde B^s$ it traverses.

It is clear from the branching structure of $B^s$ (cf. [Reference Schleimer and SegermanSS19, Remark 8.27]) that if $\ell $ is a leaf carried by $\widetilde B^s$ such that $\operatorname {\mathrm {coll}}(\ell )$ contains $\sigma $ , and if there is a descending path from $\sigma $ to another sector $\sigma '$ , then $\sigma '$ is also contained in $\operatorname {\mathrm {coll}}(\ell )$ . Consequently, we have that $\operatorname {\mathrm {coll}}(\ell )$ contains the descending set of every sector traversed by $\ell $ .

We also observe that each leaf of $\widetilde {\mathcal L}^s$ traverses a sector of $\widetilde B^s$ at most once. For if $\ell $ is a leaf traversing a sector $\sigma $ twice, a short segment contained in a regular neighborhood of $\sigma $ connecting two points of $\ell $ identified under $\operatorname {\mathrm {coll}}$ may be homotoped to lie entirely in $\ell $ . Since the branched surface $B^s$ is laminar (as observed in [Reference Schleimer and SegermanSS19]) and hence essential, this contradicts [Reference Gabai and OertelGO89, Theorem 1.d] (see also [Reference Gabai and OertelGO89, Lemma 2.7]). We conclude that for any leaf $\ell $ of $\widetilde {\mathcal L}^s$ , $\operatorname {\mathrm {coll}}(\ell )$ is a plane embedded in $\widetilde B^s\kern-1.2pt $ .

Proof of Lemma 3.1

We begin by using the above discussion to prove part (a). Let $\ell $ be any leaf of $\widetilde {\mathcal L}^s$ that traverses $\sigma $ . Then $P = \operatorname {\mathrm {coll}}(\ell )$ is a plane tessellated by sectors of $\widetilde B^s$ that contains the descending set $\Delta (\sigma )$ . From the local picture of P around vertices of $\widetilde \Gamma $ shown in Figure 7, we see that for each vertex w of P, P contains the negative subrays of both branch lines through w. So if v is the vertex at the top of $\sigma $ , then the branch lines through v are proper lines contained in P and determine a quarter plane Q as in the statement of part (a). Hence, it suffices to show that $Q = \Delta (\sigma )$ .

Figure 6 The descending set of a sector $\sigma $ , and part of its intersection with $\widetilde \Gamma $ (in colour online).

Figure 7 Local pictures of vertices in Q and incident $\widetilde \Gamma $ -edges (red) and $\widetilde \Phi $ -edges (green) in the proof of Lemma 3.1. All edges are directed upward. Note that there are always incoming $\widetilde \Phi $ -edges (the number may vary) and a unique outgoing $\widetilde \Phi $ -edge (in colour online).

Clearly, $\Delta (\sigma ) \subset Q$ , since no descending paths starting at $\sigma $ can cross the branch lines through v.

For the reverse containment, let $S_n$ denote the set of $\widetilde B^s$ -sectors reachable from $\sigma $ by a descending path traversing at most n sectors. Then $\sigma = S_1\subset S_2\subset S_3\subset \cdots $ is an exhaustion of $\Delta (\sigma )$ .

Claim 3.2. If w is a vertex in the boundary of $S_n$ , then either w is in the interior of $S_{n+1}$ or it lies on one of the two branch lines through v and hence on the boundary of Q.

Proof of claim

The proof is by induction with the case of $S_1 = \sigma $ being by inspection (see Figure 6).

Now suppose that w is in the boundary of both $S_n$ and $S_{n+1}$ . By the inductive hypothesis, we may assume that w is a vertex of a sector $\sigma ' \subset S_n \smallsetminus S_{n-1}$ . In particular, w is not the top vertex of $\sigma '$ . If w is not joined by a $\widetilde \Gamma $ -edge to the top of $\sigma '$ , then again it is clear from the picture (Figure 6) that w is in the interior of $S_{n+1}$ , which contradicts our assumption.

Otherwise, w is joined by an edge e to the top vertex $w'$ of $\sigma '$ and we say that w is one of the two side vertices of $\sigma '$ . Note that e is in the boundary of $S_n$ , since otherwise we would again have that w is in the interior of $S_{n+1}$ .

It suffices to show that e is an edge of a branch line through the vertex v. Note that $w'$ is a vertex of some sector $\sigma "$ in $S_{n-1}$ since any descending path from $\sigma $ to $\sigma '$ passes through one of the two top edges of $\sigma '$ . Since w is not in the interior of $S_{n+1}$ , we must have that $\sigma '$ is attached to $\sigma "$ along the edge $e'$ , where $e'$ is the $\widetilde \Gamma $ -edge at the top of $\sigma '$ that is not e. By the induction hypothesis, either $w'$ is in the interior of $S_{n}$ or $w'$ is contained in a branch line through v. However, if $w'$ is in the interior of $S_n$ , then e is also in the interior of $S_n$ , which is a contradiction. Hence, we must have that $w'$ lies along a branch line though v. Since $e'$ is in the interior of $S_n$ , this branch line continues along e, establishing that it contains w. This completes the proof of Claim 3.2.

We conclude that $\Delta (\sigma )$ is a subcomplex of the quarter plane Q (with its locally finite tessellation by sectors) and that $\partial \Delta (\sigma ) = \partial Q$ . It follows easily that $Q = \Delta (\sigma )$ as required.

For part (b), again let $S_n$ denote the set of $\widetilde B^s$ -sectors reachable from $\sigma $ by a descending path traversing at most n sectors. If w is a vertex of $S_n$ , then any $\widetilde \Gamma $ -path in $\Delta (\sigma )$ remains within $S_n$ . Since $S_n$ has finitely many vertices and $\widetilde \Gamma $ -rays are simple, each $\widetilde \Gamma $ -ray starting at w eventually meets $\partial \Delta (\sigma )$ . This proves part (b).

Considering a picture makes the first claim of part (c) clear; see Figure 7.

The same argument as that for part (b) shows that the $\widetilde \Phi $ -ray starting from any point in $\Delta (\sigma )$ must meet $\partial \Delta (\sigma )$ . This completes the proof of Lemma 3.1.

Next we describe and analyze a canonical set associated to a $\widetilde \Gamma $ -ray. For a vertex v or directed edge e of $\widetilde \Gamma $ , we set $\sigma (v)$ and $\sigma (e)$ to be the sector into which the maw vector field points at v or along the interior of e, respectively. So if v is the terminal vertex of the edge e and $\sigma $ is the unique sector of $\widetilde B^s$ whose top vertex is v, then $\sigma (v)= \sigma (e) = \sigma $ .

Remark 3.4. The descending path that the proof of Lemma 3.3 produces can be obtained by pushing the dual path from u to v slightly in the direction of the maw vector field and reversing orientation, as shown in Figure 8.

Figure 8 If there is a path in $\widetilde \Gamma $ from u to v (shown in black), then there is a descending path in $\widetilde B^s$ from $\sigma (v)$ to $\sigma (u)$ (shown in blue) (in colour online).

Let $\gamma $ be a $\widetilde \Gamma $ -ray, and set

$$ \begin{align*} D(\gamma)=\bigcup_{v \in\gamma}\Delta(\sigma(v)), \end{align*} $$

where the unions are taken over all $\widetilde \Gamma $ -vertices v traversed by $\gamma $ . By Lemma 3.3, it follows that $D(\gamma )$ is a nested union of quarter planes. If $\gamma $ is not eventually a branch ray of $\widetilde \Gamma $ , then $D(\gamma )$ is evidently diffeomorphic to $\mathbb {R}^2$ and we say that $D(\gamma )$ is the dynamic plane associated to $\gamma $ . By construction, $D(\gamma )$ is properly embedded in $\widetilde B^s$ . If $\gamma $ is eventually a branch ray, then $D(\gamma )$ is diffeomorphic to a half plane and we say that $D(\gamma )$ is the dynamic half plane associated to $\gamma $ .

We remark that any dynamic plane D is tessellated by sectors of $\widetilde B^s$ and hence it makes sense to speak of $\widetilde \Gamma $ -paths or $\widetilde \Phi $ -paths in D. Moreover, $\widetilde \Gamma \cup \widetilde \Phi $ determine a triangulation of D.

Remark 3.5. Since $\sigma (v) = \sigma (e)$ for an edge e of $\widetilde \Gamma $ with terminal vertex v, we also have

$$ \begin{align*} D(\gamma) = \bigcup_{e \in\gamma}\Delta(\sigma(e)), \end{align*} $$

where the union is over edges traversed by $\gamma $ .

Recall that if $\gamma $ is a $\widetilde \Gamma $ -path or ray contained in D which makes only AB turns, we say $\gamma $ is an AB path or AB ray. Then each two consecutive $\widetilde \Gamma $ -edges of $\gamma $ determine a triangle in the triangulation of D by edges of $\widetilde \Phi $ and $\widetilde \Gamma $ , and the third edge of this triangle is a $\widetilde \Phi $ -edge. The union of all these triangles is a subset S of D diffeomorphic to $[0,1]\times [0,1]$ or $[0,1]\times [0,\infty )$ called an AB strip or infinite AB strip, respectively. See Figure 9.

Figure 9 An AB strip (left) and an infinite AB strip (right), with $\widetilde \Gamma $ -edges in red and $\widetilde \Phi $ -edges in green. Edges are directed upward (in colour online).

The following lemma essentially says that $\widetilde \Phi $ -rays in a dynamic plane either converge or are separated by AB strips.

Proof. Suppose that $D = D(\gamma )$ , where $\gamma $ is a $\widetilde \Gamma $ -ray which is not eventually a branch ray, and let $p_1,p_2,\ldots $ be the sequence of vertices where $\gamma $ makes AB turns. Define $\Delta _i=\Delta (\sigma (p_i))$ , so that $\Delta _1\subset \Delta _2\subset \Delta _3\subset \cdots $ is the exhaustion of $D(\gamma )$ by descending sets. Let a and b be vertices of $\alpha $ and $\beta $ , respectively. By truncating and reindexing the exhaustion $\{\Delta _i\}$ , we can assume that a and b lie in $\Delta _1$ . Hence by Lemma 3.1(b), there exist vertices $a_1$ of $\alpha $ and $b_1$ of $\beta $ lying on the boundary of $\Delta _1$ , and $a_1$ and $b_1$ are a finite distance apart in the combinatorial metric on $\partial \Delta _1$ .

Let $q_1,r_1\in \partial \Delta _1$ be vertices that are connected by an edge from $q_1$ to $r_1$ with $\widetilde \Phi $ -rays intersecting $\partial \Delta _2$ in $q_2$ and $r_2$ respectively. We claim that $ q_2\ne r_2$ if and only if the corresponding ray segments cobound an AB-strip (see Figure 10). Indeed, if s is the sector in D above the $\widetilde \Gamma $ -edge connecting $q_1$ and $r_1$ , then the $\widetilde \Phi $ -rays from $q_1$ and $r_1$ immediately converge unless $r_1$ is a corner vertex of s as in Figure 11. Applying this analysis repeatedly proves the claim. In other words, ‘flowing’ forwards in D weakly contracts distance in $\partial \Delta _i$ , with equality if and only if the flow segments are separated by a union of AB strips. This implies that either the rays from $a_1$ and $b_1$ eventually coincide, or they both eventually meet AB rays.

Figure 10 AB strips are precisely the obstruction to contraction under ‘flowing’ forward in a dynamic plane (in colour online).

Figure 11 The flow rays from $q_1$ and $r_1$ immediately collide (left) unless $r_1$ is a corner vertex of the sector s above $q_1$ (right) (in colour online).

We next require a basic lemma about the structure of $\widetilde B^s$ . Recall that each $\widetilde B^s$ -sector has two sides, a top vertex and bottom vertex, and two corner vertices. Each side of a sector is composed of two branch segments, one which begins at the bottom vertex and terminates at a corner vertex, and one which consists of precisely one $\widetilde \Gamma $ -edge which begins at the corner vertex and terminates at the top vertex. Each of these vertices corresponds to a triple point of $\widetilde B^s$ , and has a right or left veer as shown in Figure 12. This veer agrees with the veer of the edge atop the unique tetrahedron containing the triple point (compare with Figure 3).

Figure 12 Each triple point in $B^s$ comes with a veer (in colour online).

In this language, we have the following lemma, which is a reformulation of [Reference Landry, Minsky and TaylorLMT20, Fact 1]. For an illustration of the behavior described in the lemma, see Figure 13.

Figure 13 Some situations allowed by Lemma 3.8, where x and y denote opposite veers. Note that the veers of top vertices are not governed by the lemma (in colour online).

Recall from above that an infinite AB strip is a subset of a dynamic plane homeomorphic to $[0,1]\times [0,\infty )$ , determined by an AB ray. The subsets of an infinite AB strip corresponding to $\{0,1\}\times [0,\infty )$ are $\widetilde \Phi $ -rays. Similarly, we define a bi-infinite AB strip to be a subset of a dynamic plane homeomorphic to $[0,1]\times \mathbb {R}$ determined by an AB line. The boundary components of a bi-infinite AB strip are $\widetilde \Phi $ -lines.

Recall that each $\widetilde \tau $ -edge e has two fans, consisting of the tetrahedra for which e is a $0$ -edge lying on a particular side of e. The length of a fan in $\widetilde \tau $ is the number of tetrahedra it contains. We can also define fans and their lengths for edges of $\tau $ by lifting to $\widetilde \tau $ .

We denote the length of the longest fan in $\tau $ by $\delta _{\tau }$ .

The union $D_{\text {AB}}$ of all bi-infinite AB strips in D, as in Proposition 3.10, will be called the AB region of D.

Proof. By Lemma 3.7, if D contains n or more infinite AB strips, then there is a subset S of D that is obtained by gluing n infinite AB strips along their $[0,\infty )$ boundaries. See Figure 14. By Lemma 3.8, all the vertices in the interior of S have identical veer, so if v is a vertex in the interior of S, then v lives in a tetrahedron whose top and bottom edges have the same veer. In other words, every vertex in the interior of S lies in a non-hinge tetrahedron. Any branch line traversing S therefore must pass through $n-1$ consecutive non-hinge tetrahedra.

Figure 14 A picture of the set S in the proof of Proposition 3.10. The $\widetilde \Phi $ -edges are shown in green. By Lemma 3.8, each vertex in the interior of S has the same veer. The highlighted branching segment passes through five consecutive non-hinge tetrahedra, corresponding to the vertices colored black (in colour online).

We now need a combinatorial fact about veering triangulations.

Claim 3.11. If a branch line $\gamma $ passes through consecutive non-hinge tetrahedra $T_1,\ldots , T_k$ , then the $T_i$ all lie in the fan of a single edge e.

Proof. Suppose without loss of generality that the top and bottom edges of $T_1$ are right veering. The branching line $\gamma $ passes through two faces of $T_1$ which meet along a left veering edge E of $T_1$ by [Reference Landry, Minsky and TaylorLMT20, Lemma 4.5]. We call these faces $f_1$ and $f_2$ , where $f_1$ is a bottom face for $T_1$ and $f_2$ is a top face for $T_1$ . If $f_3$ is the next face passed through by $\gamma $ , [Reference Landry, Minsky and TaylorLMT20, Lemma 4.5] again implies that $f_2$ and $f_3$ meet along a left veering edge of $T_2$ . Since e is the only left veering edge of $f_2$ , we conclude that $f_1, f_2, f_3$ are all incident to e, and that $T_1$ and $T_2$ lie in the same fan of e. Continuing in this way shows that each $T_i$ is in this fan. This completes the proof of Claim 3.11.

Returning to the proof of Proposition 3.10: since any fan of length $>1$ containing non-hinge tetrahedra contains two distinct hinge tetrahedra and any non-hinge tetrahedron is part of such a fan (see e.g. [Reference Futer and GuéritaudFG13, Observation 2.6]), Claim 3.11 implies that there exists a fan of $\tau $ with size at least $n+1$ . Hence, the number of bi-infinite AB strips in D is less than the length $\delta _{\tau }$ of the longest fan in $\tau $ , proving item (i).

Now suppose that D contains exactly n bi-infinite AB strips, and let g be the generator of the stabilizer of D. Let s and $s'$ be two adjacent such strips in the sense that there are no strips between them. It is clear that g permutes the set of bi-infinite AB strips in D; let $g'$ be a power of g preserving s and $s'$ . By Lemma 3.7, two of the boundary $\widetilde \Phi $ -lines of s and $s'$ eventually coincide. Since $s\cup s'$ is g-invariant, these boundary $\widetilde \Phi $ -lines must be equal. Applying this argument $n-1$ times establishes item (ii).

We say that two $\widetilde \Phi $ -rays are asymptotic if they eventually agree. It is clear that asymptoticity is an equivalence relation on $\widetilde \Phi $ -rays. We define the width of a dynamic plane D, denoted $w(D)$ , to be the number of asymptotic classes of $\Phi $ -rays contained in D. Lemma 3.7 and Proposition 3.10 imply the following.

Corollary 3.13. (Width of dynamic planes)

Let D be a dynamic plane. The width $w(D)$ of D satisfies

$$ \begin{align*} w(D)&=1+(\# \text{ of bi-infinite AB strips in } D)\\ &\le \delta_{\tau}. \end{align*} $$

The following lemma characterizes when the quotient of a dynamic plane is an annulus in terms of the veering combinatorics.

The following proposition is a key technical result of this section.

In particular, the dual cycles that are not homotopic to flow cycles form a finite set of homotopy classes, up to positive multiples.

3.2 Homotopy in dynamic planes

We conclude this section with an additional fact, Lemma 3.17, about dynamic planes that will be necessary in §6. In its proof, we will use the following lemma.

As a consequence, a path in $\widetilde \Gamma $ that deviates from a branch line can never return to that branch line.

Let D be a dynamic plane stabilized by some $g \in \pi _1(M)$ . As before, let L denote the quotient $D/ \langle g \rangle $ . Consider a $\Gamma $ -cycle $\gamma $ contained in L. If there is a sector $\sigma $ of L such that $\gamma $ runs along a side of $\sigma $ from its bottom vertex to its top vertex, then we may perform a homotopy of $\gamma $ , supported on $\sigma $ , that pushes $\gamma $ from one side of $\sigma $ to the other side of $\sigma $ . We refer to this homotopy as sweeping across the sector $\sigma $ . See Figure 27, where it is shown that sweeping across a sector is a homotopy through curves that are transverse to $\tau ^{(2)}$ .

Proof. The embedded dual cycles $\gamma _1, \gamma _2$ in L either intersect or not. If they intersect and are distinct, there is at least one connected component U of $L-(\gamma _1\cup \gamma _2)$ with closure homeomorphic to a disk. Let $p_1$ and $p_2$ be the two segments of $\gamma _1$ and $\gamma _2$ which cobound U. By Lemma 3.16, each of $p_1$ and $p_2$ contains an anti-branching turn.

Let $(d_1, d_2)$ be the first anti-branching turn of, say, $p_1$ . Since U is tiled by sectors, it must be the case that $p_1$ traverses an entire side of $\sigma (d_2)$ . When we sweep $\gamma _1$ across $\sigma (d_2)$ , we shrink the region U by $1$ sector. It follows that after sweeping across finitely many sectors, we can homotope $\gamma _1$ to $\gamma _2$ .

Next, suppose that $\gamma _1$ and $\gamma _2$ do not intersect. Note that this is not possible if L is a Möbius band and so we may assume that L is an annulus. Then there is a unique component of $L-(\gamma _1\cup \gamma _2)$ with compact closure, which we also call U; note that U is an annulus with boundary components $\gamma _1$ and $\gamma _2$ .

Let $(e_1,e_2)$ be an AB turn of $\gamma _2$ such that $\sigma (e_2)\subset U$ , that is, the maw vector field points into U along $e_2$ . Such a turn exists since $\gamma _2$ has a non-zero even number of anti-branching turns by Lemma 3.14. Let $\ell $ be the branch line through $e_1$ . We claim that $\ell $ intersects $\gamma _1$ in addition to $\gamma _2$ . To see this, first note that the negative subray of $\ell $ from $e_2$ (which is entirely contained in L) is not entirely contained in U, since U is tiled by finitely many sectors. Further, this negative subray cannot return to $\gamma _2$ by Lemma 3.16.

Therefore, the negative subray of $\ell $ from $e_1$ must intersect $\gamma _1$ , as in Figure 15. Let p be the first vertex of intersection between this negative ray and $\gamma _1$ . Let $(f_1,f_2)$ be the first anti-branching turn of $\gamma _1$ after p, and let $\sigma =\sigma (f_2)$ . If $\ell '$ is the branching line through $f_1$ , then the bottom of $\sigma $ can lie no lower on $\ell '$ than p. Hence, $\gamma _1$ traverses the entire left side of $\sigma $ and can be homotoped across $\sigma $ , shrinking the size of U by one sector. After applying this argument finitely many times, we are finished.

Figure 15 Notation from the proof of Lemma 3.17 (in colour online).

4.1 The Agol–Guéritaud construction

Let $\varphi $ be a pseudo-Anosov flow on $\overline M$ with no perfect fits. Here we briefly describe the Agol–Guéritaud construction of a veering triangulation on a manifold homeomorphic to M. We will not dwell on the details here since in the next section, we establish the stronger fact that the veering triangulation can be realized on M so that it is positively transverse to flow lines.

Associate to each maximal rectangle R in the completed flow space $\mathcal P$ a taut ideal tetrahedron $t_R$ . We identify the ideal vertices of $t_R$ with the singularities of R so that the edge rectangles contained in R correspond to edges of $t_R$ and faces rectangles correspond to faces of $t_R$ . The two angle $\pi $ edges of $t_R$ are the ones that correspond to edge rectangles spanning the singularities in $\partial _h R$ and $\partial _v R$ , respectively. Moreover, the coorientations on the faces of $t_R$ are determined by declaring that the two bottom faces of $t_R$ are the ones which contain the $\pi $ edge spanning the singularities in $\partial _v R$ . This convention is indicated in Figure 19 by drawing the edge joining the singularities in $\partial _h R$ above the edge joining the singularities in $\partial _v R$ .

Figure 19 From a maximal rectangle to an ideal tetrahedron. The coorientation convention is indicated by drawing the tetrahedron in the flow space as shown and coorienting its faces to point out of the page. The ideal edges are curvy so as to emphasize that there is no canonical way to draw them.

If faces $f_1$ of $t_{R_1}$ and $f_2$ of $t_{R_2}$ determine the same face rectangle in $\mathcal P$ (that is, the rectangles spanned by their vertices are equal), then we glue together the corresponding faces. Since each face rectangle is contained in exactly two maximal rectangles, the resulting space $\widetilde N$ is a manifold away from its $1$ -skeleton. By examining the ways that an edge rectangle can be extended to a maximal rectangle, one similarly verifies that $\widetilde N$ is a manifold. It is also the case that $\widetilde N$ is contractible.

Since the action of $\pi _1(M)$ on $\mathcal P$ preserves maximal, face, and edge rectangles, it induces a simplicial action on $\widetilde N$ which is cofinite on simplicies (Lemma 4.5). Because distinct singularity stabilizers have trivial intersection (and $\pi _1(M)$ is torsion free), each ideal simplex of $\widetilde N$ has a trivial stabilizer and the action $\pi _1(M) \curvearrowright \widetilde N$ is discontinuous. Moreover, because the peripheral subgroups of $\pi _1(M)$ precisely correspond to the stabilizers of singularities in $\mathcal P$ , it follows that each of these subgroups acts peripherally on $\widetilde N$ . Hence, by a theorem of Waldhausen [Reference WaldhausenWal68, Corollary 6.5], the manifolds M and $N = \widetilde N / \pi _1(M)$ are homeomorphic by a homeomorphism that is the identity on $\pi _1(M)$ .

Let $\tau $ be the induced ideal triangulation of N. It is now straightforward to see that $\tau $ is naturally a veering triangulation. The coorientations on the faces of $\tau $ come from the convention discussed above and the taut structure on each tetrahedron comes from lifting to $\widetilde N$ and ‘projecting’ the tetrahedron to its corresponding maximal rectangle. Note that we are not claiming that there is a single coherent projection from $\widetilde N$ to $\mathring {\mathcal P}$ , although we will establish this in the next section. An edge is declared to be right veering if its lift to $\widetilde N$ determines an edge rectangle in $\mathcal P$ whose singularities are at its SW and NE corners. Otherwise, it is left veering.

We summarize this as follows.

If the veering triangulation $\tau $ comes from the above construction, we say that $\tau $ is associated or dual to the flow $\varphi $ .

5.1 Fibration on $\widetilde N$

We first produce an equivariant fibration $p: \widetilde N \to \mathring {\mathcal P}$ , which is an orientation-preserving embedding on each face of $\widetilde \tau ^{(2)}$ . The goal is to complete this diagram with an equivariant homeomorphism:

where $q:\widetilde M\to \mathring {\mathcal P}$ is the map to the flow space of the flow.

The key step is Proposition 5.2, which gives an embedding of each edge of $\widetilde \tau $ in its associated rectangle in $\mathring {\mathcal P}$ , so that the three edges of every face have disjoint interiors. In the suspension-flow case, this is simple because the flow space admits an invariant affine structure in which every rectangle is Euclidean, and we may simply use straight lines in Figure 19 (indeed, this is how the original veering picture is obtained). In the general setting, there is no obvious way to do this—equivariance produces some tricky constraints which are reflected in the argument we give in §5.5. Most of the effort of the proof goes into this step. The map p is then produced in Proposition 5.11.

5.2 Compactification and a fiberwise map

We next build a preliminary map that takes p-fibers in $\widetilde N$ to q-fibers in $\widetilde M$ . However, to have uniform control of it, we compactify N and M and extend the map. We compactify N to a manifold $\overline N$ with toral boundary and construct a map $\overline h: \overline N \to \overline M$ that takes p-fibers to flow orbits, and boundary tori to singular orbits. Thus, we obtain the diagram

(5.1)

where $\widehat N$ and $\widehat M$ are completions of $\widetilde N$ and $\widetilde M$ obtained by lifting the compactifications.

The restriction of $\widehat h$ to each fiber may not be an embedding, but we show in Lemma 5.12 that it is proper and degree $1$ to its image fiber.

5.3 Straightening the fibers

An averaging step, convolving with a fiberwise bump function, produces a map which is an embedding on the fibers and hence a global homeomorphism. This is carried out in Proposition 5.14, and gives us a topological version of our main result, Proposition 5.15.

5.4 Smoothing

Finally, we address the issue of making the branched surface smooth, and furthermore making sure that the images of edges in the flow space are smooth and transverse to both foliations. The first of these is explained in Proposition 5.16, and the second in Proposition 5.17.

We next turn to carrying out the details.

5.5 Step 1: drawing diagonals and building a fibration

For any points $p,q \in \mathcal P$ lying in a maximal rectangle R, but not in a single leaf of $\mathcal F^{s}$ or $\mathcal F^u$ , we denote by $R(p,q) \subset R$ the unique rectangle with opposite vertices at p and q. So if $p,q$ are singularities of R, then $Q = R(p,q)$ is their edge rectangle. Recall that each edge rectangle corresponds to an edge of $\widetilde \tau $ by construction, and the veer of an edge rectangle is the veer of its associated edge.

A veering diagonal is a topological arc in an edge rectangle $Q = R(p,q)$ which connects p to q and is topologically transverse to the stable and unstable foliations, meaning that the path intersects each leaf in $R(p,q)$ at most, and so exactly, once.

Our first step is to prove the following.

5.5.1 Drawing diagonals given anchors

We say that the pair $(A,\alpha )$ is an anchor system if $\alpha $ is a bijection from the set of edge rectangles in $\mathcal P$ onto a subset $A \subset \mathcal P$ with the following properties:

• • containment—for each edge rectangle Q, $\alpha (Q)$ lies in the interior of Q;

• • equivariance— $g \cdot \alpha (Q)=\alpha (g \cdot Q)$ for each edge rectangle Q and each $g\in \pi _1(M)$ ; and

• • staircase monotonicity—for edge rectangles $Q_1$ and $Q_2$ that share a singular corner s, if $Q_1$ is wider than $Q_2$ , then $R(s,\alpha (Q_1))$ is wider and no taller than $R(s,\alpha (Q_2))$ .

When working with a given anchor system, we will refer to $\alpha (Q)$ as the anchor for Q and call A the set of anchors. Note that in the description of staircase monotonicity, $Q_2$ must be taller than $Q_1$ . However, we do not require the same of $R(s,\alpha (Q_2))$ and $R(s,\alpha (Q_1))$ , meaning $\alpha (Q_1)$ and $\alpha (Q_2)$ are allowed to live in the same horizontal leaf.

Let $(A,\alpha )$ be an anchor system. Let $F\subset \mathcal P$ be a face rectangle and let p denote the unique singularity lying at a corner of F. Let x be the singularity lying on a horizontal edge of $\partial F$ and let y be the last singularity, which necessarily lies on a vertical edge of F. Let $a_x=\alpha (R(p,x))$ and $a_y=\alpha (R(p,y))$ . If $R=R(a_x,x)$ and $Q=R(a_y,y)$ intersect non-trivially, we say F is busy. If F is busy, let $R'$ be the maximal subrectangle of R with the property that the stable and unstable leaves through each point in $R'$ do not intersect the interior of Q (see the right side of Figure 20). This subrectangle exists by staircase monotonicity. A point in $R'$ which corresponds to a periodic orbit is called an F-buoy. Because the points corresponding to periodic orbits are dense in $\mathcal P$ (Lemma 4.2), any busy face rectangle F has an F-buoy.

Figure 20 The right veering case in the definition of busy, with anchors shown in green. The face rectangle on the left is not busy, and the face rectangle on the right is busy. Note that it is possible that the two anchors lie in the same horizontal line (in colour online).

Lemma 5.3. If there exists an anchor system for $\mathcal {P}$ , then there exists an equivariant family of veering diagonals so that the three veering diagonals of every face rectangle have disjoint interiors.

That is, if an anchor system exists, then Proposition 5.2 holds.

Proof. Let $(A,\alpha )$ be the given anchor system, which determines busy face rectangles. For each $\pi _1(M)$ -orbit of busy face rectangle F, choose an F-buoy $b_F$ and let $B_F$ be the $\pi _1(M)$ -orbit of $b_F$ . There are finitely many orbits of face rectangles, so there are finitely many sets $B_F$ . Let $B=\bigcup B_F$ be their union, which we call the set of buoys. Note that B is $\pi _1(M)$ -invariant and discrete, since orbits of periodic points are discrete by Lemma 4.2.

Let $S=\{\ldots ,Q_{-1}, Q_0, Q_1,\ldots \}$ be a staircase with corner singularity p. If g generates the stabilizer of S, choose a $\langle g\rangle $ -equivariant family of continuous paths from p to the anchors of elements of S with the following three properties:

1. (1) for each $Q_i$ , the path from p to $\alpha (Q_i)$ is homotopic rel endpoints in $R(p,\alpha (Q_i)) \smallsetminus B$ to the first-horizontal-then-vertical path from p to $\alpha (Q_i)$ ;

2. (2) the paths are disjoint except at p; and

3. (3) the paths are topologically transverse to the stable and unstable foliations, meaning that no path intersects a leaf more than once.

See Figure 21. We can choose such a family by staircase monotonicity and the discreteness of B. We call each of these paths a half-diagonal.

Figure 21 Drawing half-diagonals (orange) satisfying properties (1)–(3) in the proof of Proposition 5.2. The green points are anchors and the pink points are buoys (in colour online).

Having chosen such a family of half-diagonals for each $\pi _1(M)$ -orbit of staircase in $\mathcal P$ , we can specify a veering diagonal for each edge rectangle $R(p,q)$ as the union of the two half-diagonals from p and q to the anchor for $R(p,q)$ . Let $\mathcal D$ denote the union of all these veering diagonals.

Let F be a face rectangle with corner singularity p, and let $e,f\subset F$ be the two diagonals in $\mathcal D$ of the same veer. The two half-diagonals incident to p are disjoint by property (2) above. The two half-diagonals not incident to p are also disjoint since F is either not busy, in which case disjointness is clear; or busy, in which case property (1) above guarantees disjointness. Therefore, $e\cap f=\{p\}$ , and it is clear that e and f are the only pair of diagonals of F whose interiors could intersect. This completes the proof.