2.1 Restricting taut foliations to JSJ components

Questions about taut foliations on graph manifolds can often be reduced to questions about taut foliations on Seifert fibered spaces, due in part to the following result.

As noted by Brittenham, Naimi, and Roberts [Reference Brittenham, Naimi and RobertsBNR97], this result has major consequences for graph manifolds.

When a closed graph manifold has positive first Betti number, the question of existence of taut foliations becomes trivial, since a result of Gabai states that any such manifold admits a co-oriented taut foliation [Reference GabaiGab83]. Correspondingly, any closed oriented three-manifold with $b_{1}>0$ has non-trivial Heegaard Floer homology, hence is not an L-space. We therefore restrict attention to rational homology sphere graph manifolds, hence to oriented Seifert fibered spaces over $S^{2}$ or $\mathbb{R}\mathbb{P}^{2}\!$ , and regular fiber complements thereof.

2.2 Conventions for Seifert fibered spaces

If $\hat{M}$ denotes the trivial circle fibration over the ( $n+1$ )-punctured two-sphere,

(2) $$\begin{eqnarray}\hat{M}:=S^{1}\times \biggl(S^{2}\Big\backslash\coprod _{i=0}^{n}D_{i}^{2}\biggr),\quad \unicode[STIX]{x2202}_{i}\hat{M}:=-\unicode[STIX]{x2202}(S^{1}\times D_{i}^{2}),\quad i\in \{0,\ldots ,n\},\end{eqnarray}$$

then writing $-\tilde{h}_{i}:\in H_{1}(\unicode[STIX]{x2202}_{i}\hat{M})$ for the meridian of each excised solid torus $S^{1}\times D_{i}^{2}$ , we have

(3) $$\begin{eqnarray}-\mathop{\sum }_{i=1}^{n}\tilde{h}_{i}=p_{S^{1}}\times \unicode[STIX]{x2202}(S^{2}\setminus D_{0}^{2})=p_{S^{1}}\times -\unicode[STIX]{x2202}D_{0}^{2}=\tilde{h}_{0}\end{eqnarray}$$

for any point class $p_{S^{1}}\in H_{0}(S^{1})$ of the circle fiber. For each $i\in \{0,\ldots ,n\}$ , if we write $\unicode[STIX]{x1D704}_{i}:H_{1}(\unicode[STIX]{x2202}_{i}\hat{M})\rightarrow H_{1}(\hat{M})$ for the map induced by inclusion, then there is a lift $\tilde{f}_{i}\in \unicode[STIX]{x1D704}_{i}^{-1}(f)$ of the regular fiber class $f\in H_{1}(\hat{M})$ satisfying $(-\tilde{h}_{i}\cdot \tilde{f}_{i})|_{\unicode[STIX]{x2202}_{i}\hat{M}}=1$ . The reverse-oriented basis $(\tilde{f}_{i},-\tilde{h}_{i})$ for $H_{1}(\unicode[STIX]{x2202}_{i}\hat{M})$ induces a projectivization map

(4) $$\begin{eqnarray}\unicode[STIX]{x1D70B}_{i}:H_{1}(\unicode[STIX]{x2202}_{i}\hat{M})\setminus \{0\}\rightarrow \mathbb{P}(H_{1}(\unicode[STIX]{x2202}_{i}\hat{M}))\stackrel{{\sim}}{\rightarrow }\mathbb{Q}\cup \{\infty \},\quad r_{i}\tilde{f}_{i}-s_{i}\tilde{h}_{i}\mapsto \frac{r_{i}}{s_{i}},\end{eqnarray}$$

by which we identify Seifert invariants with slopes, and slopes with $\mathbb{Q}\cup \{\infty \}$ .

The Seifert fibered space $M_{S^{2}}(r_{\ast }/s_{\ast }):=M_{S^{2}}(e_{0}=r_{0}/s_{0};r_{1}/s_{1},\ldots ,r_{n}/s_{n})$ over $S^{2}$ is the Dehn filling of $\hat{M}$ along the slopes $\unicode[STIX]{x1D707}_{i}:=r_{i}\tilde{f}_{i}-s_{i}\tilde{h}_{i}$ , with $\unicode[STIX]{x1D70B}_{i}(\unicode[STIX]{x1D707}_{i})=r_{i}/s_{i}$ , subject to the convention that $e_{0}=r_{0}/s_{0}\in \mathbb{Z}$ and $r_{i}/s_{i}\notin \mathbb{Z}\cup \{\infty \}$ . Setting each $h_{i}:=\unicode[STIX]{x1D704}_{i}(\tilde{h}_{i})$ then gives the presentation

(5) $$\begin{eqnarray}H_{1}\biggl(M_{S^{2}}\biggl(\frac{r_{\ast }}{s_{\ast }}\biggr)\biggr)=\langle f,h_{0},\ldots ,h_{n}\biggm\vert\mathop{\sum }_{i=0}^{n}h_{i}=e_{0}f-h_{0}=r_{1}f-s_{1}h_{1}=\cdots =r_{n}f-s_{n}h_{n}=0\rangle .\end{eqnarray}$$

Likewise, if we respectively lift $f$ and each $h_{i}$ to generators $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D702}_{i}$ for $\unicode[STIX]{x1D70B}_{1}(M_{S^{2}}(r_{\ast }/s_{\ast }))$ and substitute $\unicode[STIX]{x1D719}^{e_{0}}$ for $\unicode[STIX]{x1D702}_{0}$ , then we obtain the fundamental group presentation

(6) $$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}\biggl(M_{S^{2}}\biggl(\frac{r_{\ast }}{s_{\ast }}\biggr)\biggr)=\langle \unicode[STIX]{x1D719},\unicode[STIX]{x1D702}_{1},\ldots ,\unicode[STIX]{x1D702}_{n}\biggm\vert\unicode[STIX]{x1D719}~\text{central, }\mathop{\prod }_{i=1}^{n}\!\unicode[STIX]{x1D702}_{i}\!=\!\unicode[STIX]{x1D719}^{-e_{0}},\,\unicode[STIX]{x1D702}_{1}^{s_{1}}\!=\!\unicode[STIX]{x1D719}^{r_{1}},\,\ldots ,\,\unicode[STIX]{x1D702}_{n}^{s_{n}}\!=\!\unicode[STIX]{x1D719}^{r_{n}}\rangle .\end{eqnarray}$$

For a manifold $M_{\mathbb{R}\mathbb{P}^{2}}(r_{\ast }/s_{\ast })$ Seifert fibered over $\mathbb{R}\mathbb{P}^{2}$ , we adopt the same homology and slope conventions for the boundary of a regular fiber complement, but the global homology is slightly different. Since, this time, $\hat{M}$ is the twisted circle bundle over a punctured $\mathbb{R}\mathbb{P}^{2}$ , the fiber class $f$ is now 2-torsion. Also, since puncturing $\mathbb{R}\mathbb{P}^{2}$ once gives a Möbius strip instead of a disk, the sum $\sum _{i=1}^{n}\tilde{h}_{i}$ differs from $p_{\mathbb{R}\mathbb{P}^{2}}\times \unicode[STIX]{x2202}(\mathbb{R}\mathbb{P}^{2}\setminus D_{0}^{2})$ by twice the one-cell $c$ glued to the disk to make $\mathbb{R}\mathbb{P}^{2}$ , yielding a homology presentation of the form

(7) $$\begin{eqnarray}H_{1}\biggl(M_{\mathbb{R}\mathbb{P}^{2}}\biggl(\frac{r_{\ast }}{s_{\ast }}\biggr)\biggr)=\langle f,c,h_{0},\ldots ,h_{n}\biggm\vert2f=0,2c+\mathop{\sum }_{i=0}^{n}h_{i}=0,\,\unicode[STIX]{x1D704}_{0}(\unicode[STIX]{x1D707}_{0})=\cdots =\unicode[STIX]{x1D704}_{n}(\unicode[STIX]{x1D707}_{n})=0\rangle .\end{eqnarray}$$

For either type of base, the Seifert fibration is invariant under any reparameterization $M(r_{0}/s_{0},\ldots ,r_{n}/s_{n})\rightarrow M(r_{0}/s_{0}+z_{0},\ldots ,z_{n}+r_{n}/s_{n})$ with $\sum _{i=0}^{n}z_{i}=0$ and each $z_{i}\in \mathbb{Z}$ . The manifold also admits orientation-reversing homeomorphism, $M(r_{0}/s_{0},\ldots ,r_{n}/s_{n})\rightarrow M(-r_{0}/s_{0},\ldots ,-r_{n}/s_{n})$ .

A regular fiber complement $Y:=M\setminus \unicode[STIX]{x1D708}(f)$ in a rational homology sphere Seifert fibered space has $b_{1}(Y)=1$ , hence has a well-defined rational longitude.

It is straightforward to show (see, for example, [Reference Rasmussen and RasmussenRR15]) that

(8) $$\begin{eqnarray}M_{S^{2}}\biggl(\frac{r_{\ast }}{s_{\ast }}\biggr)\setminus \unicode[STIX]{x1D708}(f)\text{ has rational longitude }l=-\mathop{\sum }_{i=0}^{n}\frac{r_{i}}{s_{i}}.\end{eqnarray}$$

A mild generalization of the calculation in [Reference Rasmussen and RasmussenRR15] shows that the above result also holds if each solid torus $S^{1}\times D_{i}^{2}$ is replaced with an arbitrary compact oriented three-manifold $Y_{i}$ with torus boundary and $b_{1}(Y_{i})=1$ , with $r_{i}/s_{i}$ the image of the rational longitude of $Y_{i}$ . By contrast, if the JSJ component containing $\unicode[STIX]{x2202}Y$ has non-orientable base, then $l=\unicode[STIX]{x1D70B}(\tilde{f})=\infty$ .

Primality is especially important in the context of foliations, since, by Novikov [Reference NovikovNov65], no reducible manifold except $S^{1}\times S^{2}$ admits a co-oriented taut foliation. On the other hand, not all connected sums are L-spaces, so any correspondence between being an L-space and failing to admit a co-oriented taut foliation breaks down beyond the realm of prime manifolds.

2.3 Rotation number, shift, and foliation slope

One of the key insights of Jankins and Neumann into the work of Eisenbud, Hirsch, and Neumann on taut foliations on Seifert fibered spaces was the need for a better invariant on $\widetilde{\text{Homeo}_{+}}S^{1}$ . Whereas the latter group relied on the invariants $\text{}\underline{m},\overline{m}:\widetilde{\text{Homeo}_{+}}S^{1}\rightarrow \mathbb{R}$ , with $\text{}\underline{m}(\unicode[STIX]{x1D6FE}):=\min _{t\in \mathbb{R}}\unicode[STIX]{x1D6FE}(t)-t$ and $\overline{m}(\unicode[STIX]{x1D6FE}):=\max _{t\in \mathbb{R}}\unicode[STIX]{x1D6FE}(t)-t$ , Jankins and Neumann introduced the problem to a more precise invariant of circle actions: a conjugacy invariant called the (Poincaré) rotation number,

(9) $$\begin{eqnarray}\text{rot}:\widetilde{\text{Homeo}_{+}}(S^{1})\rightarrow \mathbb{R},\quad \text{rot}(\unicode[STIX]{x1D6FE})=\lim _{k\rightarrow \infty }{\displaystyle \frac{1}{k}}(\unicode[STIX]{x1D6FE}^{k}(t)-t),\end{eqnarray}$$

which is independent of $t\in \mathbb{R}$ , and rational if and only if $\unicode[STIX]{x1D6FE}$ has some closed orbit [Reference GhysGhy01]. The rotation number is not, in general, a homomorphism. However, it restricts to a homomorphism on any amenable, hence any abelian, subgroup [Reference GhysGhy01]. In particular, it restricts to a homomorphism on any representation of the fundamental group of a torus.

The simplest element of $\widetilde{\text{Homeo}_{+}}(S^{1})$ is a rotation, or shift,

(10) $$\begin{eqnarray}\text{sh}(s):t\mapsto t+s,\quad t\in \mathbb{R}.\end{eqnarray}$$

Whereas $\text{rot}\circ \text{sh}=\text{id}$ , not every element of $\widetilde{\text{Homeo}_{+}}(S^{1})$ is conjugate to a rotation. It is a classic result, however, that every element of $\widetilde{\text{Homeo}_{+}}(S^{1})$ with irrational rotation number is left and right semiconjugate to a shift of the same rotation number [Reference GhysGhy01].

Rotation numbers can also be used to associate slopes to taut foliations on tori.

Definition 2.4. For the two-torus $T$ , there is a canonical map

(11) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}:\{C^{0}\text{ codimension-one foliations on }T\}\rightarrow \mathbb{P}(H_{1}(T;\mathbb{R})),\end{eqnarray}$$

constructed below, which respects isotopy. We call $\unicode[STIX]{x1D6FC}(F)$ the slope of $F$ .

If $F$ has Reeb components, then $\unicode[STIX]{x1D6FC}(F)$ is given by the class of any closed leaf of $F$ . All Reebless foliations on tori are taut. Thus, if $F$ is Reebless, then there is a curve, say $C_{\unicode[STIX]{x1D706}}$ of primitive class $\unicode[STIX]{x1D706}\in H_{1}(T)$ , which intersects every leaf transversely, and $F$ can be realized as the suspension of a circle homeomorphism $\unicode[STIX]{x1D6FE}_{F,\unicode[STIX]{x1D706}}\in \text{Homeo}_{+}S^{1}$ from $C_{\unicode[STIX]{x1D706}}$ to itself [Reference Hector and HirschHH81]. A choice of $\unicode[STIX]{x1D707}\in H_{1}(T)$ with $\unicode[STIX]{x1D707}\cdot \unicode[STIX]{x1D706}=1$ induces a lift of this suspension to a suspension from a universal cover $\tilde{C}_{\unicode[STIX]{x1D706}}$ of $C_{\unicode[STIX]{x1D706}}$ to its translate by $\unicode[STIX]{x1D707}$ in the universal cover of $T$ . That is, if we regard $\tilde{C}_{\unicode[STIX]{x1D706}}$ as the real vector space $\{t\unicode[STIX]{x1D706}\}_{t\in \mathbb{R}}$ spanned by $\unicode[STIX]{x1D706}$ , with $C_{\unicode[STIX]{x1D706}}\cong \tilde{C}_{\unicode[STIX]{x1D706}}/\unicode[STIX]{x1D706}\mathbb{Z}$ , then we can lift the foliation $F$ to the universal cover of $T$ by iteratively suspending the map $t\unicode[STIX]{x1D706}\mapsto \tilde{\unicode[STIX]{x1D6FE}}_{F,\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}}(t)\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}$ , for an appropriate lift $\tilde{\unicode[STIX]{x1D6FE}}_{F,\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}}\in \widetilde{\text{Homeo}}_{+}S^{1}\subset \text{Homeo}_{+}\mathbb{R}$ of $\unicode[STIX]{x1D6FE}_{F,\unicode[STIX]{x1D706}}\in \text{Homeo}_{+}S^{1}$ .

This lifted suspension, in turn, induces a representation

(12) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D706}}^{F}:\unicode[STIX]{x1D70B}_{1}(T)\rightarrow \widetilde{\text{Homeo}}_{+}S^{1},\quad [\unicode[STIX]{x1D706}]\mapsto \text{sh}(1),\quad [-\unicode[STIX]{x1D707}]\mapsto \tilde{\unicode[STIX]{x1D6FE}}_{F,\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}},\end{eqnarray}$$

where $[\unicode[STIX]{x1D706}]$ and $[\unicode[STIX]{x1D707}]$ denote the lifts of $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D707}$ to $\unicode[STIX]{x1D70B}_{1}(T)$ . One can regard $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D706}}^{F}$ as describing how to traverse the line $\{t\unicode[STIX]{x1D706}\}_{t\in \mathbb{R}}\subset \langle \unicode[STIX]{x1D707},\unicode[STIX]{x1D706}\rangle _{\mathbb{R}}$ by traveling only along foliation leaves or integer multiples of $\unicode[STIX]{x1D706}$ or $\unicode[STIX]{x1D707}$ . That is, if one starts at some $t_{0}\unicode[STIX]{x1D706}$ , hops by $a\unicode[STIX]{x1D706}+b\unicode[STIX]{x1D707}$ for some $a,b\in \mathbb{Z}$ , takes the foliation leaf intersecting this new point, and follows this leaf back to the line $\{t\unicode[STIX]{x1D706}\}_{t\in \mathbb{R}}$ , then one will arrive at $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D706}}^{F}(a\unicode[STIX]{x1D706}+b\unicode[STIX]{x1D707})(t_{0})\unicode[STIX]{x1D706}$ . Note that while $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D706}}^{F}$ is independent of the choice of $\unicode[STIX]{x1D707}$ , and is determined up to conjugacy by a choice of $\unicode[STIX]{x1D706}$ , it still depends on $\unicode[STIX]{x1D706}$ .

On the other hand, when we define the slope $\unicode[STIX]{x1D6FC}(F)$ of $F$ to be

(13) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}(F):=\ker (\text{rot}\circ \unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D706}}^{F})=\langle (\text{rot}(\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D706}}^{F}\!(-\unicode[STIX]{x1D707})))\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}\rangle \in \mathbb{P}(H_{1}(T;\mathbb{R})),\end{eqnarray}$$

then the rotation number washes out all dependence on $\unicode[STIX]{x1D706}$ and choice of suspension. That is, one can use the definition of rotation number to compute $\text{rot}(\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D706}}^{F}\!(-\unicode[STIX]{x1D707}))$ in terms of the rotation number associated to a different choice of basis and suspension for $F$ , and obtain the same answer for $\unicode[STIX]{x1D6FC}(F)$ in both cases. Alternatively, any suspension homeomorphism with rational rotation number has a periodic orbit, hence realizes a foliation with a compact leaf of slope $\unicode[STIX]{x1D6FC}(F)$ . If the suspension homeomorphism has irrational rotation number, then it is semiconjugate to a shift of matching rotation number [Reference GhysGhy01], giving rise to a linear foliation of slope $\unicode[STIX]{x1D6FC}(F)$ .

2.4 Restricting Seifert fibered space foliations to torus foliations

If a compact oriented three-manifold $Y$ admits a co-oriented taut foliation transverse to $\unicode[STIX]{x2202}Y$ , then Gabai tells us that $\unicode[STIX]{x2202}Y$ can only have toroidal components [Reference GabaiGab83]. Thus, we often encounter foliations on tori as boundary restrictions of foliations on three-manifolds. Moreover, on a Seifert fibered space, any taut foliation transverse to the boundary restricts to taut foliations on boundary components.

Suppose $F$ is a co-oriented taut foliation transverse to the fibration of the Seifert fibered Dehn filling $M_{S^{2}}(r_{\ast }/s_{\ast })$ along the slopes $r_{\ast }/s_{\ast }=(r_{0}/s_{0},\ldots ,r_{n}/s_{n})$ of the trivial circle fibration $\hat{M}$ over an ( $n+1$ )-punctured $S^{2}$ , according to the conventions of § 2.2. For each boundary component $\unicode[STIX]{x2202}_{i}\hat{M}$ , we regard the foliation $F\cap \unicode[STIX]{x2202}_{i}\hat{M}$ as a suspension of a homeomorphism of the curve of class $\tilde{f}_{i}$ to itself, and since the class $-\tilde{h}_{i}$ satisfies $-\tilde{h}_{i}\cdot \tilde{f}_{i}=1$ , it specifies a lift of this suspension to a suspension of an element $\unicode[STIX]{x1D6FE}_{F,\tilde{f}_{i},-\tilde{h}_{i}}\in \widetilde{\text{Homeo}_{+}}S^{1}$ . To this suspension we associate the representation

(14) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{i}:=\unicode[STIX]{x1D70C}_{\tilde{f}_{i}}^{F\cap \unicode[STIX]{x2202}_{i}\hat{M}}:\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x2202}_{i}\hat{M})\rightarrow \widetilde{\text{Homeo}_{+}}S^{1},\quad [\tilde{f}_{i}]\rightarrow \text{sh}(1),\quad [\tilde{h}_{i}]\rightarrow \unicode[STIX]{x1D6FE}_{F,\tilde{f}_{i},-\tilde{h}_{i}},\end{eqnarray}$$

allowing us to express the slope $\unicode[STIX]{x1D6FC}(F\cap \unicode[STIX]{x2202}_{i}\hat{M})$ of $F\cap \unicode[STIX]{x2202}_{i}\hat{M}$ as

(15) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}(F\cap \unicode[STIX]{x2202}_{i}\hat{M})=\unicode[STIX]{x1D70B}_{i}(\text{rot}(\unicode[STIX]{x1D70C}_{i}([\tilde{h}_{i}]))\tilde{f}_{i}-\tilde{h}_{i})=\text{rot}(\unicode[STIX]{x1D70C}_{i}([\tilde{h}_{i}])).\end{eqnarray}$$

The construction of Eisenbud, Hirsch, and Neumann [Reference Eisenbud, Hirsch and NeumannEHN81] associating a representation $\unicode[STIX]{x1D70C}:\unicode[STIX]{x1D70B}_{1}(M(r_{i}/s_{i}))\rightarrow \widetilde{\text{Homeo}_{+}}S^{1}$ to $F$ , sending the fiber class $\unicode[STIX]{x1D719}=[f]$ to $\text{sh}(1)$ , is sufficiently compatible with the construction of each $\unicode[STIX]{x1D70C}_{i}$ above that, possibly after conjugation of each $\unicode[STIX]{x1D70C}_{i}$ , $\unicode[STIX]{x1D70C}$ can be chosen to satisfy $\unicode[STIX]{x1D70C}_{i}=\unicode[STIX]{x1D70C}\,\circ \,\unicode[STIX]{x1D704}_{i}^{\unicode[STIX]{x1D70B}_{1}}$ , with $\unicode[STIX]{x1D704}_{i}^{\unicode[STIX]{x1D70B}_{1}}:\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x2202}_{i}\hat{M})\rightarrow \unicode[STIX]{x1D70B}_{1}(M_{S^{2}}(r_{\ast }/s_{\ast }))$ the homomorphism induced by inclusion. The presentation (6) for $\unicode[STIX]{x1D70B}_{1}(M_{S^{2}}(r_{\ast }/s_{\ast }))$ then places the following restrictions on $\unicode[STIX]{x1D70C}$ , as observed by Jankins and Neumann [Reference Jankins and NeumannJN85]:

(16) $$\begin{eqnarray}\displaystyle \mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D702}_{i}=\unicode[STIX]{x1D719}^{-e_{0}} & \;\Longrightarrow \; & \displaystyle \bullet \;\text{rot}\biggl(\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}_{i})\biggr)=-e_{0}=-\frac{r_{0}}{s_{0}},\end{eqnarray}$$

(17) $$\begin{eqnarray}\displaystyle i\in \{0,\ldots ,n\}\!:\quad \unicode[STIX]{x1D702}_{i}^{s_{i}}=\unicode[STIX]{x1D719}^{r_{i}} & \;\Longrightarrow \; & \displaystyle \left\{\begin{array}{@{}l@{}}\bullet \;\text{rot}(\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}_{i}))={\displaystyle \frac{r_{i}}{s_{i}}},\quad \\ \bullet \;\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}_{i})\text{ is conjugate to }\text{sh}\biggl({\displaystyle \frac{r_{i}}{s_{i}}}\biggr).\quad \end{array}\right.\end{eqnarray}$$

Jankins and Neumann mostly focused on the case of $n=3$ , $e_{0}=-1$ , and $0<r_{1}/s_{1},r_{2}/s_{2},r_{3}/s_{3}<1$ , but their above observation holds in general.

Whereas the first condition enforces a global restriction on $F$ , the latter two conditions provide local restrictions at each $F\cap \unicode[STIX]{x2202}_{i}\hat{M}$ , which we could recover simply by considering Dehn fillings. The solid torus admits only one taut foliation, namely, the product foliation with slope given by the rational longitude. As a consequence, the co-oriented taut foliation $F\,\cap \,\unicode[STIX]{x2202}_{i}\hat{M}$ extends to a co-oriented taut foliation on the Dehn filling $\unicode[STIX]{x2202}_{i}\hat{M}(r_{i}/s_{i})$ if and only if $F\cap \unicode[STIX]{x2202}_{i}\hat{M}$ is the product foliation of slope $r_{i}/s_{i}$ , which occurs if and only if $\unicode[STIX]{x1D70C}_{i}([\tilde{h}_{i}])\!=\!\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}_{i})$ is conjugate to $\text{sh}(r_{i}/s_{i})$ , a condition which requires $\unicode[STIX]{x1D6FC}(F\cap \unicode[STIX]{x2202}_{i}\hat{M})=\text{rot}(\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}_{i}))=r_{i}/s_{i}$ . In particular, the $j\text{th}$ shift conjugacy condition, that $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}_{j})$ be conjugate to $\text{sh}(r_{j}/s_{j})$ , is due solely to the fact that $\unicode[STIX]{x2202}_{j}\hat{M}$ is glued to a solid torus. As emphasized by Boyer and Clay [Reference Boyer and ClayBC14], when one relaxes the $j\text{th}$ shift conjugacy condition, one can still find manifolds $Y$ with torus boundary for which $F$ extends to a taut foliation on $\hat{M}\cup _{\unicode[STIX]{x2202}_{j}\hat{M}}Y\!$ , for a suitable choice of gluing map.

It is presumably for this reason that Jankins and Neumann focused on the more general condition of $J$ -realizability for an ( $n+1$ )-tuple $r_{\ast }/s_{\ast }:=(e_{0}=r_{0}/s_{0},r_{1}/s_{1},\ldots ,r_{n}/s_{n})$ , given a subset $J\subset \{1,\ldots ,n\}$ . They deem $r_{\ast }/s_{\ast }$ $J$ -realizable if there is a representation $\unicode[STIX]{x1D70C}^{J}:\langle \unicode[STIX]{x1D702}_{1},\ldots ,\unicode[STIX]{x1D702}_{n}\rangle \rightarrow \widetilde{\text{Homeo}_{+}}S^{1}$ such that $\unicode[STIX]{x1D70C}^{J}$ meets the $r_{\ast }/s_{\ast }$ rotation number condition, that $\text{rot}(\prod _{i=1}^{n}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}_{i}))=-e_{0}$ with $\text{rot}(\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}_{i}))\!=\!r_{i}/s_{i}$ for each $i\in \{1,\ldots ,n\}$ , and such that $\unicode[STIX]{x1D70C}^{J}$ meets the $j\text{th}$ shift conjugacy condition for each $j\in J$ . We have already shown that $J$ -realizability is a necessary condition for $\hat{M}$ to admit a taut foliation $F$ of slopes $\unicode[STIX]{x1D6FC}(F\cap \unicode[STIX]{x2202}_{i}\hat{M})=r_{i}/s_{i}$ which extends to a taut foliation on the partial Dehn filling of $\hat{M}$ along the slopes $r_{j}/s_{j}\in \mathbb{P}(H_{1}(\unicode[STIX]{x2202}_{j}\hat{M}))$ for all $j\in \{0\}\cup J$ . With the help of Naimi, Jankins and Neumann showed that the condition is also sufficient [Reference Jankins and NeumannJN85, Reference NaimiNai94].

2.5 Solutions for $J$ -realizability

Jankins and Neumann conjectured that a slope $r_{\ast }/s_{\ast }$ is $J$ -realizable in $\widetilde{\text{Homeo}_{+}}S^{1}$ if and only if it is $J$ -realizable in a smooth Lie subgroup of $\widetilde{\text{Homeo}_{+}}S^{1}$ . Observing that any smooth Lie subgroup of $\widetilde{\text{Homeo}_{+}}S^{1}$ is conjugate to $\widetilde{\text{PSL}}_{k}(2,\mathbb{R})$ for some $k\!\in \!\mathbb{Z}_{{>}0}$ , where $\widetilde{\text{PSL}}_{k}(2,\mathbb{R})=\unicode[STIX]{x1D713}_{k}^{-1}\widetilde{\text{PSL}}(2,\mathbb{R})\unicode[STIX]{x1D713}_{k}$ for $(\unicode[STIX]{x1D713}_{k}:t\mapsto kt)\in \text{Homeo}_{+}\mathbb{R}$ , they computed

(18) $$\begin{eqnarray}\max \;\text{rot}\biggl(\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D6FE}_{i}\biggr)=\frac{1}{k}\biggl(-1+\mathop{\sum }_{i=1}^{n}\biggl(\biggl\lfloor\frac{r_{i}k}{s_{i}}\biggr\rfloor+1\biggr)\biggr)\end{eqnarray}$$

for the maximum rotation number of a product of elements $\unicode[STIX]{x1D6FE}_{i}\in \widetilde{\text{PSL}}_{k}(2,\mathbb{R})$ with each $\text{rot}(\unicode[STIX]{x1D6FE}_{i})=r_{i}/s_{i}$ . They then proved the above conjecture in all but one case, later proven by Naimi [Reference NaimiNai94]. More recently, in [Reference Calegari and WalkerCW11, Theorem 3.9] (appropriately generalized from 2 to $n$ ), Calegari and Walker rederived (18) (with the maximum taken over $k\!\in \!\mathbb{Z}_{{>}0}$ ) for $\widetilde{\text{Homeo}_{+}}S^{1}$ , without appealing to $\widetilde{\text{PSL}}_{k}(2,\mathbb{R})$ , by using dynamical techniques similar to those of Naimi.

One obtains the analogous minimum rotation number of a product by sending $r_{i}/s_{i}\mapsto -r_{i}/s_{i}$ in (18). Demanding that $-e_{0}$ lie between the minimum and maximum rotation numbers for $\prod _{i=1}^{n}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D702}_{i})$ , and multiplying the resulting inequality by $-1$ , implies that a representation in $\widetilde{\text{Homeo}_{+}}S^{1}$ can only satisfy the rotation number condition for $r_{\ast }/s_{\ast }=(e_{0},r_{1}/s_{1},\ldots ,r_{n}/s_{n})$ if

(19) $$\begin{eqnarray}\min _{k>0}-{\displaystyle \frac{1}{k}}\biggl(-1+\mathop{\sum }_{i=1}^{n}\biggl(\biggl\lfloor{\displaystyle \frac{r_{i}k}{s_{i}}}\biggr\rfloor+1\biggr)\biggr)\leqslant e_{0}\leqslant \max _{k>0}-{\displaystyle \frac{1}{k}}\biggl(1+\mathop{\sum }_{i=1}^{n}\biggl(\lceil {\displaystyle \frac{r_{i}k}{s_{i}}}\rceil -1\biggr)\biggr),\end{eqnarray}$$

a criterion which Jankins and Neumann prove is also sufficient [Reference Jankins and NeumannJN85]. Moreover, $r_{\ast }/s_{\ast }$ is $J$ -realizable in $\widetilde{\text{PSL}}_{k}(2,\mathbb{R})$ , for some $k\in \mathbb{Z}_{{>}0}$ , if and only if

(20) $$\begin{eqnarray}-{\displaystyle \frac{1}{k}}\biggl(-1+\mathop{\sum }_{i=0}^{n}\biggl(\biggl\lfloor{\displaystyle \frac{r_{i}k}{s_{i}}}\biggr\rfloor+1\biggr)\biggr)\leqslant 0\leqslant -\mathop{\sum }_{i=0}^{n}{\displaystyle \frac{r_{i}}{s_{i}}}\text{ or }-\mathop{\sum }_{i=0}^{n}{\displaystyle \frac{r_{i}}{s_{i}}}\leqslant 0\leqslant -{\displaystyle \frac{1}{k}}\biggl(1+\mathop{\sum }_{i=0}^{n}\biggl(\lceil {\displaystyle \frac{r_{i}k}{s_{i}}}\rceil -1\biggr)\biggr).\end{eqnarray}$$

The shift conjugacy condition is easier to apply: one can approximate an element of $\widetilde{\text{Homeo}_{+}}S^{1}$ with a shift-conjugate element of arbitrarily close rotation number. Jankins and Neumann used this fact to show that if one fixes all $r_{i}/s_{i}$ with $i\neq j$ , for some fixed $j\in \{1,\ldots ,n\}$ , then imposing the $j\text{th}$ shift conjugacy condition is equivalent to restricting to the interior of the interval of $r_{j}/s_{j}\in \mathbb{R}$ for which the $r_{\ast }/s_{\ast }$ rotation number condition is satisfied. More generally, Calegari and Walker have shown that the same principle holds for the rotation number condition associated to any fixed positive word [Reference Calegari and WalkerCW11, Lemma 3.31].

Since for any $r\in \mathbb{R}$ and $z\in \mathbb{Z}$ , we have

(21) $$\begin{eqnarray}z\leqslant \lfloor r\rfloor ~\;\Longleftrightarrow \;~z\leqslant r,\quad \lceil r\rceil \leqslant z~\;\Longleftrightarrow \;~r\leqslant z,\end{eqnarray}$$

it follows, for any $j\in \{1,\ldots ,n\}$ , that if we fix all $r_{i}/s_{i}$ with $i\neq j$ , then the interval of $r_{j}/s_{j}\in \mathbb{R}$ satisfying (19) is closed. On the other hand, for any $r\in \mathbb{R}$ and $z\in \mathbb{Z}$ , we know that

(22) $$\begin{eqnarray}z\leqslant \lceil r\rceil -1~\;\Longleftrightarrow \;~z<r,\quad \lfloor r\rfloor +1\leqslant z~\;\Longleftrightarrow \;~r<z.\end{eqnarray}$$

These identities led Jankins and Neumann to produce the formulas in the following result.

[JN85] and Naimi [Nai94];

Theorem 2.5 (Jankins and Neumann [Reference Jankins and NeumannJN85] and Naimi [Reference NaimiNai94]; cf. Calegari and Walker [Reference Calegari and WalkerCW11]).

For any $n\geqslant 2$ , partition $J\amalg \bar{J}=\{0,\ldots ,n\}$ with $0\in J$ , and $(n+1)$ -tuple $r_{\ast }/s_{\ast }\!:=\!(r_{0}/s_{0},\ldots ,r_{n}/s_{n})\in \mathbb{Q}^{n+1}$ with $r_{0}/s_{0}\in \mathbb{Z}$ and $r_{i}/s_{i}\notin \mathbb{Z}$ for $i>0$ , the trivial circle fibration $\hat{M}$ over an $(n+1)$ -punctured $S^{2}\!$ admits a co-oriented taut foliation $F$ transverse to the boundary, with slopes $\unicode[STIX]{x1D6FC}(F\cap \unicode[STIX]{x2202}_{i}\hat{M})=r_{i}/s_{i}$ for $i\in \{0,\ldots ,n\}$ , and with $F$ extending to a co-oriented taut foliation on the Dehn filling of $\unicode[STIX]{x2202}_{j}\hat{M}$ of slope $r_{j}/s_{j}$ for each $j\in J$ , if and only if $0=y_{-}=y_{+}$ or $0\in \langle y_{+},y_{-}\rangle$ , with

(23) $$\begin{eqnarray}\begin{array}{@{}l@{}}\displaystyle y_{-}:=\max _{1\leqslant k\leqslant s}-{\displaystyle \frac{1}{k}}\biggl(1+\mathop{\sum }_{j\in J}\!\biggl\lfloor\!{\displaystyle \frac{r_{j}k}{s_{j}}}\!\biggr\rfloor+\mathop{\sum }_{\bar{\jmath }\in \bar{J}}\!\biggl(\lceil \!{\displaystyle \frac{r_{\bar{\jmath }}k}{s_{\bar{\jmath }}}}\!\rceil -1\biggr)\!\biggr),\\ \displaystyle y_{+}:=\min _{1\leqslant k\leqslant s}-{\displaystyle \frac{1}{k}}\biggl(-1+\mathop{\sum }_{j\in J}\!\lceil \!{\displaystyle \frac{r_{j}k}{s_{j}}}\!\rceil +\mathop{\sum }_{\bar{\jmath }\in \bar{J}}\!\biggl(\biggl\lfloor\!{\displaystyle \frac{r_{\bar{\jmath }}k}{s_{\bar{\jmath }}}}\!\biggr\rfloor+1\biggr)\!\biggr),\end{array}\end{eqnarray}$$

where $s$ is the least common positive multiple of the $s_{i}$ .

2.6 Dehn fillings and $\bar{N}$ -fillings

For any particular $i\in \{1,\ldots ,n\}$ in the above theorem, if one fixes the remaining slopes, one finds that the space of slopes in $\mathbb{P}(H_{1}(\unicode[STIX]{x2202}_{i}\hat{M}))$ for which the desired taut foliation exists is often an interval. We now introduce some notation to describe such spaces of slopes in general.

Definition 2.6. If $Y$ is a compact oriented three-manifold with torus boundary, then we define the sets ${\mathcal{F}}^{L}(Y)\subset {\mathcal{F}}^{D}(Y)\subset {\mathcal{F}}(Y)\subset \mathbb{P}(H_{1}(\unicode[STIX]{x2202}Y;\mathbb{Z}))$ of rational foliation slopes as follows:

$$\begin{eqnarray}\displaystyle {\mathcal{F}}^{L}(Y) & := & \displaystyle \left\{\unicode[STIX]{x1D6FC}(F\cap \unicode[STIX]{x2202}Y)\biggm\vert\begin{array}{@{}c@{}}\displaystyle F\text{ is a co-oriented taut foliation on }Y\!\text{ transverse to }\unicode[STIX]{x2202}Y,\\ \displaystyle \text{restricting to a rational co-oriented }linear\text{foliation on }\unicode[STIX]{x2202}Y\end{array}\right\},\nonumber\\ \displaystyle \displaystyle {\mathcal{F}}^{D}(Y) & := & \displaystyle \{\unicode[STIX]{x1D707}\mid \text{The Dehn filling }Y(\unicode[STIX]{x1D707})\text{ admits a co-oriented taut foliation}\},\nonumber\\ \displaystyle \displaystyle {\mathcal{F}}(Y) & := & \displaystyle \{\unicode[STIX]{x1D6FC}(F\cap \unicode[STIX]{x2202}Y)\mid F\text{ is a co-oriented taut foliation on }Y\!\text{ transverse to }\unicode[STIX]{x2202}Y\}.\nonumber\end{eqnarray}$$

All linear foliations, even irrational ones, are taut, but rational linear foliations are product foliations, hence extend to co-oriented taut foliations on Dehn fillings of matching slope, implying ${\mathcal{F}}^{L}(Y)\subset {\mathcal{F}}^{D}(Y)\subset {\mathcal{F}}(Y)$ . In fact, the work of Jankins and Neumann tells us that ${\mathcal{F}}^{L}={\mathcal{F}}^{D}$ for manifolds Seifert fibered over the disk, and that the analogous result holds for manifolds Seifert fibered over a punctured $S^{2}$ . Since the same also holds for manifolds Seifert fibered over a punctured $\mathbb{R}\mathbb{P}^{2}$ [Reference Boyer and ClayBC14], and since Corollary 2.2 tells us that taut foliations on homology sphere graph manifolds isotop to restrict to taut foliations transverse to boundaries on Seifert fibered JSJ components, we additionally have ${\mathcal{F}}^{L}={\mathcal{F}}^{D}$ for any graph manifold with torus boundary and $b_{1}=1$ .

In this latter case, it is natural to ask whether ${\mathcal{F}}(Y)$ admits a description analogous to the Dehn filling characterization for ${\mathcal{F}}^{L}(Y)$ . That is, can ${\mathcal{F}}(Y)$ be characterized in terms of taut foliations on some closed union of $Y$ with some other manifold? Boyer and Clay answer this question affirmatively [Reference Boyer and ClayBC14], as we shall see.

Let $\bar{N}$ denote the regular fiber complement

(24) $$\begin{eqnarray}\bar{N}:=M_{S^{2}}(0,-{\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}})\setminus (S^{1}\!\times \!D_{0}^{2}),\end{eqnarray}$$

which Boyer and Clay call the ‘twisted $I$ -bundle over the Klein bottle’, or $N_{2}$ . The manifold $\bar{N}$ can play a role analogous to that of the solid torus for Dehn fillings.

We then have the following result for $\bar{N}$ -fillings.

In particular, for a graph manifold $Y$ with torus boundary, we have $\unicode[STIX]{x1D707}\in {\mathcal{F}}(Y)$ if and only if an $\bar{N}$ -filling $Y^{\bar{N}}(\unicode[STIX]{x1D707})$ admits a co-oriented taut foliation.

3.1 L-space Dehn fillings and $\bar{N}$ -fillings

Definition 3.1. If $Y$ is a compact oriented three-manifold with torus boundary, then we define the L-space interval of $Y$ to be

(25) $$\begin{eqnarray}{\mathcal{L}}(Y):=\{\unicode[STIX]{x1D707}\in \mathbb{P}(H_{1}(\unicode[STIX]{x2202}Y))|\,Y(\unicode[STIX]{x1D707})\text{ is an L-space}\}.\end{eqnarray}$$

We shall write ${\mathcal{L}}^{\circ }(Y)$ for the interior of ${\mathcal{L}}(Y)$ in $\mathbb{P}(H_{1}(\unicode[STIX]{x2202}Y))$ .

Thus, ${\mathcal{L}}(Y)$ is analogous to, and often complementary to, ${\mathcal{F}}^{D}(Y)$ , especially when $Y$ has no reducible non-L-space Dehn fillings. Moreover, since the set of slopes of co-oriented taut foliations meeting a generalized rotation number condition is often the closure of the set of product foliation slopes meeting that condition [Reference Calegari and WalkerCW11, Lemma 3.31], it is natural to ask if ${\mathcal{F}}(Y)$ bears any relation to the complement of ${\mathcal{L}}^{\circ }(Y)$ . In fact, we have the following result.

Thus ${\mathcal{L}}^{\circ }(Y)$ can be regarded as the L-space $\bar{N}$ -filling interval of $Y$ .

3.2 L-space gluing

Our primary tool for characterizing when a union of three-manifolds along a torus boundary gives an L-space is the following joint result of Rasmussen and the author [Reference Rasmussen and RasmussenRR15]. Hanselman and Watson have proven a similar result in [Reference Hanselman and WatsonHW15].

We shall later show that in the case of non-solid-torus graph manifolds $Y_{i}$ with torus boundary, various foliation results allow us to drop some of the above hypotheses, so that one obtains an L-space if and only if $\mathbb{P}(H_{1}(\unicode[STIX]{x2202}Y_{2}))=\unicode[STIX]{x1D711}_{\ast }^{\mathbb{P}}({\mathcal{L}}^{\circ }(Y_{1}))\cup {\mathcal{L}}^{\circ }(Y_{2})$ .

In rather the opposite direction, if $Y$ is Seifert fibered over a punctured $S^{2}$ or $\mathbb{R}\mathbb{P}^{2}$ , then a Dehn filling $Y^{\prime }$ of $Y$ fails to be a graph manifold if and only if $Y^{\prime }$ fails to be prime, if and only if $Y^{\prime }\neq S^{1}\times S^{2}$ and the Dehn filling, in some $\unicode[STIX]{x2202}_{i}Y$ , was along the fiber lift $\tilde{f}_{i}\in H_{1}(\unicode[STIX]{x2202}_{i}Y)$ of slope $\unicode[STIX]{x1D70B}_{i}(\tilde{f}_{i})=\infty$ (see the remark near the end of § 2.2). In this case, $Y^{\prime }$ is neither a graph manifold nor a habitat for taut foliations, but since it has compressible boundary, its L-space gluing properties simplify, due to the following result.

The above result explains why not every graph manifold $Y$ with torus boundary satisfies ${\mathcal{F}}^{D}(Y)\amalg {\mathcal{L}}(Y)=\mathbb{P}(H_{1}(\unicode[STIX]{x2202}Y))$ . That is, no reducible manifold (besides $S^{1}\times D^{2}$ ) admits a co-oriented taut foliation, but there exist graph manifolds with reducible Dehn fillings which are not $S^{1}\times D^{2}$ or an L-space.

3.3 Floer simple manifolds and L-space intervals

It is not known, in general, what forms the sets ${\mathcal{F}}(Y)$ or ${\mathcal{F}}^{D}(Y)$ can take for an arbitrary compact oriented three-manifold $Y$ with torus boundary, but the situation for L-spaces is better understood. As shown in [Reference Rasmussen and RasmussenRR15] by Rasmussen and the author, ${\mathcal{L}}(Y)$ can only be empty, the set of a single point, a closed interval, or the complement of the rational longitude in $\mathbb{P}(H_{1}(\unicode[STIX]{x2202}Y))$ . For historical reasons, we call $Y$ Floer simple in the latter two cases. Equivalently, we could define Floer simple manifolds as follows.

In particular, if $Y$ is Floer simple, then its space ${\mathcal{L}}(Y)$ of L-space Dehn filling slopes can be specified entirely in terms of the left-hand and right-hand endpoints of ${\mathcal{L}}(Y)$ , in a sense we can make precise, prefaced with the introduction of an abbreviative notation for the closed interval with infinite endpoint.

Definition 3.6. For $y\in \mathbb{Q}$ , we shall write $[-\infty ,y]$ , $[y,+\infty ]$ , $[-\infty ,y\rangle$ , and $\langle y,+\infty ]$ for the following intervals in $(\mathbb{Q}\cup \{\infty \})\subset (\mathbb{R}\cup \{\infty \})$ :

$$\begin{eqnarray}\begin{array}{@{}l@{}}\displaystyle \,[-\infty ,y]:=\{\infty \}\cup \langle -\infty ,y],\quad [-\infty ,y\rangle :=\{\infty \}\cup \langle -\infty ,y\rangle ,\\ \displaystyle \,[y,+\infty ]:=[y,+\infty \rangle \cup \{\infty \},\quad \langle y,+\infty ]:=\langle y,+\infty \rangle \cup \{\infty \}.\end{array}\end{eqnarray}$$

Definition 3.7. If $y_{-},y_{+}\!\in \mathbb{Q}\cup \{\infty \}$ , then we define the L-space interval from $y_{-}$ to $y_{+}$ , denoted $[[y_{-},y_{+}]]\subset \mathbb{Q}\cup \{\infty \}$ , as follows:

(26) $$\begin{eqnarray}[[y_{-},y_{+}]]:=\left\{\begin{array}{@{}ll@{}}\langle -\infty ,+\infty \rangle ,\quad & \infty =y_{-},y_{+}=\infty ,\\ \langle y_{-},+\infty ]\cup [-\infty ,y_{+}\rangle ,\quad & \mathbb{Q}\ni y_{-}=y_{+}\in \mathbb{Q},\\ \text{}[y_{-},+\infty ]\cup [-\infty ,y_{+}],\quad & \mathbb{Q}\ni y_{-}>y_{+}\in \mathbb{Q},\\ \text{}[y_{-},+\infty ]\cap [-\infty ,y_{+}],\quad & \mathbb{Q}\ni y_{-}<y_{+}\in \mathbb{Q},\\ \text{}[-\infty ,y_{+}],\quad & \infty =y_{-},y_{+}\in \mathbb{Q},\\ \text{}[y_{-},+\infty ],\quad & \,\mathbb{Q}\ni y_{-},y_{+}=\infty .\end{array}\right.\end{eqnarray}$$

In other words, $[[y_{-},y_{+}]]$ is the unique interval with left-hand endpoint $y_{-}$ and right-hand endpoint $y_{+}$ which is closed if $y_{-}\neq y_{+}$ and open otherwise.

The above follows from the aforementioned result, proven in [Reference Rasmussen and RasmussenRR15], that if ${\mathcal{L}}(Y)$ contains more than one point, then ${\mathcal{L}}(Y)$ is either a closed interval or the complement of a point in $\mathbb{P}(H_{1}(\unicode[STIX]{x2202}Y))$ . The following computation of L-space intervals for Seifert fibered spaces demonstrates one use of this ‘ $[[\cdot ,\cdot ]]$ ’ notation.

Proposition 3.9. If $Y$ is a regular fiber complement in a Seifert fibered rational homology sphere, then ${\mathcal{L}}^{\circ }(Y)\amalg {\mathcal{F}}(Y)=\mathbb{P}(H_{1}(\unicode[STIX]{x2202}Y))$ . If $Y$ has non-orientable base, we have ${\mathcal{L}}(Y)=\langle -\infty ,+\infty \rangle$ and ${\mathcal{F}}^{D}(Y)=\emptyset$ (unless $Y$ is the twisted $S^{2}$ -bundle over the Möbius strip, in which case ${\mathcal{F}}^{D}(Y)=\{\infty \}$ ). If $Y$ is a regular fiber complement in $M_{S^{2}}(y_{\ast })$ , for some $y_{\ast }=(y_{0},\ldots ,y_{n})\in \mathbb{Q}^{n+1}$ , then $\mathbb{P}(H_{1}(\unicode[STIX]{x2202}Y))\setminus {\mathcal{F}}^{D}(Y)={\mathcal{L}}(Y)=[[y_{-},y_{+}]]$ , where

(27) $$\begin{eqnarray}y_{-}:=\max _{k>0}-{\displaystyle \frac{1}{k}}\biggl(1+\mathop{\sum }_{i=0}^{n}\lfloor y_{i}k\rfloor \biggr),\quad y_{+}:=\min _{k>0}-{\displaystyle \frac{1}{k}}\biggl(-1+\mathop{\sum }_{i=0}^{n}\lceil y_{i}k\rceil \biggr),\end{eqnarray}$$

unless $Y$ is a solid torus, in which case $y_{-}:=y_{+}:=-\!\sum _{i=0}^{n}y_{i}$ is the rational longitude of $Y$ .

4.1 L/NTF-equivalence and gluing

For a pair of manifolds spliced together along torus boundaries, we can often prove stronger gluing results about the existence of co-oriented taut foliations or non-trivial Heegaard Floer homology if we are able to use gluing theorems from both areas of mathematics. In general, however, this strategy only works if we know that each manifold behaves in a suitably complementary manner with respect to co-oriented taut foliations and L-space Dehn fillings, a notion which we now make precise.

In certain circumstances, one can characterize L/NTF-equivalence in terms of $\bar{N}$ -fillings.

There are some classes of manifold which we already know to be L/NTF-equivalent.

We are now ready to state our main gluing result.

4.2 Graph manifold conventions

Any closed graph manifold can be regarded as a Dehn filling of a graph manifold $Y$ with torus boundary. If $b_{1}(Y)>1$ , then ${\mathcal{L}}(Y)={\mathcal{L}}^{\circ }(Y)=\emptyset$ and ${\mathcal{F}}(Y)=\mathbb{P}(H_{1}(\unicode[STIX]{x2202}Y))$ (see Proposition 4.3), which is not very interesting.

If $b_{1}(Y)=1$ , then we call $Y$ a tree manifold, since $Y$ admits a rational homology sphere Dehn filling, corresponding to a tree graph. Rooting the tree graph for $Y$ at the Seifert fibered piece containing $\unicode[STIX]{x2202}Y$ provides a recursive construction for $Y$ ,

(28) $$\begin{eqnarray}Y=\hat{M}\cup \biggl(\coprod _{i=1}^{n_{{\rm \small{d}}}}(S^{1}\times D_{i}^{2})\;\amalg \;\coprod _{i=1}^{n_{{\rm \small{g}}}}Y_{i}\biggr),\end{eqnarray}$$

where each of $Y_{1},\ldots ,Y_{n_{{\rm \small{g}}}}$ is a non-solid-torus tree manifold with torus boundary. Since $b_{1}(Y)=1$ , $\hat{M}$ is the trivial circle fibration over an ( $n+1$ )-punctured $S^{2}$ or the twisted circle fibration over an ( $n+1$ )-punctured $\mathbb{R}\mathbb{P}^{2}\!$ , with boundary components $\unicode[STIX]{x2202}_{i}^{{\rm \small{d}}}\hat{M}:=\unicode[STIX]{x2202}_{i}\hat{M}$ for $i\in \{1,\ldots ,n_{{\rm \small{d}}}\}$ , $\unicode[STIX]{x2202}_{i}^{{\rm \small{g}}}\hat{M}:=\unicode[STIX]{x2202}_{n_{{\rm \small{d}}}+i}\hat{M}$ for $i\in \{1,\ldots ,n_{{\rm \small{g}}}\}$ , and $\unicode[STIX]{x2202}Y:=\unicode[STIX]{x2202}_{n+1}\hat{M}=:\unicode[STIX]{x2202}_{n_{{\rm \small{g}}}+1}^{{\rm \small{g}}}\hat{M}=:\unicode[STIX]{x2202}_{n_{{\rm \small{d}}}+1}^{{\rm \small{d}}}\hat{M}$ , with $n:=n_{{\rm \small{d}}}+n_{{\rm \small{g}}}$ . We shall sometimes call $\hat{M}$ the ‘foundation’ for $Y$ .

Since edges in the graph for $Y$ correspond to gluings along incompressible tori, each gluing map $\unicode[STIX]{x1D711}_{i}:\unicode[STIX]{x2202}Y_{i}\rightarrow -\unicode[STIX]{x2202}_{i}^{{\rm \small{g}}}\hat{M}$ , for $i\in \{1,\ldots ,n_{{\rm \small{g}}}\}$ , labels one of the $n_{{\rm \small{g}}}$ edges descending from the root of the graph for $Y$ . We call $Y_{1},\ldots ,Y_{n_{{\rm \small{g}}}}$ the daughter subtrees of $Y$ . Each $Y_{i}$ is a tree manifold with torus boundary and $b_{1}(Y_{i})=1$ , with tree rooted at the Seifert fibered piece containing $\unicode[STIX]{x2202}Y_{i}$ , giving rise to a recursive description for $Y_{i}$ analogous to that for $Y$ in (28). For any $i\in \{1,\ldots ,n_{{\rm \small{g}}}\}$ for which $Y_{i}$ is Floer simple, i.e. for which ${\mathcal{L}}^{\circ }(Y_{i})\neq \emptyset$ , we invoke Proposition 3.8 to write

(29) $$\begin{eqnarray}[[y_{i-}^{{\rm \small{g}}},y_{i+}^{{\rm \small{g}}}]]:=\unicode[STIX]{x1D711}_{i\ast }^{\mathbb{P}}({\mathcal{L}}(Y_{i})).\end{eqnarray}$$

If, instead, ${\mathcal{L}}(Y_{i})\neq \emptyset$ for some non-Floer-simple $Y_{i}$ , then we write

(30) $$\begin{eqnarray}\{y_{i-}^{{\rm \small{g}}}\}:=\{y_{i+}^{{\rm \small{g}}}\}:=\unicode[STIX]{x1D711}_{i\ast }^{\mathbb{P}}({\mathcal{L}}(Y_{i})).\end{eqnarray}$$

Since the $n_{{\rm \small{d}}}$ solid tori glued to $\hat{M}$ create the exceptional fibers of the Seifert fibered ‘root’ $\hat{M}\cup \coprod _{i=1}^{n_{{\rm \small{d}}}}(S^{1}\times D_{i}^{2})$ of our tree, we record the Seifert data of this Seifert fibered space by labeling the root vertex with the Dehn filling slopes $y_{1}^{{\rm \small{d}}},\ldots ,y_{n_{{\rm \small{d}}}}^{{\rm \small{d}}}\in \mathbb{Q}$ . More explicitly, to each solid torus $S^{1}\times D_{i}^{2}$ in (28) we associate the gluing map $\unicode[STIX]{x1D711}_{i}^{{\rm \small{d}}}:\unicode[STIX]{x2202}(S^{1}\times D_{i}^{2})\rightarrow -\unicode[STIX]{x2202}_{i}^{{\rm \small{d}}}\hat{M}$ , and set $y_{i}^{{\rm \small{d}}}:=\unicode[STIX]{x1D711}_{i\ast }^{{\rm \small{d}}\mathbb{P}}(l_{i})$ , for $l_{i}$ the rational longitude of $S^{1}\times D_{i}^{2}$ . As usual, we demand that each $y_{i}^{{\rm \small{d}}}\neq \infty$ . We stray slightly from our earlier convention by allowing $y_{i}^{{\rm \small{d}}}\in \mathbb{Z}$ , but this allows us to fix $e_{0}:=y_{0}^{{\rm \small{d}}}:=0$ and then forget the zeroth fiber complement altogether, without loss of generality.

4.3 Statement of main results

We first show that all graph manifolds with torus boundary are L/NTF-equivalent, making our main gluing tool, Proposition 4.4, applicable for all such non-solid-torus graph manifolds. We then can make the inductive gluing arguments necessary to calculate L-space intervals for graph manifolds with torus boundary.

The above also implies that the following calculation of ${\mathcal{L}}(Y)$ for graph manifolds $Y\!$ with torus boundary completely determines both ${\mathcal{F}}(Y)$ and ${\mathcal{F}}^{D}(Y)$ .

Theorem 4.6. Suppose $Y$ is a graph manifold with torus boundary and non-empty ${\mathcal{L}}(Y)$ . If the Seifert fibered component of $Y$ containing $\unicode[STIX]{x2202}Y$ has non-orientable base, then ${\mathcal{L}}(Y)=\langle -\infty ,+\infty \rangle$ . Otherwise, we have

$$\begin{eqnarray}{\mathcal{L}}(Y)=\left\{\begin{array}{@{}ll@{}}[[y_{-},y_{+}]],\quad & \quad Y\,\text{Floer simple},\\ \{y_{-}\}=\{y_{+}\},\quad & \quad Y\text{ not Floer simple},\end{array}\right.\end{eqnarray}$$

where, for $n_{{\rm \small{d}}},n_{{\rm \small{g}}},y_{i}^{{\rm \small{d}}},y_{i-}^{{\rm \small{g}}},$ and $y_{i+}^{{\rm \small{g}}}$ as defined in § 4.2, we define $y_{-},y_{+}\in \mathbb{Q}\cup \{\infty \}$ as

$$\begin{eqnarray}\displaystyle y_{-} & := & \displaystyle \,\max _{k>0}-{\displaystyle \frac{1}{k}}\biggl(1+\mathop{\sum }_{i=1}^{n_{{\rm \small{d}}}}\lfloor y_{i}^{{\rm \small{d}}}k\rfloor +\mathop{\sum }_{i=1}^{n_{{\rm \small{g}}}}(\lceil y_{i+}^{{\rm \small{g}}}k\rceil -1)\biggr),\nonumber\\ \displaystyle y_{+} & := & \displaystyle \,\min _{k>0}-{\displaystyle \frac{1}{k}}\biggl(-1+\mathop{\sum }_{i=1}^{n_{{\rm \small{d}}}}\lceil y_{i}^{{\rm \small{d}}}k\rceil +\mathop{\sum }_{i=1}^{n_{{\rm \small{g}}}}(\lfloor y_{i-}^{{\rm \small{g}}}k\rfloor +1)\biggr),\nonumber\end{eqnarray}$$

unless $Y\!$ is a solid torus, in which case $y_{-}\!:=y_{+}\!:=-\!\sum _{i=1}^{n_{{\rm \small{d}}}}y_{i}^{{\rm \small{d}}}$ .

Unlike the case of oriented Seifert fibered spaces over the Möbius strip or disk, not all graph manifolds with torus boundary are Floer simple. We therefore need a companion result to characterize precisely when ${\mathcal{L}}^{\circ }$ or ${\mathcal{L}}$ is non-empty.

Note that all eight $({\rm \small{fs}})$ and $({\rm \small{nfs}})$ conditions are mutually exclusive. Note also that the isolated L-space fillings described in $({\rm \small{nfs3}})$ and $({\rm \small{nfs4}})$ are not graph manifolds.

We now proceed to prove our main results, starting with that of L/NTF-equivalence.

4.4 Proof of Theorem 4.5

Proposition 4.3 gives the desired result for $b_{1}(Y)>1$ .

We therefore restrict attention to graph manifolds $Y$ with $b_{1}(Y)=1$ and torus boundary $\unicode[STIX]{x2202}Y$ , so that $Y$ admits the recursive description in (28), with tree graph rooted at the Seifert fibered piece containing $\unicode[STIX]{x2202}Y$ . Inductively assume that any such $Y\!$ with tree height less than or equal to $k-1$ is L/NTF-equivalent and satisfies ${\mathcal{F}}(Y)\amalg ({\mathcal{L}}(Y)\cup {\mathcal{R}}(Y))=\mathbb{P}(H_{1}(\unicode[STIX]{x2202}Y))$ , noting that Proposition 3.9 covers the case of trees of height zero. Fix an arbitrary tree manifold $Y$ with $b_{1}(Y)=1$ , torus boundary, and tree height $k>0$ , and parameterize its data as in § 4.2.

To each $j\in \{1,\ldots ,n_{{\rm \small{g}}}\}$ , $y_{\ast }^{j}:=(y_{j+1}^{j},\ldots ,y_{n_{{\rm \small{g}}}+1}^{j})\in (\mathbb{Q}\cup \{\infty \})^{n_{{\rm \small{g}}}+1-j}$ , and $\unicode[STIX]{x1D703}\in \{0,1\}$ we associate a manifold $Y_{\unicode[STIX]{x1D703}}^{j}[y_{\ast }^{j}]$ , constructed as follows. Starting with $\hat{M}$ , first perform Dehn fillings of slopes $y_{1}^{{\rm \small{d}}},\ldots ,y_{n_{{\rm \small{d}}}}^{{\rm \small{d}}}$ along the respective boundary components $\unicode[STIX]{x2202}_{1}^{{\rm \small{d}}}\hat{M},\ldots ,\unicode[STIX]{x2202}_{n_{{\rm \small{d}}}}^{{\rm \small{d}}}\hat{M}$ in $\hat{M}$ . Next, attach the graph manifolds $Y_{1},\ldots ,Y_{j-1}$ , via the respective gluing maps $\unicode[STIX]{x1D711}_{1},\ldots ,\unicode[STIX]{x1D711}_{j-1}$ , to the resulting manifold. Leaving the boundary component $\unicode[STIX]{x2202}_{j}^{{\rm \small{g}}}\hat{M}=:\unicode[STIX]{x2202}Y^{j}[y_{\ast }^{j}]$ unfilled, lastly perform $\bar{N}$ -fillings of slopes $y_{j+1}^{j},\ldots ,y_{n_{{\rm \small{g}}}+1}^{j}$ along the respective boundary components $\unicode[STIX]{x2202}_{j+1}^{{\rm \small{g}}}\hat{M},\ldots ,\unicode[STIX]{x2202}_{n_{{\rm \small{g}}}+1}^{{\rm \small{g}}}\hat{M}$ of the resulting manifold, and call the result $Y_{0}^{j}[y_{\ast }^{j}]$ . To form $Y_{1}^{j}[y_{\ast }^{j}]$ , replace the $\bar{N}$ -filling of slope $y_{n_{{\rm \small{g}}}+1}^{j}$ in $\unicode[STIX]{x2202}_{n_{{\rm \small{g}}}+1}^{{\rm \small{g}}}\hat{M}\subset Y_{0}^{j}[y_{\ast }^{j}]$ with the Dehn filling of slope $y_{n_{{\rm \small{g}}}+1}^{j}$ . In addition, set $y_{\ast }^{n_{{\rm \small{g}}}+1}:=\emptyset$ and $Y_{0}^{n_{{\rm \small{g}}}+1}[\emptyset ]:=Y$ .

For positive $j\leqslant n_{{\rm \small{g}}}$ , inductively assume that for any $y_{\ast }^{j}\in (\mathbb{Q}\cup \infty )^{n_{{\rm \small{g}}}+1-j}$ and $\unicode[STIX]{x1D703}\in \{0,1\}$ , any manifold of the form $Y_{\unicode[STIX]{x1D703}}^{j}[y_{\ast }^{j}]$ is L/NTF-equivalent if it is prime, noting that Proposition 4.3 covers the base case of $j=1$ . For any $y\in \mathbb{Q}\cup \{\infty \}$ , $y_{\ast }^{j+1}:=(y_{j+2}^{j},\ldots ,y_{n_{{\rm \small{g}}}+1}^{j})\in (\mathbb{Q}\cup \{\infty \})^{n_{{\rm \small{g}}}+1-j}$ , $\unicode[STIX]{x1D703}\in \{0,1\}$ , and manifold of the form $Y_{\unicode[STIX]{x1D703}}^{j+1}[y_{\ast }^{j+1}]$ , we can make matching choices of $\bar{N}$ -filling gluing maps to obtain

(31) $$\begin{eqnarray}Y_{\unicode[STIX]{x1D703}}^{j+1}[y_{\ast }^{j+1}]^{\bar{N}}(y)=Y_{j}\cup Y_{\unicode[STIX]{x1D703}}^{j}[(y,y_{\ast }^{j+1})].\end{eqnarray}$$

Suppose $Y_{\unicode[STIX]{x1D703}}^{j+1}[y_{\ast }^{j+1}]^{\bar{N}}(y)$ is prime, implying $Y_{\unicode[STIX]{x1D703}}^{j}[(y,y_{\ast }^{j+1})]$ is prime and hence L/NTF-equivalent by inductive assumption. Since $Y_{\unicode[STIX]{x1D703}}^{j}[(y,y_{\ast }^{j+1})]$ and $Y_{j}$ are L/NTF-equivalent non-solid-torus graph manifolds with torus boundary, Proposition 4.4 makes $Y_{\unicode[STIX]{x1D703}}^{j+1}[y_{\ast }^{j+1}]^{\bar{N}}(y)$ an L-space if and only if it fails to admit a co-oriented taut foliation. Since this holds for arbitrary $y\in \mathbb{Q}\cup \{\infty \}$ , Proposition 4.2 tells us $Y_{\unicode[STIX]{x1D703}}^{j+1}[y_{\ast }^{j+1}]$ is L/NTF-equivalent.

Completing our induction on $j$ , we conclude that $Y:=Y_{0}^{n_{{\rm \small{g}}}+1}[\emptyset ]$ is L/NTF-equivalent, and that $Y_{1}^{n_{{\rm \small{g}}}}[y]$ is L/NTF-equivalent for any $y\in \mathbb{Q}\cup \{\infty \}$ for which $Y_{1}^{n_{{\rm \small{g}}}}[y]$ is prime. Thus, for any prime Dehn filling $Y(y)$ , Proposition 4.4 tells us that the union

(32) $$\begin{eqnarray}Y(y)=Y_{n_{{\rm \small{g}}}}\cup _{\unicode[STIX]{x1D711}_{n_{{\rm \small{g}}}}}Y_{1}^{n_{{\rm \small{g}}}}[y]\end{eqnarray}$$

is an L-space if and only if it fails to admit a co-oriented taut foliation, and so

(33) $$\begin{eqnarray}\mathbb{P}(H_{1}(\unicode[STIX]{x2202}Y))\setminus ({\mathcal{L}}(Y)\cup {\mathcal{R}}(Y))={\mathcal{F}}(Y)\setminus {\mathcal{R}}(Y)={\mathcal{F}}(Y).\end{eqnarray}$$

Inducting on tree height $k$ then completes the proof.◻

We next prove Theorem 4.6 and Proposition 4.7 in tandem over the course of §§ 4.5–4.8. The inductive program laid out in § 4.5 spans all three of the subsequent subsections.

4.5 Inductive set-up for proof of Theorem 4.6 and Proposition 4.7

Both results hold automatically when $b_{1}(Y)\!>\!1$ . This leaves the case of $b_{1}(Y)\!=\!1$ , so that $Y$ admits the recursive description in (28), with tree rooted at the Seifert fibered piece containing $\unicode[STIX]{x2202}Y$ .

Inductively assume that both Theorem 4.6 and Proposition 4.7 hold for all tree manifolds with torus boundary, $b_{1}=1$ , and tree height less than or equal to $k-1$ , noting that Proposition 3.9 covers the height zero case. In addition, inductively assume that Theorem 4.6 and Proposition 4.7 hold for any tree manifold with torus boundary, $b_{1}=1$ , tree height less than or equal to $k$ , and up to $n_{{\rm \small{g}}}-1$ daughter subtrees, noting that Proposition 3.9 also covers the case of zero daughter subtrees.

For the remainder of the proof, we fix an arbitrary height $k$ tree manifold $Y$ with torus boundary and $b_{1}(Y)=1$ , as described in § 4.2. Thus, $Y$ has $n_{{\rm \small{g}}}$ daughter subtrees $Y_{1},\ldots ,Y_{n_{{\rm \small{g}}}}$ , attached via respective gluing maps $\unicode[STIX]{x1D711}_{1},\ldots ,\unicode[STIX]{x1D711}_{n_{{\rm \small{g}}}}$ , and the Seifert fibered piece containing $\unicode[STIX]{x2202}Y$ , at which we root the tree for $Y$ , is the Dehn filling of slope $y_{\ast }^{{\rm \small{d}}}=(y_{1}^{{\rm \small{d}}},\ldots ,y_{n_{{\rm \small{d}}}}^{{\rm \small{d}}})$ of the ‘foundation’ $\hat{M}$ of $Y,$ where $\hat{M}$ is either the trivial $S^{1}$ -fibration over an ( $n:=n_{{\rm \small{d}}}+n_{{\rm \small{g}}}+1$ )-punctured $S^{2}$ or the twisted $S^{1}$ -fibration over an $n$ -punctured $\mathbb{R}\mathbb{P}^{2}$ . For any $y\in \mathbb{Q}\cup \{\infty \}$ , let ${\hat{Y}}[y]$ denote the complement of $Y_{n_{{\rm \small{g}}}}\setminus \unicode[STIX]{x2202}Y_{n_{{\rm \small{g}}}}$ in the Dehn filling $Y(y)$ , so that we regard $Y(y)$ as the union

(34) $$\begin{eqnarray}Y(y)={\hat{Y}}[y]\cup _{\unicode[STIX]{x1D711}_{n_{{\rm \small{g}}}}}Y_{n_{{\rm \small{g}}}}.\end{eqnarray}$$

For any $y\in \mathbb{Q}$ , our inductive assumptions make Theorem 4.6 and Proposition 4.7 hold for ${\hat{Y}}[y]$ and $Y_{n_{{\rm \small{g}}}}$ , since $Y_{n_{{\rm \small{g}}}}$ has tree height less than or equal to $k-1$ , and since for $y\neq \infty$ , ${\hat{Y}}[y]$ is a $b_{1}=1$ tree manifold with torus boundary, $n_{{\rm \small{g}}}-1$ daughter subtrees, and tree height less than or equal to $k$ .

4.6 Non-orientable base

Consider the case in which $\hat{M}$ is $S^{1}$ -fibered over a punctured $\mathbb{R}\mathbb{P}^{2}$ . First note that since the regular fiber class is torsion, its primitive lift $\tilde{f}_{n_{{\rm \small{g}}}}^{{\rm \small{g}}}\in H_{1}(\unicode[STIX]{x2202}Y)$ , of slope $\infty$ , is the rational longitude, which means that $\infty \notin {\mathcal{L}}(Y)$ .

Suppose there is some $y\neq \infty$ for which ${\mathcal{L}}({\hat{Y}}[y])\neq \emptyset$ . Then, by inductive assumption, Proposition 4.7 tells us that the daughter subtrees $Y_{1},\ldots ,Y_{n_{{\rm \small{g}}}-1}$ are Floer simple, with $\infty \neq y_{i-}^{{\rm \small{g}}}\geqslant y_{i+}^{{\rm \small{g}}}\neq \infty$ for all $i\in \{1,\ldots ,n_{{\rm \small{g}}}-1\}$ , where $[[y_{i-}^{{\rm \small{g}}},y_{i+}^{{\rm \small{g}}}]]:=\unicode[STIX]{x1D711}_{i}({\mathcal{L}}(Y_{i}))$ . This conversely implies that ${\hat{Y}}[y]$ is Floer simple for all $y\in \langle -\infty ,+\infty \rangle$ . Now, for each $y\in \langle -\infty ,+\infty \rangle$ , ${\hat{Y}}[y]$ is a non-solid-torus graph manifold, and so Proposition 4.4 tells us that the union $Y(y)={\hat{Y}}[y]\cup Y_{n_{{\rm \small{g}}}}$ in (34) is an L-space if and only if

(35) $$\begin{eqnarray}{\mathcal{L}}^{\circ }({\hat{Y}}[y])\cup \unicode[STIX]{x1D711}_{n_{{\rm \small{g}}}\ast }^{\mathbb{P}}({\mathcal{L}}^{\circ }(Y_{n_{{\rm \small{g}}}}))=\mathbb{P}(H_{1}(\unicode[STIX]{x2202}_{n_{{\rm \small{g}}}}^{{\rm \small{ g}}}\hat{M})).\end{eqnarray}$$

Since, by inductive assumption, Theorem 4.6 implies ${\mathcal{L}}^{\circ }({\hat{Y}}[y])\!=\!\langle -\infty ,+\infty \rangle$ for all $y\!\in \!\langle -\infty ,+\infty \rangle$ , (35) holds if and only if $Y_{n_{{\rm \small{g}}}}$ is Floer simple and has $\infty \neq y_{n_{{\rm \small{g}}}-}^{{\rm \small{g}}}\geqslant y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}\neq \infty$ , in which case we consequently have ${\mathcal{L}}(Y)=\langle -\infty ,+\infty \rangle$ .

Conversely, suppose that ${\mathcal{L}}({\hat{Y}}[y])=\emptyset$ for all $y\in \mathbb{Q}$ . Then for all $y\in \mathbb{Q}$ , Proposition 4.4 implies that $Y(y)={\hat{Y}}[y]\cup Y_{n_{{\rm \small{g}}}}$ is not an L-space, and so ${\mathcal{L}}(Y)=\emptyset$ . Moreover, since there is $y\in \mathbb{Q}$ with ${\mathcal{L}}({\hat{Y}}[y])=\emptyset$ , our inductive assumption for Proposition 4.7 tells us that either there is some $i\in \{1,\ldots ,n_{{\rm \small{g}}}-1\}$ for which $Y_{i}$ is not Floer simple, or there is some Floer simple $Y_{i}$ failing to satisfy $\infty \neq y_{i-}^{{\rm \small{g}}}\geqslant y_{i+}^{{\rm \small{g}}}\neq \infty$ .

Thus, in either case, both Theorem 4.6 and Proposition 4.7 hold for $Y$ .

4.7 Orientable base: cases in which ${\hat{Y}}[y]$ is never a solid torus

From now on, we assume that the JSJ component of $Y$ containing $\unicode[STIX]{x2202}Y$ has orientable base.

In this subsection of the proof, we consider the case in which ${\hat{Y}}[y]$ is not a solid torus for any $y\in \mathbb{Q}\cup \{\infty \}$ . More precisely, we consider the case of a fixed tree manifold $Y$ with torus boundary and $b_{1}(Y)=1$ , parameterized as in § 4.2, with tree height $k>0$ and $n_{{\rm \small{g}}}>0$ daughter subtrees, where we demand that if $n_{{\rm \small{g}}}-1=0$ , then $y_{i}^{{\rm \small{d}}}\notin \mathbb{Z}$ for at least two distinct values of $i\in \{1,\ldots ,n_{{\rm \small{d}}}\}$ .

We begin by fixing some notation. For all $k\in \mathbb{Z}_{{>}0}$ , define ${\hat{y}}_{-}^{0}(k),{\hat{y}}_{+}^{0}(k)\in \mathbb{Q}\cup \{\infty \}$ by

(36) $$\begin{eqnarray}\begin{array}{@{}l@{}}\displaystyle {\hat{y}}_{-}^{0}(k):=-{\displaystyle \frac{1}{k}}\biggl(1+\mathop{\sum }_{i=1}^{n_{{\rm \small{d}}}}\lfloor y_{i}^{{\rm \small{d}}}k\rfloor +\mathop{\sum }_{i=1}^{n_{{\rm \small{g}}}-1}(\lceil y_{i+}^{{\rm \small{g}}}k\rceil -1)\biggr),\\ \displaystyle {\hat{y}}_{+}^{0}(k):=-{\displaystyle \frac{1}{k}}\biggl(-1+\mathop{\sum }_{i=1}^{n_{{\rm \small{d}}}}\lceil y_{i}^{{\rm \small{d}}}k\rceil +\mathop{\sum }_{i=1}^{n_{{\rm \small{g}}}-1}(\lfloor y_{i-}^{{\rm \small{g}}}k\rfloor +1)\biggr).\end{array}\end{eqnarray}$$

The endpoints $y_{-},y_{+}\in \mathbb{Q}\cup \{\infty \}$ defined in Theorem 4.6 are then given by

(37) $$\begin{eqnarray}y_{-}:=\max _{k>0}\biggl({\hat{y}}_{-}^{0}(k)-\frac{1}{k}(\lceil y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}k\rceil -1)\biggr),\quad y_{+}:=\min _{k>0}\biggl({\hat{y}}_{+}^{0}(k)-\frac{1}{k}(\lfloor y_{n_{{\rm \small{g}}}-}^{{\rm \small{g}}}k\rfloor +1)\biggr).\end{eqnarray}$$

Moreover, if we define the functions ${\hat{y}}_{-},{\hat{y}}_{+}\in \mathbb{Q}\cup \{\infty \}$ of $y\in \mathbb{Q}\cup \{\infty \}$ by

(38) $$\begin{eqnarray}{\hat{y}}_{-}:=\max _{k>0}\biggl(-\frac{1}{k}\lfloor yk\rfloor +{\hat{y}}_{-}^{0}(k)\biggr),\quad {\hat{y}}_{+}:=\min _{k>0}\biggl(-\frac{1}{k}\lceil yk\rceil +{\hat{y}}_{+}^{0}(k)\biggr),\end{eqnarray}$$

then by inductive assumption, we have

(39) $$\begin{eqnarray}{\mathcal{L}}({\hat{Y}}[y])=\left\{\begin{array}{@{}ll@{}}\{{\hat{y}}_{-}\}=\{{\hat{y}}_{+}\},\quad & {\hat{Y}}[y]\text{ not Floer simple},\\ \text{}[[{\hat{y}}_{-},{\hat{y}}_{+}]],\quad & {\hat{Y}}[y]\;\text{Floer simple},\end{array}\right.\end{eqnarray}$$

for all $y\in \mathbb{Q}$ for which ${\mathcal{L}}({\hat{Y}}[y])\neq \emptyset$ . Since ${\hat{Y}}[y]$ and $Y_{n_{{\rm \small{g}}}}$ are each non-solid-torus graph manifolds for all $y\in \mathbb{Q}$ , Proposition 4.4 then implies, for each $y\in \mathbb{Q}$ , that

(40) $$\begin{eqnarray}y\in {\mathcal{L}}(Y)\quad \;\Longleftrightarrow \;\quad {\mathcal{L}}^{\circ }({\hat{Y}}[y])\cup \unicode[STIX]{x1D711}_{n_{{\rm \small{g}}}\ast }^{\mathbb{P}}({\mathcal{L}}^{\circ }(Y_{n_{{\rm \small{g}}}}))=\mathbb{Q}\cup \{\infty \}\end{eqnarray}$$

for all $y\in \mathbb{Q}$ . Note that ${\hat{Y}}[\infty ]$ is not a graph manifold, being a non-solid-torus with compressible boundary, hence not prime.

We next prove some basic rules about the behavior of $y_{-}$ and $y_{+}$ .

Claim 1. If $y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}},{\hat{y}}_{-},y_{-},y\in \mathbb{Q}$ , then

(41) $$\begin{eqnarray}y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}>{\hat{y}}_{-}\quad \;\Longleftrightarrow \;\quad y\in [y_{-},+\infty ].\end{eqnarray}$$

If $y_{n_{{\rm \small{g}}}-}^{{\rm \small{g}}},{\hat{y}}_{+},y_{+},y\in \mathbb{Q}$ , then

(42) $$\begin{eqnarray}y_{n_{{\rm \small{g}}}-}^{{\rm \small{g}}}<{\hat{y}}_{+}\quad \;\Longleftrightarrow \;\quad y\in [-\infty ,y_{+}].\end{eqnarray}$$

Proof of Claim 1.

Suppose that $y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}},{\hat{y}}_{-},y_{-},y\in \mathbb{Q}$ . Then

(43) $$\begin{eqnarray}\displaystyle & \displaystyle y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}>{\hat{y}}_{-} & \displaystyle \nonumber\\ \displaystyle \displaystyle \;\Longleftrightarrow \; & y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}>-\frac{1}{k}\lfloor yk\rfloor +{\hat{y}}_{-}^{0}(k) & \displaystyle \forall k\in \mathbb{Z}_{{>}0}\end{eqnarray}$$

(44) $$\begin{eqnarray}\displaystyle \displaystyle \;\Longleftrightarrow \; & \lceil y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}k\rceil -1\geqslant -\lfloor yk\rfloor +{\hat{y}}_{-}^{0}(k)k & \displaystyle \forall k\in \mathbb{Z}_{{>}0}\end{eqnarray}$$

(45) $$\begin{eqnarray}\displaystyle \displaystyle \;\Longleftrightarrow \; & yk\geqslant {\hat{y}}_{-}^{0}(k)k-(\lceil y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}k\rceil -1) & \displaystyle \forall k\in \mathbb{Z}_{{>}0}\end{eqnarray}$$

(46) $$\begin{eqnarray}\displaystyle \displaystyle \;\Longleftrightarrow \; & y\geqslant y_{-}, & \displaystyle\end{eqnarray}$$

where (37) implies (43), (22) implies (44), (21) implies (45), and (38) implies (46). The proof of (42) is nearly identical, but with signs reversed.◻

Claim 2. If $y_{1}^{{\rm \small{d}}},\ldots ,y_{n_{{\rm \small{d}}}}^{{\rm \small{d}}},y_{1+}^{{\rm \small{g}}},\ldots ,y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}\in \mathbb{Q},$ then

(47) $$\begin{eqnarray}y_{-}>-\!\mathop{\sum }_{i=1}^{n_{{\rm \small{d}}}}y_{i}^{{\rm \small{d}}}-\mathop{\sum }_{i=1}^{n_{{\rm \small{g}}}}y_{i+}^{{\rm \small{g}}}.\end{eqnarray}$$

If $y_{1}^{{\rm \small{d}}},\ldots ,y_{n_{{\rm \small{d}}}}^{{\rm \small{d}}},y_{1-}^{{\rm \small{g}}},\ldots ,y_{n_{{\rm \small{g}}}-}^{{\rm \small{g}}}\in \mathbb{Q},$ then

(48) $$\begin{eqnarray}y_{+}<-\!\mathop{\sum }_{i=1}^{n_{{\rm \small{d}}}}y_{i}^{{\rm \small{d}}}-\mathop{\sum }_{i=1}^{n_{{\rm \small{g}}}}y_{i-}^{{\rm \small{g}}}.\end{eqnarray}$$

Proof of Claim 2.

Writing $[\cdot ]:\mathbb{Q}\rightarrow [0,1\rangle$ for the map sending $q\mapsto [q]:=q-\lfloor q\rfloor$ , define

(49) $$\begin{eqnarray}y_{-}^{\prime }(k):={\displaystyle \frac{1}{k}}\biggl(-1+\mathop{\sum }_{i=1}^{n_{{\rm \small{d}}}}[y_{i}^{{\rm \small{d}}}k]+\mathop{\sum }_{i=1}^{n_{{\rm \small{g}}}}(1-[-y_{i+}^{{\rm \small{g}}}k])\biggr)\end{eqnarray}$$

for each $k\in \mathbb{Z}_{{>}0}$ , so that

(50) $$\begin{eqnarray}y_{-}=-\!\mathop{\sum }_{i=1}^{n_{{\rm \small{d}}}}y_{i}^{{\rm \small{d}}}-\!\mathop{\sum }_{i=1}^{n_{{\rm \small{g}}}}y_{i+}^{{\rm \small{g}}}+\max _{k>0}y_{-}^{\prime }(k).\end{eqnarray}$$

In addition, let

(51) $$\begin{eqnarray}s_{+}:=\min \{k\in \mathbb{Z}_{{>}0}\mid y_{1}^{{\rm \small{d}}}k,\ldots ,y_{n_{{\rm \small{d}}}}^{{\rm \small{ d}}}k,y_{1+}^{{\rm \small{g}}}k,\ldots ,y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}k\in \mathbb{Z}\}\end{eqnarray}$$

denote the least common positive multiple of the denominators of the $\{y_{i}^{{\rm \small{d}}}\}$ and $\{y_{i+}^{{\rm \small{g}}}\}$ .

Suppose (47) fails to hold. Then since $y_{-}^{\prime }(k)\leqslant 0$ for all $k\in \mathbb{Z}_{{>}0}$ , we have

(52) $$\begin{eqnarray}\displaystyle 0 & {\geqslant} & \displaystyle y_{-}^{\prime }(1)+(s_{+}-1)y_{-}^{\prime }(s_{+}-1)\nonumber\\ \displaystyle & = & \displaystyle -2+\mathop{\sum }_{i=1}^{n_{{\rm \small{d}}}}([y_{i}^{{\rm \small{d}}}]+[-y_{i}^{{\rm \small{d}}}])+\mathop{\sum }_{i=1}^{n_{{\rm \small{g}}}}(1+(1-[-y_{i+}^{{\rm \small{g}}}]-[y_{i+}^{{\rm \small{g}}}]))\nonumber\\ \displaystyle & {\geqslant} & \displaystyle -2+|\{i:y_{i}^{{\rm \small{d}}}\in \mathbb{Q}\setminus \mathbb{Z}\}|+n_{{\rm \small{g}}},\end{eqnarray}$$

and likewise, we have

(53) $$\begin{eqnarray}0\geqslant s_{+}y_{-}^{\prime }(s_{+})=-1+\mathop{\sum }_{i=1}^{n_{{\rm \small{g}}}}1=n_{{\rm \small{g}}}-1.\end{eqnarray}$$

The hypotheses of § 4.7, however, demand that either $|\{i:y_{i}^{{\rm \small{d}}}\in \mathbb{Q}\setminus \mathbb{Z}\}|\geqslant 2$ and $n_{{\rm \small{g}}}=1$ , contradicting (52), or $n_{{\rm \small{g}}}\!>\!1$ , contradicting (53). Thus (47) holds, and a similar argument proves (48).◻

For the proof that Theorem 4.6 and Proposition 4.7 hold for $Y$ , we divide our argument into two main cases, depending on whether or not $\infty \in {\mathcal{L}}(Y)$ .

Proof. Since ${\hat{Y}}[\infty ]$ is not a graph manifold, our inductive assumptions fail to hold for ${\hat{Y}}[\infty ]$ , but fortunately, $Y(\infty )$ has a simple structure. The remark in § 2.2 implies

(54) $$\begin{eqnarray}Y(\infty )=\biggl(\#_{i=1}^{n_{{\rm \small{d}}}}L(s_{i}^{{\rm \small{d}}},r_{i}^{{\rm \small{d}}})\biggr)\#\biggl(\#_{i=1}^{n_{{\rm \small{g}}}}Y_{i}((\unicode[STIX]{x1D711}_{i\ast }^{\mathbb{P}})^{-1}(\infty ))\biggr),\end{eqnarray}$$

where we have written $y_{i}^{{\rm \small{d}}}=r_{i}^{{\rm \small{d}}}/s_{i}^{{\rm \small{d}}}$ , $L(s_{i}^{{\rm \small{d}}},r_{i}^{{\rm \small{d}}})$ denotes the lens space of slope $s_{i}^{{\rm \small{d}}}/r_{i}^{{\rm \small{d}}}$ , and $Y_{i}((\unicode[STIX]{x1D711}_{i\ast }^{\mathbb{P}})^{-1}(\infty ))$ is the Dehn filling of the inverse image of the slope $\infty$ . Thus,

(55) $$\begin{eqnarray}\infty \in {\mathcal{L}}(Y)~\;\Longleftrightarrow \;~\infty \in \unicode[STIX]{x1D711}_{i\ast }^{\mathbb{P}}({\mathcal{L}}(Y_{i}))\quad \text{for all }i\in \{1,\ldots ,n_{{\rm \small{g}}}\}.\end{eqnarray}$$

Conditions $({\rm \small{nfs3}})$ and $({\rm \small{fs3}})$ jointly exhaust the cases in which the right-hand condition of (55) holds and all daughter subtrees $Y_{1},\ldots ,Y_{n_{{\rm \small{g}}}}$ are Floer simple. Condition $({\rm \small{nfs4}})$ , on the other hand, describes all cases in which the right-hand condition of (55) holds and at least one $Y_{j}$ is not Floer simple.

If $({\rm \small{nfs4}})$ holds, then, permuting the daughter subtrees without loss of generality so that $\unicode[STIX]{x1D711}_{n_{{\rm \small{g}}}\ast }^{\mathbb{P}}({\mathcal{L}}(Y_{n_{{\rm \small{g}}}}))=\{\infty \}$ , we have ${\mathcal{L}}^{\circ }(Y_{n_{{\rm \small{g}}}})=\emptyset$ , which, by (40), implies ${\mathcal{L}}(Y)\cap \mathbb{Q}=\emptyset$ , so that ${\mathcal{L}}(Y)=\{\infty \}$ . Moreover, the fact that $\unicode[STIX]{x1D711}_{n_{{\rm \small{g}}}\ast }^{\mathbb{P}}({\mathcal{L}}(Y_{n_{{\rm \small{g}}}}))=\{\infty \}$ implies, by inductive assumption, that $y_{n_{{\rm \small{g}}}-}^{{\rm \small{g}}}=y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}=\infty$ . Thus $y_{-}$ and $y_{+}$ each have infinite summands, and so $y_{-}=y_{+}=\infty$ .

Next suppose that $({\rm \small{nfs3}})$ holds, and set

(56) $$\begin{eqnarray}I_{-\infty }:=\{i:[[y_{i-}^{{\rm \small{g}}},y_{i+}^{{\rm \small{g}}}]]=[-\infty ,y_{i+}^{{\rm \small{g}}}]\},\quad I_{+\infty }:=\{i:[[y_{i-}^{{\rm \small{g}}},y_{i+}^{{\rm \small{g}}}]]=[y_{i-}^{{\rm \small{g}}},+\infty ]\}.\end{eqnarray}$$

We can also define the analogous sets for ${\hat{Y}}[y]$ :

(57) $$\begin{eqnarray}I_{-\infty }^{{\hat{Y}}[y]}:=I_{-\infty }\cap \{1,\ldots ,n_{{\rm \small{g}}}-1\},\quad I_{+\infty }^{{\hat{Y}}[y]}:=I_{+\infty }\cap \{1,\ldots ,n_{{\rm \small{g}}}-1\}.\end{eqnarray}$$

Since $I_{-\infty }$ and $I_{+\infty }$ are non-empty for $({\rm \small{nfs3}})$ , we know that $y_{-}$ and $y_{+}$ each have infinite summands, implying $y_{-}=y_{+}=\infty$ . Assume without loss of generality that $n_{{\rm \small{g}}}\in I_{+\infty }$ . Thus $I_{-\infty }^{{\hat{Y}}[y]}\neq \emptyset$ , and $I_{+\infty }^{{\hat{Y}}[y]}$ is either empty or not. By inductive assumption, Proposition 4.7 holds for ${\hat{Y}}[y]$ for all $y\in \mathbb{Q}$ . Thus, for each $y\in \mathbb{Q}$ , either $I_{-\infty }^{{\hat{Y}}[y]}$ and $I_{+\infty }^{{\hat{Y}}[y]}$ are both non-empty, making $({\rm \small{nfs3}})$ hold for ${\hat{Y}}[y]$ , so that ${\mathcal{L}}({\hat{Y}}[y])=\{\infty \}$ ; or $I_{-\infty }^{{\hat{Y}}[y]}\neq \emptyset$ and $I_{+\infty }^{{\hat{Y}}[y]}=\emptyset$ , making $({\rm \small{fs3}})$ hold for ${\hat{Y}}[y]$ , so that ${\mathcal{L}}({\hat{Y}}[y])=[{\hat{y}}_{-},+\infty ]$ . In both cases, the right-hand side of (40) fails to hold for all $y\in \mathbb{Q}$ , and we are left with ${\mathcal{L}}(Y)=\{\infty \}=\{y_{-}\}=\{y_{+}\}$ .

This leaves us with the case in which $({\rm \small{fs3}})$ holds, for which we first consider the subcase $I_{+\infty }=\emptyset$ and $I_{-\infty }\neq \emptyset$ , implying $[[y_{-},y_{+}]]=[y_{-},+\infty ]$ . Without loss of generality, assume $n_{{\rm \small{g}}}\in I_{-\infty }$ , so that $[[y_{n_{{\rm \small{g}}}-}^{{\rm \small{g}}},y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}]]=[-\infty ,y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}]$ . Since ${\mathcal{L}}({\hat{Y}}[y])=[[{\hat{y}}_{-},{\hat{y}}_{+}]]$ for all $y\in \mathbb{Q}$ by inductive assumption, we know that either:

1. (a) $I_{-\infty }^{{\hat{Y}}[y]}=I_{+\infty }^{{\hat{Y}}[y]}=\emptyset$ ; or

2. (b) $I_{-\infty }^{{\hat{Y}}[y]}\neq \emptyset$ , with ${\mathcal{L}}({\hat{Y}}[y])=[{\hat{y}}_{-},+\infty ]$ .

In case (a), the condition $I_{-\infty }^{{\hat{Y}}[y]}=I_{+\infty }^{{\hat{Y}}[y]}=\emptyset$ , together with (55), implies

(58) $$\begin{eqnarray}\infty \neq y_{i-}^{{\rm \small{g}}}\geqslant y_{i+}^{{\rm \small{g}}}\neq \infty \quad \text{for all }i\in \{1,\ldots ,n_{{\rm \small{ g}}}-1\}.\end{eqnarray}$$

Thus, if we apply Claim 2 to (38), respectively substituting ${\hat{y}}_{-}$ , ${\hat{y}}_{+}$ , $n_{{\rm \small{d}}}+1$ , $n_{{\rm \small{g}}}-1$ , and $(y,y_{1}^{{\rm \small{d}}},\ldots ,y_{n_{{\rm \small{d}}}}^{{\rm \small{d}}})$ for $y_{-}$ , $y_{+}$ , $n_{{\rm \small{d}}}$ , $n_{{\rm \small{g}}}$ and $(y_{1}^{{\rm \small{d}}},\ldots ,y_{n_{{\rm \small{d}}}}^{{\rm \small{d}}})$ in the statement of the claim, then we obtain

(59) $$\begin{eqnarray}\infty \neq {\hat{y}}_{-}~>-y-\mathop{\sum }_{i=1}^{n_{{\rm \small{d}}}}y_{i}^{{\rm \small{d}}}-\mathop{\sum }_{i=1}^{n_{{\rm \small{g}}}-1}y_{i+}^{{\rm \small{g}}}\;\geqslant \;-y-\mathop{\sum }_{i=1}^{n_{{\rm \small{d}}}}y_{i}^{{\rm \small{d}}}-\mathop{\sum }_{i=1}^{n_{{\rm \small{g}}}-1}y_{i-}^{{\rm \small{g}}}\;>~{\hat{y}}_{+}\neq \infty ,\end{eqnarray}$$

with the outer, strict, inequalities resulting from Claim 2, and the middle, non-strict, inequality resulting from (58). Thus, for any $y\in \mathbb{Q}$ , we have

(60) $$\begin{eqnarray}{\mathcal{L}}^{\circ }({\hat{Y}}[y])\cup \unicode[STIX]{x1D711}_{n_{{\rm \small{g}}}\ast }^{\mathbb{P}}({\mathcal{L}}^{\circ }(Y_{n_{{\rm \small{g}}}}))=(\langle {\hat{y}}_{-},+\infty ]\cup [-\infty ,{\hat{y}}_{+}\rangle )\cup \langle -\infty ,y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}\rangle\end{eqnarray}$$

by inductive assumption, and so (40) implies that $y\in {\mathcal{L}}(Y)$ if and only if $y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}>{\hat{y}}_{-}$ , which, by Claim 1, occurs if and only if $y\in [y_{-},+\infty ]$ . On the other hand, in case (b), with ${\mathcal{L}}^{\circ }({\hat{Y}}[y])=\langle {\hat{y}}_{-},+\infty \rangle$ , (40) again implies, for any $y\in \mathbb{Q}$ , that $y\in {\mathcal{L}}(Y)$ if and only if $y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}>{\hat{y}}_{-}$ , which again occurs if and only if $y\in [y_{-},+\infty ]$ . Thus, whether case (a) or (b) holds, we always have $\infty \in {\mathcal{L}}(Y)=[y_{-},+\infty ]=[[y_{-},y_{+}]]$ .

If $({\rm \small{fs3}})$ holds with $I_{+\infty }\neq \emptyset$ and $I_{-\infty }=\emptyset$ , then an argument precisely analogous to that in the preceding paragraph shows that $\infty \in {\mathcal{L}}(Y)=[-\infty ,y_{+}]=[[y_{-},y_{+}]]$ .

Lastly, suppose that $({\rm \small{fs3}})$ holds with $I_{+\infty }=I_{-\infty }=\emptyset$ . This, as well as the fact that $\infty \in$ $[[y_{i-}^{{\rm \small{g}}},y_{i+}^{{\rm \small{g}}}]]$ for each $i\in \{1,\ldots ,n_{{\rm \small{g}}}\}$ , implies that $\infty \neq y_{i-}^{{\rm \small{g}}}\geqslant y_{i+}^{{\rm \small{g}}}\neq \infty$ for all $i\in \{1,\ldots ,n_{{\rm \small{g}}}\}$ , which, by Claim 2, implies that $\infty \neq y_{-}>y_{+}\neq \infty$ . Moreover, applying Claim 2 to (38) as in case (a) above yields (59), so that we also have $\infty \neq {\hat{y}}_{-}>{\hat{y}}_{+}\neq \infty$ . By inductive assumption, we then have

(61) $$\begin{eqnarray}{\mathcal{L}}^{\circ }({\hat{Y}}[y])\cup \unicode[STIX]{x1D711}_{n_{{\rm \small{g}}}\ast }^{\mathbb{P}}({\mathcal{L}}^{\circ }(Y_{n_{{\rm \small{g}}}}))=(\langle {\hat{y}}_{-},+\infty ]\cup [-\infty ,{\hat{y}}_{+}\rangle )\cup (\langle y_{n_{{\rm \small{g}}}-}^{{\rm \small{g}}},+\infty ]\cup [-\infty ,y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}\rangle )\end{eqnarray}$$

for any $y\in \mathbb{Q}$ , and so (40) implies that $y\in {\mathcal{L}}(Y)$ if and only if $y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}>{\hat{y}}_{-}$ or $y_{n_{{\rm \small{g}}}-}^{{\rm \small{g}}}<{\hat{y}}_{+}$ which, by Claim 1, occurs if and only if $y\in [y_{-},+\infty ]\cup [-\infty ,y_{+}]$ . We therefore have $\infty \in {\mathcal{L}}(Y)=$ $[[y_{-},y_{+}]]=[y_{-},+\infty ]\cup [-\infty ,y_{+}]$ , completing our proof.◻

Proof. We first observe that if any daughter subtree of $Y$ fails to be Floer simple, then ${\mathcal{L}}(Y)=\emptyset$ . That if we choose $Y_{n_{{\rm \small{g}}}}$ to be non-Floer-simple, then by (40), ${\mathcal{L}}^{\circ }(Y_{n_{{\rm \small{g}}}})=\emptyset$ implies ${\mathcal{L}}(Y)\cap \mathbb{Q}=\emptyset$ , which, since $\infty \notin {\mathcal{L}}(Y)$ , implies ${\mathcal{L}}(Y)=\emptyset$ . Thus, we henceforth assume the daughter subtrees $Y_{1}$ , …, $Y_{n_{{\rm \small{g}}}}$ are all Floer simple.

Let $I_{\pm \infty },I_{\cap }\subset \{1,\ldots ,n_{{\rm \small{g}}}\}$ denote the sets

(62) $$\begin{eqnarray}\begin{array}{@{}rcl@{}}\displaystyle I_{\pm \infty } & \!\!\!:=\!\!\!\! & \{i:[[y_{i-}^{{\rm \small{g}}},y_{i+}^{{\rm \small{g}}}]]=\langle -\infty ,+\infty \rangle \},\\ \displaystyle I_{\cap } & \!\!\!:=\!\!\!\! & \{i:[[y_{i-}^{{\rm \small{g}}},y_{i+}^{{\rm \small{g}}}]]=[y_{i-}^{{\rm \small{g}}},+\infty ]\cap [-\infty ,y_{i+}^{{\rm \small{g}}}]\}.\end{array}\end{eqnarray}$$

Then by (55) from the proof of Proposition 4.8, we know that

(63) $$\begin{eqnarray}I_{\pm \infty }\cup I_{\cap }\neq \emptyset .\end{eqnarray}$$

Suppose $I_{\pm \infty }\neq \emptyset$ . We claim that in this case, ${\mathcal{L}}(Y)\neq \emptyset$ if and only if $({\rm \small{fs1}})$ holds for $Y$ , in which case ${\mathcal{L}}(Y)=[[y_{-},y_{+}]]$ . First note that $Y$ fails to satisfy any of the conditions for non-empty ${\mathcal{L}}(Y)$ in Proposition 4.7, except possibly $({\rm \small{fs1}})$ . Suppose that $Y$ satisfies $({\rm \small{fs1}})$ . Then for each $i\in \{1,\ldots ,n_{{\rm \small{g}}}-1\}$ , we have $\infty \in [[y_{i-}^{{\rm \small{g}}},y_{i+}^{{\rm \small{g}}}]]^{\circ }$ , implying $\infty \neq y_{i-}^{{\rm \small{g}}}\geqslant y_{i+}^{{\rm \small{g}}}\neq \infty$ , which by Claim 2 implies $\infty \neq {\hat{y}}_{-}>{\hat{y}}_{+}\neq \infty$ for all $y\in \mathbb{Q}$ . By inductive assumption, we then have $\infty \in {\mathcal{L}}^{\circ }({\hat{Y}}[y])$ for all $y\in \mathbb{Q}$ . Thus, since (40) tells us that

(64) $$\begin{eqnarray}{\mathcal{L}}^{\circ }({\hat{Y}}[y])\cup \langle -\infty ,+\infty \rangle =\mathbb{Q}\cup \{\infty \}\quad \;\Longleftrightarrow \;\quad y\in {\mathcal{L}}(Y),\end{eqnarray}$$

we have ${\mathcal{L}}(Y)=\mathbb{Q}=\langle -\infty ,+\infty \rangle =[[y_{-},y_{+}]]$ . Suppose instead we are given that ${\mathcal{L}}(Y)\neq \emptyset$ . Then (64) tells us that $\infty \in {\mathcal{L}}^{\circ }({\hat{Y}}[y])$ for all $y\in {\mathcal{L}}(Y)$ . By inductive assumption this implies, for each $y\in {\mathcal{L}}(Y)$ , that $({\rm \small{fs3}})$ holds for ${\hat{Y}}[y]$ , with $\infty \in [[{\hat{y}}_{-},{\hat{y}}_{+}]]^{\circ }$ . Since $\infty \neq \{{\hat{y}}_{-},{\hat{y}}_{+}\}$ , we know that ${\hat{y}}_{-}$ and ${\hat{y}}_{+}$ cannot have infinite summands, and so $I_{-\infty }^{{\hat{Y}}[y]}=I_{+\infty }^{{\hat{Y}}[y]}=\emptyset$ . This, in addition to the fact that $({\rm \small{fs3}})$ holds for ${\hat{Y}}[y]$ , implies that $\infty \in [[y_{i-}^{{\rm \small{g}}},y_{i+}^{{\rm \small{g}}}]]^{\circ }$ for all $i\in \{1,\ldots ,n_{{\rm \small{g}}}-1\}$ , and thus $({\rm \small{fs1}})$ holds for $Y$ .

Next, consider the case in which $I_{\pm \infty }=\emptyset$ , so that by (63), we have $I_{\cap }\neq \emptyset$ . Assume, without loss of generality, that $n_{{\rm \small{g}}}\in I_{\cap }$ . Then since $\infty \notin [[y_{n_{{\rm \small{g}}}-}^{{\rm \small{g}}},y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}]]^{\circ }$ , the L-space gluing condition in (40) again tells us that $\infty \in {\mathcal{L}}^{\circ }({\hat{Y}}[y])$ for all $y\in {\mathcal{L}}(Y)$ . Just as in the preceding paragraph, we deduce from this that $\infty \in [[y_{i-}^{{\rm \small{g}}},y_{i+}^{{\rm \small{g}}}]]^{\circ }$ for all $i\in \{1,\ldots ,n_{{\rm \small{g}}}-1\}$ , and that this implies that ${\mathcal{L}}^{\circ }({\hat{Y}}[y])=[[{\hat{y}}_{-},{\hat{y}}_{+}]]$ , with $\infty \neq {\hat{y}}_{-}>{\hat{y}}_{+}\neq \infty$ , for all $y\in \mathbb{Q}$ . Since $y_{-}$ and $y_{+}$ have only finite summands, we also know that $y_{-},y_{+}\in \mathbb{Q}$ . We therefore have

(65) $$\begin{eqnarray}\displaystyle y\in {\mathcal{L}}(Y) & \;\Longleftrightarrow \; & \displaystyle (\langle {\hat{y}}_{-},+\infty ]\cup [-\infty ,{\hat{y}}_{+}\rangle )\cup (\langle y_{n_{{\rm \small{g}}}-}^{{\rm \small{g}}},+\infty ]\cap [-\infty ,y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}\rangle )=\mathbb{Q}\cup \{\infty \}\nonumber\\ \displaystyle & \;\Longleftrightarrow \; & \displaystyle y_{n_{{\rm \small{g}}}+}^{{\rm \small{g}}}>{\hat{y}}_{-}\quad \text{and}\quad y_{n_{{\rm \small{g}}}-}^{{\rm \small{g}}}<{\hat{y}}_{+}\nonumber\\ \displaystyle & \;\Longleftrightarrow \; & \displaystyle y\in [y_{-},+\infty ]\cap [-\infty ,y_{+}],\end{eqnarray}$$

where the first line is due to (40), and the third line is due to Claim 1. Thus, if $y_{-}>y_{+}$ , then ${\mathcal{L}}(Y)=\emptyset$ ; if $y_{-}=y_{+}$ , then $({\rm \small{nfs2}})$ holds for $Y$ , with ${\mathcal{L}}(Y)=\{y_{-}\}=\{y_{+}\}$ ; and if $y_{-}<y_{+}$ , then $({\rm \small{fs2}})$ holds for $Y$ , with ${\mathcal{L}}(Y)=[[y_{-},y_{+}]]=[y_{-},y_{+}]$ .◻