2.1 Conventions for Lagrangian disks with Legendrian boundary in $\mathbb{R}^{2n}$

In this section we introduce a specific convention for Lagrangian disks that is convenient for our study of associated homotopy classes. We consider a more general setting for Lagrangian disks in $\mathbb{R}_{\text{st}}^{2n}$ that generalizes that considered above. Let $\unicode[STIX]{x1D6EC}$ be a Legendrian sphere in $\mathbb{R}_{\text{st}}^{2n-1}=T^{\ast }\mathbb{R}^{n-1}\times \mathbb{R}$ with contact form $dz-y\cdot dx$ , where $(x,y)$ are standard coordinates on $T^{\ast }\mathbb{R}^{n-1}$ and $z$ on $\mathbb{R}$ . The symplectization of $\mathbb{R}_{\text{st}}^{2n-1}$ is the symplectic manifold $\mathbb{R}\times \mathbb{R}_{\text{st}}^{2n-1}$ with symplectic form $d(e^{t}(dz-y\cdot dx))$ , where $t$ is a coordinate on the additional $\mathbb{R}$ -factor. The symplectization then contains the Lagrangian cone $\mathbb{R}\times \unicode[STIX]{x1D6EC}$ on $\unicode[STIX]{x1D6EC}$ .

Consider embeddings of half of the symplectization into $\mathbb{R}^{2n}$ of the following form:

$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}_{a}:[0,\infty )\times \mathbb{R}_{\text{st}}^{2n-1}~\rightarrow ~~\mathbb{R}_{\text{st}}^{2n}\cap \{x_{1}\geqslant 1\}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D713}_{a}(t,x,y,z)=(e^{t},z,x,e^{t}y)+(a,0,0,0),\quad a>0. & \displaystyle \nonumber\end{eqnarray}$$

Note that the image of $[0,\infty )\times \unicode[STIX]{x1D6EC}$ is a Lagrangian cylinder in $\{x_{1}\geqslant a\}$ . (Here $\{x_{1}=a\}$ is a contact hypersurface and the Lagrangian cylinder lies in the corresponding cylindrical end.)

We will consider Lagrangian disks in $\mathbb{R}^{2n}$ with Legendrian boundary, which we define as Lagrangian embeddings of $\mathbb{R}^{n}$ that agree with a Lagrangian cylinder on a Legendrian sphere outside any disk of sufficiently large radius.

More precisely, let $\unicode[STIX]{x1D706}:S^{n-1}\rightarrow \mathbb{R}^{2n-1}$ be a Legendrian embedding. Let $g:\mathbb{R}^{n}\rightarrow \mathbb{R}^{2n}$ be a Lagrangian embedding and let $(r,\unicode[STIX]{x1D709})\in [0,\infty )\times S^{n-1}$ be polar coordinates on $\mathbb{R}^{n}$ . If there exists  $r_{0}>0$ and $a>0$ such that

$$\begin{eqnarray}g(r,\unicode[STIX]{x1D709})=\unicode[STIX]{x1D713}_{a}(\log (r/r_{0}),\unicode[STIX]{x1D706}(\unicode[STIX]{x1D709}))\quad \text{for}~r\geqslant r_{0},\end{eqnarray}$$

then $g$ is a Lagrangian disk with Legendrian boundary $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D706}(S^{n-1})$ .

Let $L=g(\mathbb{R}^{n})$ be a Lagrangian disk with Legendrian boundary such that $g(\mathbb{R}^{n})\cap T_{S^{k}}^{\ast }\mathbb{R}^{n}=\varnothing$ . Let $h(r)=r/r_{0}+a$ . Then for, $r_{1}>0$ sufficiently large, $\operatorname{pr}\circ g|_{\{r\leqslant r_{1}\}}$ gives a map from a disk $D$ into $\mathbb{R}^{n}\setminus S^{k}$ such that $D$ maps into $\{x_{1}\leqslant h(r_{1})\}$ , and $\unicode[STIX]{x2202}D=\{r=r_{1}\}$ maps to $\{x_{1}=h(r_{1})\}$ . Furthermore, there exists a constant $K>0$ , depending on the Legendrian boundary of $L$ such that if $r_{1}$ is sufficiently large then $\operatorname{pr}(g(D))$ lies inside the radius $K\cdot h(r_{1})$ ball $B_{K\cdot h(r_{1})}$ around the origin in $\mathbb{R}^{n}$ . In particular, we get a map

It is clear that there exists $r_{1}>0$ such that the homotopy class of $f_{L,r}$ is independent of $r$ for all $r>r_{1}$ . We write $[f_{L}]\in \unicode[STIX]{x1D70B}_{n}(S^{n-k-1})$ for this homotopy class. This agrees with the corresponding homotopy class discussed in §1.

2.2 Half rotations and suspension

Consider a Lagrangian disk $L$ with Legendrian boundary $\unicode[STIX]{x1D6EC}$ in $\mathbb{R}_{\text{st}}^{2n}$ as above. We will construct from $L$ a Legendrian submanifold $\unicode[STIX]{x1D6E4}(L)\subset \mathbb{R}_{\text{st}}^{2n+1}$ , which can be thought of as a double of $L$ and which bounds a Lagrangian disk $C(L)$ in $\mathbb{R}_{\text{st}}^{2n+2}$ .

We construct $\unicode[STIX]{x1D6E4}(L)$ by defining its front. Recall that if $\unicode[STIX]{x1D6F4}$ is a Legendrian submanifold of $J^{1}(\mathbb{R}^{n})=T^{\ast }\mathbb{R}^{n}\times \mathbb{R}$ then the front of $\unicode[STIX]{x1D6F4}$ is the 0-jet projection $f_{\unicode[STIX]{x1D6F4}}=\operatorname{pr}^{0}:\unicode[STIX]{x1D6F4}\rightarrow J^{0}(\mathbb{R}^{n})=\mathbb{R}^{n}\times \mathbb{R}$ . Furthermore, if $\unicode[STIX]{x1D6F4}$ is in general position with respect to this projection then $f_{\unicode[STIX]{x1D6F4}}$ determines $\unicode[STIX]{x1D6F4}$ as follows. Let $\operatorname{pr}_{b}:\mathbb{R}^{n}\times \mathbb{R}\rightarrow \mathbb{R}^{n}$ denote the base projection. A generic point in the image of  $\operatorname{pr}_{b}\circ f_{\unicode[STIX]{x1D6F4}}$ has a neighborhood $U$ such that $\operatorname{pr}_{b}^{-1}(U)\cap f_{\unicode[STIX]{x1D6F4}}(\unicode[STIX]{x1D6F4})$ is the graph of a finite number of local functions $f_{j}:U\rightarrow \mathbb{R}$ . The fiber coordinates $y=(y_{1},\ldots ,y_{n})$ are then determined over $U$ by the equations $y_{j}=\unicode[STIX]{x2202}f_{j}/\unicode[STIX]{x2202}x_{j}$ , and the genericity condition ensures that these solutions continue over the singular locus (which has codimension one). Similarly, we define the front of an exact Lagrangian submanifold $L\subset \mathbb{R}^{2n}$ by picking a Legendrian lift into $\mathbb{R}^{2n}\times \mathbb{R}$ and taking the front projection into $\mathbb{R}^{n}\times \mathbb{R}$ of that lift. If $L$ is connected then its front is well defined up to an overall $\mathbb{R}$ -translation.

We now turn to the actual construction, see Figure 1. Consider the front $g_{L}$ of $L\cap \{x_{1}\leqslant a\}$ , for some $a>0$ sufficiently large that $L$ is a cone on its Legendrian boundary in the region $\{x_{1}\geqslant a-1\}$ . Translating $L$ by $-a$ in the $x_{1}$ -direction we get a front $g_{L}^{\prime }$ lying over the half-space $(-\infty ,-1]\times \mathbb{R}^{n-1}\times \mathbb{R}$ . If

$$\begin{eqnarray}f_{\unicode[STIX]{x1D6EC}}:S^{n-1}\rightarrow \mathbb{R}^{n-1}\times \mathbb{R},\quad f_{\unicode[STIX]{x1D6EC}}=(f_{\unicode[STIX]{x1D6EC}}^{b},z_{\unicode[STIX]{x1D6EC}})\end{eqnarray}$$

is the front for the Legendrian boundary $\unicode[STIX]{x1D6EC}$ , then for $-2\leqslant x_{1}\leqslant -1$ the front $g_{L}^{\prime }$ on $S^{n-1}\times [-2,-1]$ is given by

$$\begin{eqnarray}g_{L}^{\prime }(\unicode[STIX]{x1D709},r)=(r,f_{\unicode[STIX]{x1D6EC}}^{b}(\unicode[STIX]{x1D709}),\unicode[STIX]{x1D713}(r)z_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D709})),\end{eqnarray}$$

for a monotone increasing positive function $\unicode[STIX]{x1D713}$ . Denote the Lagrangian disk in $\{x_{1}\leqslant -1\}\subset \mathbb{R}^{2n}$ defined by the front $g_{L}^{\prime }$ by $L^{-}$ .

Figure 1. The fronts of: $L$ (top), $\unicode[STIX]{x1D6E4}(L)$ (middle) and $C(L)$ (bottom).

The Lagrangian disk $L^{-}$ constitutes almost half of the Lagrangian projection of $\unicode[STIX]{x1D6E4}(L)$ . We next define the other half $L^{+}$ :

$$\begin{eqnarray}L^{+}=T(L^{-}),\end{eqnarray}$$

where $T:\mathbb{R}^{2n}\rightarrow \mathbb{R}^{2n}$ is the composition of reflections in the hyperplanes $\{x_{1}=0\}$ and $\{y_{1}=0\}$ . The front of $L^{+}$ is then given by the function $g_{L}^{\prime \prime }$ on $S^{n-1}\times [1,2]$ , where

$$\begin{eqnarray}g_{L}^{\prime \prime }(\unicode[STIX]{x1D709},r)=(r,f_{\unicode[STIX]{x1D6EC}}^{b}(\unicode[STIX]{x1D709}),\unicode[STIX]{x1D713}(-r)z_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D709})).\end{eqnarray}$$

With $L^{-}$ and $L^{+}$ defined we can now define the front of $\unicode[STIX]{x1D6E4}(L)$ in $\mathbb{R}^{n}\times \mathbb{R}$ as follows.

1. (i) Over $\{x_{1}\leqslant -1\}$ it agrees with the front of $L^{-}$ .

2. (ii) Over $\{x_{1}\geqslant 1\}$ it agrees with the front of $L^{+}$ .

3. (iii) In the region $\{-2\leqslant x_{1}\leqslant 2\}$ the front is given by

   $$\begin{eqnarray}g_{L}(\unicode[STIX]{x1D709},r)=(r,f_{\unicode[STIX]{x1D6EC}}^{b}(\unicode[STIX]{x1D709}),\unicode[STIX]{x1D719}(r)z_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D709})),\end{eqnarray}$$
   where $\unicode[STIX]{x1D719}:[-2,2]\rightarrow \mathbb{R}$ is a Morse function with a maximum at $r=0$ and no other critical points, and such that $\unicode[STIX]{x1D719}(r)=\unicode[STIX]{x1D713}(r)$ for $-2<r<-1$ , $\unicode[STIX]{x1D719}(r)=\unicode[STIX]{x1D713}(-r)$ for $1<r<2$ .

By construction, $\unicode[STIX]{x1D6E4}(L)$ is a Legendrian submanifold of $\mathbb{R}^{2n+1}$ , whose Reeb chords all lie in the slice $\{x_{1}=0=y_{1}\}$ , where they agree with the Reeb chords of $\unicode[STIX]{x1D6EC}$ . (There are no Reeb chords of $\unicode[STIX]{x1D6E4}(L)$ outside the slice since the Lagrangian disk $L$ , which is the projection along the Reeb direction, is embedded.) If $c$ is a Reeb chord of $\unicode[STIX]{x1D6EC}$ then we denote by ${\hat{c}}$ the corresponding Reeb chord of $\unicode[STIX]{x1D6E4}(L)$ . It follows from the ‘front formula’ for Maslov grading [Reference Ekholm, Etnyre and SullivanEES05, Lemma 3.4] that the gradings of the two Reeb chords are related by

$$\begin{eqnarray}|{\hat{c}}|=|c|+1.\end{eqnarray}$$

We next construct a Lagrangian disk $C(L)\subset \mathbb{R}_{\text{st}}^{2n+2}$ with Legendrian boundary $\unicode[STIX]{x1D6E4}(L)$ , see Figure 1. Again we construct it by defining its front. Consider the ‘left’ half of the front of $\unicode[STIX]{x1D6E4}(L)$ , $g_{L}:D^{n}\rightarrow \mathbb{R}_{+}^{n}\times \mathbb{R}$ , which lies over the half-plane $\{x_{1}\leqslant 0\}$ . Then $g_{L}$ takes the boundary $\unicode[STIX]{x2202}D$ to the front of $\unicode[STIX]{x1D6EC}$ in $0\times \mathbb{R}^{n-1}\times \mathbb{R}$ .

Consider now $\mathbb{R}^{n+1}$ , with coordinates $(x_{1},x_{2},\ldots ,x_{n},x_{n+1})$ , as an open book with binding $\mathbb{R}^{n-1}$ corresponding to the last $n-1$ coordinates and with page $\mathbb{R}_{+}^{n}$ with coordinates $(r\cos \unicode[STIX]{x1D703},r\sin \unicode[STIX]{x1D703},x_{3},\ldots ,x_{n+1})$ , $r\geqslant 0$ . Define the immersed Lagrangian disk $C^{\prime }(L)$ to be that defined by the front which lies over the pages with positive $x_{2}$ -coordinate (i.e., $0\leqslant \unicode[STIX]{x1D703}\leqslant \unicode[STIX]{x1D70B}$ ) and which in each such page agrees with the front $g_{L}$ . Then $C^{\prime }(L)$ is an immersed Lagrangian, all of whose double points lie in the binding, these double points corresponding exactly to the Reeb chords of  $\unicode[STIX]{x1D6EC}$ . For small $\unicode[STIX]{x1D716}>0$ , the part of the front of $C^{\prime }(L)$ which lies over $\{x_{2}\geqslant \unicode[STIX]{x1D716}\}$ defines an embedded Lagrangian disk, which near the boundary looks like a cone on the Legendrian sphere $\unicode[STIX]{x1D6E4}(L)$ . After small deformation of the front we may then add a half-infinite cone on the front of $\unicode[STIX]{x1D6E4}(L)$ to obtain a Lagrangian disk with Legendrian boundary in the sense defined before. This is our desired Lagrangian disk $C(L)$ .

Recall that in this situation, we associated maps $f_{L}:S^{n}\rightarrow S^{n-k-1}$ and $f_{C(L)}:S^{n+1}\rightarrow S^{n-k}$ to $L$ and to $C(L)$ , respectively.

3.1 Lagrangian sphere immersions

Let $\unicode[STIX]{x1D719}:S^{n}\rightarrow \mathbb{R}_{\text{st}}^{2n}$ be a Lagrangian immersion. The tangential Gauss map takes any point $p\in S^{n}$ to the Lagrangian tangent plane $d\unicode[STIX]{x1D719}(T_{p}S^{n})$ and thus defines a map $G_{\unicode[STIX]{x1D719}}:S^{n}\rightarrow U(n)/O(n)\rightarrow U/O$ , where $U(n)/O(n)$ is the Lagrangian Grassmannian of Lagrangian $n$ -planes in $\mathbb{R}_{\text{st}}^{2n}$ , and the map to $U/O$ is given by stabilization. After small perturbation $\unicode[STIX]{x1D719}$ is self-transverse and hence has only transverse double points and no other self intersections. If $a$ is a transverse double point of $\unicode[STIX]{x1D719}$ then there is an integer Maslov grading $|a|$ associated to $a$ , see [Reference Ekholm and SmithES14].

The standard example of a Lagrangian sphere immersion is the Whitney sphere, defined as follows. Equip $\mathbb{R}_{\text{st}}^{2n}$ with the standard complex structure $i$ and view it as complex $n$ -space $\mathbb{C}^{n}$ with coordinates $x+iy\in \mathbb{R}^{n}+i\mathbb{R}^{n}$ . Consider $S^{n}$ as the unit sphere in $\mathbb{R}^{n+1}$ :

$$\begin{eqnarray}S^{n}=\{(x,y_{1})\in \mathbb{R}^{n}\times \mathbb{R}:|x|^{2}+y_{1}^{2}=1\}.\end{eqnarray}$$

The Whitney immersion is the map $w:S^{n}\rightarrow \mathbb{C}^{n}$ given by

(3.1) $$\begin{eqnarray}w(x,y)=x(1+iy_{1}).\end{eqnarray}$$

Then $w$ has one self-transverse double point $c$ with preimages at $(0,\pm 1)$ and no other singularities. The Maslov grading of the double point is $|c|=n$ .

We will assume throughout this section that $\unicode[STIX]{x1D719}:S^{n}\rightarrow \mathbb{R}_{\text{st}}^{2n}$ is a self transverse real analytic Lagrangian immersion with exactly one double point $a$ of Maslov grading $|a|=n$ . We say that $\unicode[STIX]{x1D719}$ is tangentially standard if its stable tangential Gauss map $G_{\unicode[STIX]{x1D719}}$ is homotopic to the stable tangential Gauss map $G_{w}$ of the Whitney immersion.

Fix a primitive $\unicode[STIX]{x1D703}$ of the symplectic form, $d\unicode[STIX]{x1D703}=\unicode[STIX]{x1D714}_{\text{st}}$ . The immersed sphere $\unicode[STIX]{x1D719}:S^{n}\rightarrow \mathbb{R}_{\text{st}}^{2n}$ has a Legendrian lift $\unicode[STIX]{x1D719}\times z:S^{n}\rightarrow \mathbb{R}_{\text{st}}^{2n}\times \mathbb{R}$ , where the contact form on $\mathbb{R}_{\text{st}}^{2n}\times \mathbb{R}$ is $dz-\unicode[STIX]{x1D703}$ , defined with respect to a choice $z:S^{n}\rightarrow \mathbb{R}$ of function satisfying $\unicode[STIX]{x1D719}^{\ast }\unicode[STIX]{x1D703}=dz$ . In the case under consideration $\unicode[STIX]{x1D719}$ has a single double point and $\unicode[STIX]{x1D719}\times z$ is an embedding. We can then distinguish between small disjoint disk neighborhoods $V^{\pm }$ of the two preimages of the double point of $\unicode[STIX]{x1D719}(S^{n})$ , by declaring that $V^{+}$ lives in the upper sheet and $V^{-}$ in the lower sheet of the image of $\unicode[STIX]{x1D719}\times z$ .

3.2 Moduli spaces from displacing Hamiltonians

Fix an almost complex structure $J$ on $\mathbb{R}_{\text{st}}^{2n}$ which is standard in a sufficiently small neighborhood of $\unicode[STIX]{x1D719}(S^{n})$ . Let $H=H_{t}:\mathbb{R}^{2n}\rightarrow \mathbb{R}$ be a time dependent Hamiltonian function with associated Hamiltonian vector field $X_{H_{t}}$ and time-one flow $\unicode[STIX]{x1D713}_{H}^{1}$ . We suppose that $\unicode[STIX]{x1D713}_{H}^{1}(\unicode[STIX]{x1D719}(S^{n}))\cap \unicode[STIX]{x1D719}(S^{n})=\varnothing$ . As in [Reference Ekholm and SmithES16, §3.1, (3.3)] we fix a one-parameter family of 1-forms $\unicode[STIX]{x1D6FE}_{r}\in \unicode[STIX]{x1D6FA}^{1}(D)$ on the two-dimensional closed disk $D$ , with $r\in [0,\infty )$ , such that $\unicode[STIX]{x1D6FE}_{0}\equiv 0$ , and such that with respect to a fixed conformal isomorphism $D\setminus \{\pm 1\}\rightarrow \mathbb{R}\times [0,1]$ , and with co-ordinates $(s,t)\in \mathbb{R}\times [0,1]$ , $\unicode[STIX]{x1D6FE}_{r}$ has compact support and for $r\gg 0$ agrees with $dt$ on $[-r,r]\times [0,1]$ .

Consider the Floer equation for maps $u:(D,\unicode[STIX]{x2202}D)\rightarrow (\mathbb{R}_{\text{st}}^{2n},\unicode[STIX]{x1D719}(S^{n}))$ , such that $u|_{\unicode[STIX]{x2202}D}$ admits a continuous lift to $S^{n}$ ,

(3.2) $$\begin{eqnarray}(du+\unicode[STIX]{x1D6FE}_{r}\otimes X_{H})^{0,1}=0,\end{eqnarray}$$

where the complex anti-linear part is taken with respect to $J$ and the standard complex structure on $D$ , and where $X_{H}=X_{H}(u(z),t(z))$ for the function $t:D\rightarrow [0,1]$ defined by the second co-ordinate in our fixed conformal equivalence $D-\{\pm 1\}\rightarrow \mathbb{R}\times [0,1]$ . For fixed $r\in [0,\infty )$ , we write ${\mathcal{F}}^{r}$ for the space of solutions of (3.2) and we write

$$\begin{eqnarray}{\mathcal{F}}=\mathop{\bigcup }_{r\in [0,\infty )}{\mathcal{F}}^{r}\end{eqnarray}$$

for the corresponding parameterized moduli space.

Lemma 3.1. The space ${\mathcal{F}}^{r}$ has formal dimension $n$ ; ${\mathcal{F}}$ has formal dimension $n+1$ . For generic data the formal dimension equals the actual dimension, and ${\mathcal{F}}$ has boundary

$$\begin{eqnarray}\unicode[STIX]{x2202}{\mathcal{F}}={\mathcal{F}}^{0}{\approx}_{C^{1}}S^{n}.\end{eqnarray}$$

In order to describe the compactification of ${\mathcal{F}}$ , consider spaces ${\mathcal{F}}_{j}^{r}$ of solutions $u$ of the Floer equation (3.2) with $j$ negative boundary punctures, where the disk is asymptotic to the unique double point of $\unicode[STIX]{x1D719}(S^{n})$ . More precisely, the source of such a map is the disk $D$ with $j$ boundary punctures $\unicode[STIX]{x1D701}_{1},\ldots ,\unicode[STIX]{x1D701}_{j}$ . Consider a punctured arc $I_{j}$ in $\unicode[STIX]{x2202}D$ centered around $\unicode[STIX]{x1D701}_{j}$ and note that it is subdivided into two components $I_{j}^{-}$ and $I_{j}^{+}$ in the negative and positive direction along the boundary from $\unicode[STIX]{x1D701}_{j}$ . We require that for any sufficiently small $I_{j}$ , $u(I_{j}^{-})$ lies in the upper branch $V^{+}$ of the double point and $u(I_{j}^{+})$ in the lower branch $V^{-}$ . The following results were proved in [Reference Ekholm and SmithES14, Lemma 3.4].

Lemma 3.2. The formal dimensions of ${\mathcal{F}}_{j}^{r}$ and ${\mathcal{F}}_{j}$ are

$$\begin{eqnarray}n-j(n-1)\quad \text{and}\quad n-j(n-1)+1,\end{eqnarray}$$

respectively. For generic data these spaces are all transversely cut out. In particular, ${\mathcal{F}}_{1}$ is a closed one-dimensional manifold and ${\mathcal{F}}_{j}$ is empty for $j>1$ .

Similarly, we consider the moduli space ${\mathcal{M}}$ of unperturbed holomorphic disks with boundary on $\unicode[STIX]{x1D719}(S^{n})$ and one positive puncture at the double point. The following result is proved in [Reference Ekholm and SmithES14, Lemma 3.5].

The moduli space ${\mathcal{M}}$ is a space of holomorphic maps modulo automorphisms, and the construction of the smooth structure above relies on a gauge-fixing procedure described in detail in [Reference Ekholm and SmithES14, Appendix A.2]; the gauge-fixing involves fixing parameterizations of maps in ${\mathcal{M}}$ so that $\pm i$ map to small spheres centered on the preimages of the double point. We will not distinguish notationally between ${\mathcal{M}}$ and the corresponding space of maps after gauge-fixing; in particular, in the sequel we will make use of an evaluation map ${\mathcal{M}}\times D\rightarrow \mathbb{C}^{n}$ , where $D$ denotes the source-disk for maps in the gauge-fixed moduli space corresponding to ${\mathcal{M}}$ .

3.3 The Gromov–Floer compactification

The space ${\mathcal{F}}$ is non-compact; it has one boundary component ${\mathcal{F}}^{0}$ which is diffeomorphic to the sphere $S^{n}$ parameterizing constant solutions. It can be compactified to a $C^{1}$ -smooth compact manifold with boundary by adding broken solutions.

Lemma 3.4. For generic $J$ and $H$ , the product

$$\begin{eqnarray}{\mathcal{N}}={\mathcal{F}}_{1}\times {\mathcal{M}}\end{eqnarray}$$

is a $C^{1}$ -smooth $n$ -manifold, canonically diffeomorphic to the Gromov–Floer boundary of ${\mathcal{F}}$ .

[Reference Ekholm and SmithES14, Theorem 5.21] constructs a Floer gluing map on ${\mathcal{N}}\times [\unicode[STIX]{x1D70C}_{0},\infty )$ which, when composed with a map reparameterizing the domain, gives a smooth embedding

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}:{\mathcal{N}}\times [\unicode[STIX]{x1D70C}_{0},\infty )\rightarrow {\mathcal{F}}\end{eqnarray}$$

whose image parameterizes a neighborhood of the Gromov–Floer boundary of ${\mathcal{F}}$ . (The effect of the re-parameterization, discussed further below, is that the holomorphic disk part of the glued map, coming from ${\mathcal{M}}$ , lies in a small half-disk around the puncture of the domain of the map from ${\mathcal{F}}_{1}$ .) It follows that

$$\begin{eqnarray}\overline{{\mathcal{F}}}={\mathcal{F}}\setminus \unicode[STIX]{x1D6F9}({\mathcal{N}}\times (\unicode[STIX]{x1D70C}_{0},\infty ))\end{eqnarray}$$

is a smooth compact submanifold of ${\mathcal{F}}$ with boundary

$$\begin{eqnarray}\unicode[STIX]{x2202}\overline{{\mathcal{F}}}{\approx}_{C^{1}}S^{n}\cup {\mathcal{N}},\end{eqnarray}$$

where the diffeomorphism on the first component is given by inclusion of constant maps (with a lift of the map to the domain of the immersion $\unicode[STIX]{x1D719}$ ), and the diffeomorphism on ${\mathcal{N}}$ is given by the reparameterized gluing map $\unicode[STIX]{x1D6F9}$ .

Recall that ${\mathcal{F}}_{1}$ is a closed $1$ -manifold. As in [Reference Ekholm and SmithES14, §4] consider next the abstract filling

$$\begin{eqnarray}{\mathcal{T}}={\mathcal{D}}\times {\mathcal{M}}\end{eqnarray}$$

of ${\mathcal{N}}$ , where ${\mathcal{D}}$ is a collection of disks filling ${\mathcal{F}}_{1}$ . We let ${\mathcal{B}}=\overline{{\mathcal{F}}}\cup _{{\mathcal{N}}}{\mathcal{T}}$ , which is a $C^{1}$ -smooth compact $(n+1)$ -dimensional manifold with boundary, whose unique boundary component is canonically diffeomorphic to $S^{n}$ .

With no assumption on the stable Gauss map of $\unicode[STIX]{x1D719}$ we have the following weaker result.

3.4 Extending the evaluation map

From the Lagrangian immersion $\unicode[STIX]{x1D719}:S^{n}\rightarrow \mathbb{R}_{\text{st}}^{2n}$ we have constructed a bounding manifold ${\mathcal{B}}$ , $\unicode[STIX]{x2202}{\mathcal{B}}=S^{n}$ . The subspace $\overline{{\mathcal{F}}}\subset {\mathcal{B}}$ is a space of Floer holomorphic disks with boundary that lift via $\unicode[STIX]{x1D719}$ to $S^{n}$ , and hence there is a canonical evaluation map $\operatorname{ev}_{1}:\overline{{\mathcal{F}}}\rightarrow S^{n}$ . This evaluation map cannot extend to ${\mathcal{B}}$ as a map to $S^{n}$ , because the map has degree $1$ on the boundary $\unicode[STIX]{x2202}{\mathcal{B}}$ . However, it can be extended over the filling ${\mathcal{T}}$ as a map to $\mathbb{R}^{2n}$ . We prove our results on homotopy classes using a particular such extension coming from the holomorphic disk components of broken solutions.

Consider first the 1-manifold ${\mathcal{F}}_{1}$ . Let $\unicode[STIX]{x2202}D$ denote the boundary of the source disk $D$ of the maps in ${\mathcal{F}}_{1}$ . There is a smooth map

$$\begin{eqnarray}\unicode[STIX]{x1D701}:{\mathcal{F}}_{1}\rightarrow \unicode[STIX]{x2202}D\end{eqnarray}$$

taking a solution $u\in {\mathcal{F}}_{1}$ to the coordinate of its negative boundary puncture. After a small rotation of the domain we may assume that $\unicode[STIX]{x1D701}$ is transverse to $1\in \unicode[STIX]{x2202}D$ . Fix a small closed interval $I\subset \unicode[STIX]{x2202}D$ centered at $1$ , and let

$$\begin{eqnarray}\unicode[STIX]{x1D701}^{-1}(I)=I_{1}\cup \cdots \cup I_{m},\end{eqnarray}$$

where $I_{j}$ are the components of the preimage and where $I$ is sufficiently small that the restriction of $\unicode[STIX]{x1D701}$ to each $I_{j}$ is a diffeomorphism onto $I$ . We subdivide ${\mathcal{N}}={\mathcal{F}}_{1}\times {\mathcal{M}}$ into $m+1$ pieces:

$$\begin{eqnarray}{\mathcal{N}}=\mathop{\bigcup }_{j=1}^{m}(I_{j}\times {\mathcal{M}})~\cup ~(U\times {\mathcal{M}}),\end{eqnarray}$$

where $U={\mathcal{F}}_{1}\setminus \bigcup _{j=1}^{m}I_{j}$ . Recall ${\mathcal{F}}_{1}=\unicode[STIX]{x2202}{\mathcal{D}}$ for an abstract collection of 2-disks ${\mathcal{D}}$ . Let ${\mathcal{D}}_{j}\subset {\mathcal{D}}$ denote a small half-disk neighborhood of the center of $I_{j}$ that intersects the boundary in $I_{j}$ . Recall the neighborhoods $V^{\pm }$ of the preimages of the double point introduced in §3.1.

Lemma 3.8. Let $\unicode[STIX]{x1D70E}_{j}\in \{\pm \}$ be the sign of the derivative $d\unicode[STIX]{x1D701}$ in $I_{j}$ and let $\unicode[STIX]{x2202}_{+}I_{j}$ and $\unicode[STIX]{x2202}_{-}I_{j}$ denote the positive and negative endpoints of $I_{j}$ . For $I$ small enough

$$\begin{eqnarray}\operatorname{ev}_{1}(\unicode[STIX]{x2202}_{-}I_{j})\subset V^{\unicode[STIX]{x1D70E}_{j}}\quad \text{and}\quad \operatorname{ev}_{1}(\unicode[STIX]{x2202}_{+}I_{j})\subset V^{-\unicode[STIX]{x1D70E}_{j}}.\end{eqnarray}$$

We extend the evaluation map $\operatorname{ev}_{1}$ from $\overline{{\mathcal{F}}}$ to ${\mathcal{B}}$ in two stages. The conditions of the following lemma will arise naturally in the setting of Theorem 1.1, but can always be achieved by appropriate scaling and translation. Let $(x,y)=(x_{1},y_{1},\ldots ,x_{n},y_{n})$ be standard coordinates on $\mathbb{R}^{2n}$ . Let $L\subset \mathbb{R}^{2n}$ be a Lagrangian disk with Legendrian boundary $\unicode[STIX]{x1D6EC}$ , and suppose that $L$ agrees with the cone on $\unicode[STIX]{x1D6EC}$ in the half-space $\{x_{1}\leqslant -\unicode[STIX]{x1D716}\}$ . (Note that here we are conical in a region lying over the negative, rather than the positive, $x_{1}$ -axis; this differs from the convention at the start of §2.1, but fits with the construction of the Legendrian double $\unicode[STIX]{x1D6E4}(L)$ in §2.2, which involves translation of the front.) Assume that $\unicode[STIX]{x1D6EC}$ has a single Reeb chord, which has grading $(n-1)$ . We construct the Legendrian double $\unicode[STIX]{x1D6E4}(L)$ of $L$ , in the sense of §2.2, in such a way that: (i) $L^{+}$ lies in $\{x_{1}\geqslant -\unicode[STIX]{x1D70F}\}$ , and agrees with the front of $L$ in that region; (ii) its reflection $L^{-}$ lies in $\{x_{1}\leqslant -\unicode[STIX]{x1D70F}-2\unicode[STIX]{x1D716}\}$ ; and (iii) the cylindrical piece which joins the two lies in $\{-2\unicode[STIX]{x1D716}-\unicode[STIX]{x1D70F}\leqslant x_{1}\leqslant -\unicode[STIX]{x1D70F}\}$ . Let $\unicode[STIX]{x1D719}:S^{n}\rightarrow \mathbb{R}^{2n}$ be the Lagrangian immersion obtained by projecting $\unicode[STIX]{x1D6E4}(L)$ along the $z$ -axis. Then $\unicode[STIX]{x1D719}$ has one transverse double point, corresponding to the unique Reeb chord of $\unicode[STIX]{x1D6EC}$ ; the double point has grading $n$ and lies in the slice $\{x_{1}=-\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D716}\}$ .

As mentioned briefly above, the map

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}:{\mathcal{F}}_{1}\times {\mathcal{M}}\times [\unicode[STIX]{x1D70C}_{0},\unicode[STIX]{x1D70C}_{1}]\rightarrow \overline{{\mathcal{F}}}\end{eqnarray}$$

that gives the collar neighborhood on the boundary is constructed by first pre-gluing the disks in ${\mathcal{F}}_{1}$ and ${\mathcal{M}}$ , then applying Floer–Picard iteration to obtain an actual solution, and finally composing the result with a reparameterization of the domain so that the holomorphic disk part lies in a small half-disk of size ${\mathcal{O}}(e^{-\unicode[STIX]{x1D70C}})$ close to the boundary puncture of the disk in ${\mathcal{F}}_{1}$ . In this construction, we use an explicit gauge-fixing procedure for the holomorphic maps in ${\mathcal{M}}$ . The (pre-)gluing map in fact takes as input an element in this gauge-fixed space of smooth maps, see [Reference Ekholm and SmithES14, Appendix A.9].

For $(u,v,\unicode[STIX]{x1D70C})\in {\mathcal{F}}_{1}\times {\mathcal{M}}\times [\unicode[STIX]{x1D70C}_{0},\unicode[STIX]{x1D70C}_{1}]$ , let $\unicode[STIX]{x1D6F9}^{\prime }(u,v,\unicode[STIX]{x1D70C})$ be the ‘naive’ version of $\unicode[STIX]{x1D6F9}(u,v,\unicode[STIX]{x1D70C})$ in which we do not apply Floer–Picard iteration: $\unicode[STIX]{x1D6F9}^{\prime }(u,v,\unicode[STIX]{x1D70C})$ is constructed only by pregluing and reparameterization. The distance between the starting point for Floer–Picard iteration and the actual solution resulting from iteration is estimated in [Reference Ekholm and SmithES14, (5.10)]. In the current setting this implies that, by taking $\unicode[STIX]{x1D70C}_{0}$ sufficiently large, we can make $\unicode[STIX]{x1D6F9}(u,v,\unicode[STIX]{x1D70C})$ and $\unicode[STIX]{x1D6F9}^{\prime }(u,v,\unicode[STIX]{x1D70C})$ arbitrarily $C^{1}$ -close for all $(u,v,\unicode[STIX]{x1D70C})\in {\mathcal{F}}_{1}\times {\mathcal{M}}\times [\unicode[STIX]{x1D70C}_{0},\unicode[STIX]{x1D70C}_{1}]$ .

Proof. As indicated at the start of the section, we aim to construct an extension of $\operatorname{ev}_{1}$ using holomorphic disk components of broken curves (whose evaluation image we will be able to control, in contrast to that of disks in ${\mathcal{F}}_{1}$ , say). At this stage, the key requirement will be to construct the first step of the extension so as to have image in $\unicode[STIX]{x1D719}(S^{n})\cup \{x_{1}\leqslant -\unicode[STIX]{x1D70F}\}$ .

For glued maps $w:D\rightarrow \mathbb{R}^{2n}$ , where $w=u\#v=\unicode[STIX]{x1D6F9}(u,v,\unicode[STIX]{x1D70C})$ for $(u,v)\in U\times {\mathcal{M}}\subset {\mathcal{F}}_{1}\times {\mathcal{M}}$ , the boundary point $1\in \unicode[STIX]{x2202}D$ at which we evaluate when defining $\operatorname{ev}_{1}$ , $\operatorname{ev}_{1}(w)=w(1)$ , lies in the part of the domain $D$ of the glued map that comes from its component $u$ in the factor ${\mathcal{F}}_{1}$ . Since the glued and pre-glued maps are arbitrarily close, this means that $\operatorname{ev}_{1}(w)=w(1)$ is arbitrarily close to $u(1)$ , see Figure 2.

Figure 2. Top: the evaluation point $1$ is far from the glued in holomorphic disk $v$ , as in Lemma 3.9. Middle: the glued in holomorphic disk passes the evaluation point $1$ , as in Lemma 3.10. Bottom: the passage as viewed from the holomorphic disk, where the evaluation point sweeps the whole boundary.

Consequently, by Lemma 3.8, if $w^{\pm }=u^{\pm }\#v$ for $(u^{\pm },v)\in \unicode[STIX]{x2202}_{\pm }I_{j}\times {\mathcal{M}}$ , we can connect the two points $\operatorname{ev}_{1}(w^{\pm })\in V^{\pm \unicode[STIX]{x1D70E}}$ to $\operatorname{ev}_{1}(w^{\mp })\in V^{\mp \unicode[STIX]{x1D70E}}$ by a short straight line segment in $\mathbb{R}^{2n}$ near the double point. This gives an extension of the evaluation map to $\unicode[STIX]{x2202}({\mathcal{D}}\setminus \bigcup _{j=1}^{m}{\mathcal{D}}_{j})\times {\mathcal{M}}$ , where we map $\unicode[STIX]{x2202}({\mathcal{D}}_{j}\setminus I_{j})\times {\mathcal{M}}$ to the line segments above.

The union of $\unicode[STIX]{x1D719}(S^{n})$ and a small ball $B$ in $\mathbb{R}^{2n}$ around its double point is homotopy equivalent to the space obtained from $S^{n}$ by attaching a $1$ -cell $e$ with one endpoint in $V^{+}$ and the other in $V^{-}$ . The fundamental group of this space is generated by a loop $\unicode[STIX]{x1D6FE}\cup e$ , where $\unicode[STIX]{x1D6FE}$ is a path in $S^{n}$ connecting the endpoints of $e$ . We take $\unicode[STIX]{x1D6FE}$ to lie in the part of $S^{n}$ mapped by $\unicode[STIX]{x1D719}$ to $\{x_{1}\leqslant -\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D716}\}$ and think of $e$ as a short path in $B$ . Then it is clear that $\{x_{1}\leqslant -\unicode[STIX]{x1D70F}\}$ contains a 2-cell bounding $\unicode[STIX]{x1D6FE}\cup e$ , which means that we can find the desired extension. To see this note that $\operatorname{ev}_{1}$ of the preglued map corresponding to $u\#v\in \unicode[STIX]{x2202}({\mathcal{D}}\setminus \bigcup _{j=1}^{m}{\mathcal{D}}_{j})\times {\mathcal{M}}$ is independent of $v$ . We then first homotope the image of $\operatorname{ev}$ , in $\unicode[STIX]{x1D719}(S^{n})\cup B$ , into $\unicode[STIX]{x1D6FE}\cup e$ and then extend over the two cell $({\mathcal{D}}\setminus \bigcup _{j=1}^{m}{\mathcal{D}}_{j})\times {\mathcal{M}}$ . The actual glued map is arbitrarily close to the preglued one, and existence of the desired extension follows.◻

Assume that an extension $\widetilde{\operatorname{ev}}_{1}$ over $({\mathcal{D}}\setminus \bigcup _{j=1}^{m}{\mathcal{D}}_{j})\times {\mathcal{M}}$ as constructed in Lemma 3.9 has been fixed. Note that for this extension $\widetilde{\operatorname{ev}}_{1}(\unicode[STIX]{x2202}({\mathcal{D}}\setminus \bigcup _{j=1}^{m}{\mathcal{D}}_{j})\times {\mathcal{M}})$ maps into a small neighborhood of the double point. We use the holomorphic disks which are parameterized by ${\mathcal{M}}$ to extend the map to the remainder of the filling ${\mathcal{T}}={\mathcal{D}}\times {\mathcal{M}}$ .

Lemma 3.10. For sufficiently large $\unicode[STIX]{x1D70C}_{0}$ , there exists a further extension of $\operatorname{ev}_{1}$ to all of ${\mathcal{D}}\times {\mathcal{M}}$ such that the image of each subspace ${\mathcal{D}}_{j}\times {\mathcal{M}}$ lies in an arbitrarily small neighborhood of the total evaluation map

$$\begin{eqnarray}\operatorname{ev}:D\times {\mathcal{M}}\longrightarrow \mathbb{R}^{2n},\quad (z,u)\mapsto u(z),\end{eqnarray}$$

where $D$ denotes the source disk for maps in (the gauge-fixed model of) ${\mathcal{M}}$ .

Proof. Fix $I_{j}$ . As the location of the glued-in holomorphic disk sweeps $I_{j}$ , the location of the boundary point $1$ on the glued domain sweeps the boundary of the once punctured source disk of maps in ${\mathcal{M}}$ . More formally, a glued disk $w=u\#v$ is close to breaking, which means that the restriction of $w$ to a small half-disk around the point in $\unicode[STIX]{x2202}D$ where $u$ has a negative puncture is arbitrarily close to $v$ for sufficiently large gluing parameter. This means that as the puncture of $u$ moves through the interval $I_{j}$ around $1$ , the evaluation map $\operatorname{ev}_{1}(u\#v)$ is arbitrarily close to the evaluation map $v|_{\unicode[STIX]{x2202}D}$ , see Figure 2.

As mentioned above, by construction of the map over $\widetilde{\operatorname{ev}}_{1}$ , its image $\widetilde{\operatorname{ev}}_{1}(\unicode[STIX]{x2202}({\mathcal{D}}\setminus \bigcup _{j=1}^{m}{\mathcal{D}}_{j})\times {\mathcal{M}})$ is approximately constant and lies in a small ball around the double point of $\unicode[STIX]{x1D719}$ . Contracting this ball to the double point (and the maps into it to constants), we may identify the products,

$$\begin{eqnarray}{\mathcal{D}}_{j}\times {\mathcal{M}}\quad \text{and}\quad D\times {\mathcal{M}},\end{eqnarray}$$

in such a way that the map $\operatorname{ev}_{1}|_{I_{j}\times {\mathcal{M}}}$ corresponds to the evaluation along the boundary $\unicode[STIX]{x2202}D$ . We then naturally extend the map over ${\mathcal{D}}_{j}$ as the evaluation map over $D$ under this identification. The lemma follows.◻

4.1 Compactification

Let $(x_{1},y_{1},\ldots ,x_{n},y_{n})$ be coordinates on $T^{\ast }\mathbb{R}^{n}$ and let $x\in \mathbb{R}^{n}$ . Let $L$ be a Lagrangian disk agreeing with the fiber $T_{x}^{\ast }\mathbb{R}^{n}$ outside a compact set. We will first show how to change $L$ in a neighborhood of infinity to obtain a Lagrangian disk with Legendrian boundary in the sense of §2.1. The Legendrian boundary will in fact be the standard Legendrian unknot.

To this end, we consider a Lagrangian disk $D\subset \mathbb{R}^{2n}=T^{\ast }\mathbb{R}^{n}$ with Legendrian boundary the standard Legendrian unknot, and such that $D$ intersects the zero-section in one point and agrees with the fiber at that point in a neighborhood of this intersection point. In agreement with the conventions used after Lemma 3.8, we assume that $D$ is a cone over a Legendrian unknot over the negative $x_{1}$ -axis. Consider scaling the fiber coordinates in $T^{\ast }\mathbb{R}^{n}$ by $\unicode[STIX]{x1D706}>0$ . By choosing $\unicode[STIX]{x1D706}$ sufficiently large we make the intersection of $D$ with an arbitrarily large ball agree with the fiber at the intersection point. Removing the intersection with the ball and inserting the corresponding part of $L$ , we get the desired Lagrangian disk with Legendrian boundary equal to the standard unknot. This disk agrees with $L$ in a ball, and is a cylinder over a Legendrian unknot in the region $\{x_{1}\leqslant -\unicode[STIX]{x1D70F}\}$ for sufficiently large $\unicode[STIX]{x1D70F}>0$ .

Let $S_{k}\subset \mathbb{R}^{n}\setminus \{x\}$ be a point if $k=0$ , or a $k$ -sphere in $\mathbb{R}^{n}$ if $1\leqslant k<n-1$ , as in §1. By a small homotopy, it will be convenient to normalize choices as follows (cf. the discussion after Lemma 3.8). We take $x=0$ to be the origin, and we assume that $S_{k}\subset \{x_{1}=\unicode[STIX]{x1D700}\}$ lies in an affine hyperplane (so in particular $x=0$ does not lie in the convex hull of $S_{k}$ ), for some small $\unicode[STIX]{x1D700}>0$ . We have a Lagrangian disk $L$ that agrees with the fiber $T_{0}^{\ast }\mathbb{R}^{n}$ over $0$ outside a compact set, and which is disjoint from $T_{S_{k}}^{\ast }\mathbb{R}^{n}$ . We alter $L$ as above, inserting a cone on the unknot over the negative $x_{1}$ -axis, to obtain a Lagrangian disk with Legendrian boundary in the sense of §2.1.

Consider now the Lagrangian projection of the Legendrian sphere $\unicode[STIX]{x1D6E4}(L)\subset \mathbb{R}^{2n}\times \mathbb{R}$ , as constructed in §2.2. If we view the standard Legendrian unknot as living in the contact hypersurface $\{x_{1}=-\unicode[STIX]{x1D70F}+\unicode[STIX]{x1D70F}_{0}\}$ , then $\unicode[STIX]{x1D6E4}(L)$ agrees with the cone over that unknot in the region $\{-\unicode[STIX]{x1D70F}\leqslant x_{1}\leqslant -\unicode[STIX]{x1D70F}+\unicode[STIX]{x1D70F}_{0}\}$ , for some large $\unicode[STIX]{x1D70F}>\unicode[STIX]{x1D70F}_{0}>0$ . Furthermore, since $L$ has Legendrian boundary the standard unknot, $\unicode[STIX]{x1D6E4}(L)$ has exactly one Reeb chord, which has grading $n$ . The corresponding double point of its projection lies in $\{x_{1}=-\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D716}\}$ , where we form $\unicode[STIX]{x1D6E4}(L)$ by the doubling construction centered at $\{x_{1}=-\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D716}\}$ . Denote the resulting Lagrangian sphere immersion with one double point by $\unicode[STIX]{x1D719}:S^{n}\rightarrow \mathbb{R}^{2n}$ .

4.2 Degenerating the almost complex structure

We continue in the setting of §4.1. The map $\unicode[STIX]{x1D719}:S^{n}\rightarrow \mathbb{R}_{\text{st}}^{2n}$ satisfies the conditions from the start of §3.1, so we have the moduli spaces of Floer-holomorphic and holomorphic disks as before. Bearing in mind Lemma 3.9, the evaluation map $\operatorname{ev}_{1}:\overline{{\mathcal{F}}}_{1}\rightarrow \unicode[STIX]{x1D719}(S^{n})\subset \mathbb{R}^{2n}$ admits an extension over $({\mathcal{D}}\setminus \bigcup _{j}{\mathcal{D}}_{j})\times {\mathcal{M}}$ with image inside the half-space $\{x_{1}\leqslant -\unicode[STIX]{x1D70F}\}$ , which is disjoint from $\operatorname{pr}^{-1}(S_{k})=(T^{\ast }\mathbb{R}^{n})|_{S_{k}}$ . To construct the desired extension of the evaluation map to all of ${\mathcal{B}}$ with image outside $(T^{\ast }\mathbb{R}^{n})|_{S_{k}}$ , in light of Lemma 3.10, it suffices to prove that for a suitable almost complex structure $J$ on $T^{\ast }\mathbb{R}^{n}$ , the image of the total evaluation $\operatorname{ev}:D\times {\mathcal{M}}\rightarrow T^{\ast }\mathbb{R}^{n}$ is completely disjoint from the submanifold $\operatorname{pr}^{-1}(S_{k})\subset T^{\ast }\mathbb{R}^{n}$ .

To that end, we consider a family of almost complex structures associated to the limit in which the fibres of $\operatorname{pr}:T^{\ast }\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ are shrunk to zero volume. Let $\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D70B}_{2}(T^{\ast }\mathbb{R}^{n},\unicode[STIX]{x1D719}(S^{n}))$ denote the generator of positive area, so for any taming almost complex structure $J$ , the holomorphic discs in ${\mathcal{M}}$ have relative homotopy class $\unicode[STIX]{x1D6FD}$ and area $A(\unicode[STIX]{x1D719})=\int _{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D714}_{\text{st}}$ .

Equivalently, one can fix the Lagrangian immersion $\unicode[STIX]{x1D719}(S^{n})$ , and consider the almost complex structures $J_{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D706}}^{\ast }(J)$ . Let $\unicode[STIX]{x1D706}_{0}$ be as in Lemma 4.2 and write ${\mathcal{M}}_{\unicode[STIX]{x1D706}}$ for the moduli space of $J_{\unicode[STIX]{x1D706}}$ -holomorphic disks with one positive puncture and boundary on $\unicode[STIX]{x1D719}(S^{n})$ .

4.3 Proofs of the local results

4.3.1 Proof of Theorem 1.1

Lemma 4.1 shows that $f_{L}:S^{n}\rightarrow \mathbb{R}^{n}\setminus \{0\}$ is homotopic to the composition $\operatorname{pr}\circ \unicode[STIX]{x1D719}$ where $\operatorname{pr}:T^{\ast }\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is the bundle projection, and where the Lagrangian immersion $\unicode[STIX]{x1D719}:S^{n}\rightarrow T^{\ast }\mathbb{R}^{n}$ with one double point was constructed in §4.1. Proposition 3.6 then gives a spin $(n+1)$ -manifold ${\mathcal{B}}$ such that $\unicode[STIX]{x2202}{\mathcal{B}}=S^{n}$ (compare to the proof of Corollary 1.2 where we use that ${\mathcal{B}}$ is parallelizable) and Lemma 3.10 gives an extension of the map $\operatorname{ev}_{1}=\unicode[STIX]{x1D719}$ on $S^{n}$  to a map $\operatorname{ev}_{1}:{\mathcal{B}}\rightarrow T^{\ast }\mathbb{R}^{n}\setminus T_{0}^{\ast }\mathbb{R}^{n}$ . Composing with the projection $\operatorname{pr}$ we find that $\operatorname{pr}\circ \unicode[STIX]{x1D719}$ extends to $\operatorname{pr}\circ \operatorname{ev}_{1}:{\mathcal{B}}\rightarrow \mathbb{R}^{n}\setminus \{0\}$ . Consider now the preimage $C_{\ell }=(\operatorname{pr}\circ \unicode[STIX]{x1D719})^{-1}(\ell )$ of a ray $\ell \approx [0,\infty )$ emanating at $0\in \mathbb{R}^{n}$ which is transverse to $\operatorname{pr}\circ \unicode[STIX]{x1D719}$ . Then $C_{\ell }$ is a closed $1$ -manifold, and the tangent space of the sphere of rays at $\ell$ gives a normal framing of $C_{\ell }$ . Similarly, let $F_{\ell }\subset {\mathcal{B}}$ denote the oriented surface which is the preimage $(\operatorname{pr}\circ \operatorname{ev}_{1})^{-1}(\ell )$ of $\ell$ , similarly equipped with a normal framing.

Let $\unicode[STIX]{x1D709}_{S^{n}}$ denote the unique spin structure on $S^{n}$ . We view $\unicode[STIX]{x1D709}_{S^{n}}$ as a class in $H^{1}(\text{SO}^{\prime }(S^{n}))$ , where $\text{SO}^{\prime }(S^{n})$ denotes the bundle of oriented frames on the stabilized tangent bundle $TS^{n}\oplus \mathbb{R}$ . Similarly, we view the spin structure on ${\mathcal{B}}$ as an element $\unicode[STIX]{x1D709}_{{\mathcal{B}}}\in H^{1}(\text{SO}({\mathcal{B}}))$ , where $\text{SO}({\mathcal{B}})$ denotes the frame bundle of ${\mathcal{B}}$ . Note that $\unicode[STIX]{x1D709}_{S^{n}}$ is the restriction of $\unicode[STIX]{x1D709}_{{\mathcal{B}}}$ to the boundary $S^{n}$ .

Given a normal framing $\unicode[STIX]{x1D708}$ of a circle $\unicode[STIX]{x1D6FE}$ in $S^{n}\subset {\mathcal{B}}$ we equip it with a full framing of $T{\mathcal{B}}|_{\unicode[STIX]{x1D6FE}}$ by adding a framing $\unicode[STIX]{x1D70F}$ in the two remaining directions which rotates once compared to the framing given by the tangent vector of $S^{1}$ followed by the constant vector in $\mathbb{R}$ . Let $\vec{\unicode[STIX]{x1D6FE}}$ denote the curve $\unicode[STIX]{x1D6FE}$ lifted to $\text{SO}^{\prime }(S^{n})$ by the framing $(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D708})$ . An explicit calculation shows that $\unicode[STIX]{x1D708}$ is the null cobordant framing if and only if

$$\begin{eqnarray}\langle \unicode[STIX]{x1D709}_{S^{n}},\vec{\unicode[STIX]{x1D6FE}}\rangle =0.\end{eqnarray}$$

For the link $C_{\ell }$ it is elementary to check that the framing $(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D708})$ extends to a framing of $F_{\ell }$ . This means that $\vec{C_{\ell }}$ bounds in $\text{SO}({\mathcal{B}})$ . Hence

$$\begin{eqnarray}\langle \unicode[STIX]{x1D709}_{{\mathcal{B}}},\vec{C_{\ell }}\rangle =\langle \unicode[STIX]{x1D709}_{S^{n}},\vec{C_{\ell }}\rangle =0.\end{eqnarray}$$

This shows that $C_{\ell }$ is framed nullcobordant, which by the Pontryagin–Thom construction finishes the proof.◻

4.3.2 Proof of Corollary 1.2

The proof of Corollary 1.2 follows similar lines, but uses triviality of the tangent bundle of the filling ${\mathcal{B}}$ beyond its spin structure.

Let $L$ be the Lagrangian disk. The stable homotopy class of $f_{L}$ is the homotopy class of the $m$ -fold suspension $\unicode[STIX]{x1D6F4}^{m}f_{L}$ for any sufficiently large $m$ . By Lemma 2.3, this is also the homotopy class of the map $f_{C^{m}(L)}$ associated to the $(n+m)$ -dimensional Lagrangian disk $C^{m}(L)=C(C(\ldots C(L)))$ . We point out that although the Legendrian boundary of $C^{m}(L)$ might not be the standard Legendrian unknot, it is a knot with only one Reeb chord, and that is sufficient for the argument below to apply.

Picking $m$ sufficiently large to be in the stable range, and also so that $\unicode[STIX]{x1D70B}_{n+m}(U/O)$ vanishes (e.g. $n+m\equiv -1\hspace{0.6em}{\rm mod}\hspace{0.2em}8$ ), Proposition 3.5 gives a parallelizable manifold ${\mathcal{B}}$ with boundary $S^{n+m}$ and Lemma 3.10 gives an extension $\operatorname{pr}\circ \operatorname{ev}_{1}$ of the corresponding map $\unicode[STIX]{x1D719}:S^{n+m}\rightarrow T^{\ast }(\mathbb{R}^{n+m}\setminus S^{k})$ to ${\mathcal{B}}$ . Consider the composition $\operatorname{pr}\circ \unicode[STIX]{x1D719}$ followed by the inclusion $\mathbb{R}^{n+m}\subset S^{n+m}$ . Let $\unicode[STIX]{x1D6E4}\approx S^{n+m-k-1}\subset \mathbb{R}^{n+m}\setminus S^{k}$ be a fiber sphere in the unit normal bundle of $S^{k}$ . Then $S^{n+m}\setminus S^{k}\approx \unicode[STIX]{x1D6E4}\times D^{k+1}$ is foliated by open $(k+1)$ -disks $D_{\unicode[STIX]{x1D6FE}}$ parameterized by $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}$ . For $D_{\unicode[STIX]{x1D6FE}}$ transverse to $\operatorname{pr}\circ \operatorname{ev}_{1}:{\mathcal{B}}\rightarrow S^{n+m}\setminus S^{k}$ we find that

$$\begin{eqnarray}M_{\unicode[STIX]{x1D6FE}}=(\operatorname{pr}\circ \operatorname{ev}_{1})^{-1}(D_{\unicode[STIX]{x1D6FE}})\subset S^{n+m}\end{eqnarray}$$

is a framed submanifold that framed bounds the framed submanifold

$$\begin{eqnarray}W_{\unicode[STIX]{x1D6FE}}=(\operatorname{pr}\circ \operatorname{ev}_{1})^{-1}(D_{\unicode[STIX]{x1D6FE}})\subset {\mathcal{B}}.\end{eqnarray}$$

The manifold ${\mathcal{B}}$ is a parallelizable $(n+m+1)$ -manifold; the stability condition $k<(n+m-3)/2$ guarantees that we can perform framed surgery on ${\mathcal{B}}$ , along framed spheres in the interior and disjoint from $W_{\unicode[STIX]{x1D6FE}}$ , to yield a parallelizable manifold ${\mathcal{B}}^{\prime }$ which is $(k+2)$ -connected. Fix a Morse function on ${\mathcal{B}}^{\prime }$ with no critical points of index ${\leqslant}k+2$ , and with gradient field pointing outwards along the boundary. Via gradient flow, one can then isotope the $(k+2)$ -dimensional subset $W_{\unicode[STIX]{x1D6FE}}\subset {\mathcal{B}}^{\prime }$ into a collar neighborhood of the boundary of ${\mathcal{B}}^{\prime }$ (which is identified with a collar neighborhood of the boundary of ${\mathcal{B}}$ ), since for dimension reasons it will generically miss the ascending manifolds of all critical points. By the Pontryagin–Thom construction the result again follows.◻

Remark 4.5. In the borderline case $\{k=0,n=3\}$ , the relevant homotopy group $\unicode[STIX]{x1D70B}_{3}(S^{2})$ is not stable, but Corollary 1.2 shows vanishing of the corresponding stabilized invariant. Concretely this means that if $L\subset T^{\ast }(\mathbb{R}^{3}\backslash \{0\})$ is a Lagrangian disk which coincides with a fiber outside a compact set, then $[f_{L}]\in \unicode[STIX]{x1D70B}_{3}(S^{2})\cong \mathbb{Z}$ is even.

Note that when $n=3$ the moduli space ${\mathcal{F}}_{-2}$ has virtual dimension zero, cf. Lemma 3.2, so the construction of the bounding manifold ${\mathcal{B}}$ does not go through directly in this case.

4.4 Quasi-isomorphism type

For a closed exact Lagrangian $L\subset T^{\ast }Q$ , it is known that $L$ is isomorphic to the zero-section in the Fukaya category. We point out that the hypotheses in Theorem 1.1 imply the analogous result for the nearby Lagrangian fiber, though we know of no Floer-theoretic argument to constrain the homotopy class $f_{L}$ .

Fix a coefficient field $\mathbb{K}$ . A Liouville like manifold $(Y,\unicode[STIX]{x1D714}=d\unicode[STIX]{x1D703})$ (see [Reference Ekholm and LekiliEL17, Appendix B.3] for a more detailed description of manifolds $Y$ as considered below) then has a well-defined wrapped Fukaya category $W(Y)$ , an $A_{\infty }$ -category over $\mathbb{K}$ , whose objects are exact spin Lagrangian submanifolds which are either closed or are cylinders on Legendrian submanifolds of the ideal boundary near infinity; see [Reference Abouzaid and SeidelAS10] for the construction.

Proof. We compactify the base $\mathbb{R}^{n}\setminus \{0\}=S^{n-1}\times \mathbb{R}$ to $S^{n-1}\times S^{1}$ , and we compactify the distinguished fibre $T_{x}^{\ast }$ by adding a handle to the Legendrian unknot at infinity. This yields the plumbing $X=T^{\ast }(S^{n-1}\times S^{1})\#_{\unicode[STIX]{x2202}}T^{\ast }S^{n}$ , which is naturally a Liouville domain with an exact symplectic structure $\unicode[STIX]{x1D714}$ . The manifold $X$ contains non-compact Lagrangian thimbles $T_{p}^{\ast }$ and $T_{q}^{\ast }$ , which are cotangent fibres to points $p\in S^{n-1}\times S^{1}$ and $q\in S^{n}$ , in both cases lying near infinity (i.e. in the parts of the base added in the compactification). The Lagrangian disk $L$ extends to a Lagrangian sphere $L^{\prime }$ in $X$ , which meets the cotangent fibre $T_{q}^{\ast }$ transversely once, and is disjoint from the fibre $T_{p}^{\ast }$ . Since $n\geqslant 3$ , the results of [Reference Abouzaid and SmithAS12] show that every compact Lagrangian submanifold in $X$ can be expressed as a twisted complex on the two core components, and then the conditions

$$\begin{eqnarray}HF(L^{\prime },T_{p}^{\ast })=0,\quad HF(L^{\prime },T_{q}^{\ast })=\mathbb{K}\end{eqnarray}$$

imply that $L^{\prime }$ is quasi-isomorphic in the wrapped (and hence compact) Fukaya category $W(X)$ to the core component $S^{n}\subset X$ . Applying Viterbo functoriality [Reference Abouzaid and SeidelAS10], we infer that the original Lagrangian $L$ is quasi-isomorphic to the cotangent fibre of $\mathbb{R}^{n}\backslash \{0\}$ .◻

5.1 Neck stretching around Lagrangian submanifolds

Let $C$ be a closed Lagrangian submanifold of $\mathbb{R}_{\text{st}}^{2n}$ . Then $C$ has a neighborhood which is symplectomorphic to a disk bundle neighborhood of the zero-section in $T^{\ast }C$ . We identify this neighborhood with the union of two pieces: a collar region $[0,-1)\times U^{\ast }C$ , where $U^{\ast }C$ denotes the unit cotangent bundle equipped with the standard contact form and the product has the symplectic form of the symplectization, together with a suitably scaled version of the unit disk cotangent bundle $D^{\ast }C$ glued in. Stretching the neck then corresponds to replacing the collar region by $[0,-T)\times U^{\ast }C$ for $T\rightarrow \infty$ . In the limit the manifold separates into two pieces, one a completion of $\mathbb{R}_{\text{st}}^{2n}\setminus C$ with negative end $U^{\ast }C$ , and the other symplectomorphic to $T^{\ast }C$ with positive end $U^{\ast }C$ .

Given instead a Lagrangian submanifold $C$ with non-empty ideal Legendrian boundary $\unicode[STIX]{x1D6E4}$ , before stretching the neck around $C$ we must adjust the Liouville structure, so the Liouville vector field agrees with the Liouville field of the cotangent bundle of $C$ near $C$ . To accomplish this we proceed as in [Reference Ekholm, Ng and ShendeENS17, §6.1 and Figure 7]. Consider a large ball such that $C$ intersects its boundary sphere in $\unicode[STIX]{x1D6E4}$ . We create the new Liouville structure by attaching $D^{\ast }(\unicode[STIX]{x1D6E4}\times [0,\infty ))$ along $\unicode[STIX]{x1D6E4}$ , where $D^{\ast }\unicode[STIX]{x1D6E4}$ denotes a disk subbundle of the cotangent bundle $T^{\ast }\unicode[STIX]{x1D6E4}$ . One can interpolate between the usual radial Liouville vector field on $\mathbb{R}_{\text{st}}^{2n}$ and the field $p\cdot \unicode[STIX]{x2202}_{p}$ in the cotangent bundle neighborhood to define a Liouville field for the new structure. This gives a Liouville-like manifold, that we denote $\mathbb{R}_{C}^{2n}$ , with non-compact contact boundary. Note that the Liouville-like manifolds we consider look like ordinary Liouville manifolds except that they have additional ends where they look like the standard fiberwise Liouville structure on the cotangent bundle $T^{\ast }(\unicode[STIX]{x1D6E4}\times [0,\infty ))$ where $\unicode[STIX]{x1D6E4}$ is a closed $(n-1)$ -manifold. We refer to [Reference Ekholm and LekiliEL17, Appendix B.3] for a discussion of such domains. It still contains $C$ as an embedded Lagrangian submanifold, and in $\mathbb{R}_{C}^{2n}$ we can stretch the neck around $C$ just as in the closed case. The Maslov class of $C$ is unaffected by the passage from $\mathbb{R}_{\text{st}}^{2n}$ to $\mathbb{R}_{C}^{2n}$ , and in particular monotonicity of $C$ is unaffected. For uniform notation, we let $\mathbb{R}_{C}^{2n}=\mathbb{R}_{\text{st}}^{2n}$ when $C$ is closed.

5.2 Conley–Zehnder and Morse indices

Virtual dimensions of moduli spaces of punctured holomorphic curves are typically expressed in terms of Conley–Zehnder indices of Reeb orbits. These may be canonically defined in the unit cotangent bundle of an orientable manifold, but depend on additional choices in the non-orientable case. In this section we explain our convention for such indices.

Let $C\subset \mathbb{R}_{\text{st}}^{2n}$ be a Lagrangian submanifold. Fix a Riemannian metric on $C$ . The Reeb orbits on the unit cotangent bundle are the oriented closed geodesics for the metric. We will not distinguish the geodesic from the Reeb orbit in our notation. In case $C$ has Legendrian boundary $\unicode[STIX]{x1D6E4}$ we choose the metric in the cylindrical end $[0,\infty )\times \unicode[STIX]{x1D6E4}$ to have the form $ds^{2}=dt^{2}+f(t)g$ , where $g$ is a metric on $\unicode[STIX]{x1D6E4}$ and where $f(t)>0$ is a function with $f^{\prime }(t)>0$ . Then the $t$ -coordinate along a non-constant geodesic in the cylindrical end cannot have a local maximum. Hence, there are no closed geodesics in the end.

If $\operatorname{pr}:T^{\ast }C\rightarrow C$ is projection, then $T(T^{\ast }C)\cong \operatorname{pr}^{\ast }(TC\otimes \mathbb{C})$ . Let $\det _{\mathbb{C}}(TC)$ denote the corresponding determinant bundle, i.e. the complex line bundle which is the determinant of the complexified tangent bundle $TC\otimes \mathbb{C}$ .

We next define trivializations of $\det _{\mathbb{C}}(TC)$ . Consider the real line bundle $\det (TC)$ over $C$ and fix a section $s:C\rightarrow \det (TC)$ that is transverse to the $0$ -section. (When $C$ is orientable we can take $s$ to be non-vanishing.) Let $P=s^{-1}(0)$ .

Proof. First note that the total space $E$ of $\det (TC)$ is orientable. The normal bundle to $P$ in $E$ is the sum of the normal bundle $\unicode[STIX]{x1D708}_{1}$ to $P$ in $C$ and the normal bundle $\unicode[STIX]{x1D708}_{2}$ of $C$ in $E$ . Since $P$ is Poincaré dual to $w_{1}(TC)$ it follows that $w_{1}(\unicode[STIX]{x1D708}_{1})=w_{1}(TC)|_{P}$ . Since $\unicode[STIX]{x1D708}_{2}=\det (TC)$ by definition of $E$ , also $w_{1}(\unicode[STIX]{x1D708}_{2})=w_{1}(TC)|_{P}$ . Hence the normal bundle to $P$ in $E$ has vanishing first Stiefel–Whitney class and since $E$ is orientable, $P$ is orientable.

We claim that a neighborhood $N(P)$ of $P$ is orientable as well. To see this, note that, by orientability of $P$ , $N(P)$ is orientable provided the normal bundle $\unicode[STIX]{x1D708}_{1}$ of $P$ in $C$ is trivial. Let $\unicode[STIX]{x1D70E}:P\rightarrow \unicode[STIX]{x1D708}_{1}$ be a section of $\unicode[STIX]{x1D708}_{1}$ transverse to the $0$ -section. Since $P$ is $\mathbb{Z}/2$ -Poincaré dual to the first Stiefel–Whitney class $w_{1}(TC)$ , $\unicode[STIX]{x1D70E}^{-1}(0)$ is Poincaré dual to $w_{1}(TC)^{2}$ . But

$$\begin{eqnarray}w_{2}(TC\otimes \mathbb{C})=w_{2}(TC\oplus TC)=w_{1}(TC)^{2}=0,\end{eqnarray}$$

by Lemma 5.2. Hence $\unicode[STIX]{x1D708}_{1}$ is orientable and so is $N(P)$ , and we have a tubular neighborhood $N(P)\approx P\times (-\unicode[STIX]{x1D716},\unicode[STIX]{x1D716})$ of $P$ in $C$ as required. Finally, picking an orientation of $\unicode[STIX]{x1D708}_{1}$ allows us to define a $\mathbb{Z}$ -valued intersection number between loops in $C$ and $P$ . Since $P$ is $\mathbb{Z}/2$ -Poincaré dual to $w_{1}(TC)$ , it is clear that the resulting class in $H^{1}(C;\mathbb{Z})$ is an integral lift of $w_{1}$ .◻

To normalize sections in the construction below we use the metric on $TC$ to induce a metric $|\cdot |$ on the real line bundle $\det (TC)$ . Let $s:C\rightarrow \det (TC)$ be the section above and define

$$\begin{eqnarray}v^{\prime }(q)=\frac{1}{|s(q)|}s(q),\quad q\in C\setminus N(P).\end{eqnarray}$$

Then $v^{\prime }$ trivializes $\det _{\mathbb{C}}(TC)$ over $C\setminus N(P)$ by viewing $v^{\prime }$ as a non-zero section of $\det _{\mathbb{C}}(TC)$ .

By the above discussion also $\det (TC)|_{N(P)}$ is trivial. The boundary of $N(P)\approx P\times (-\unicode[STIX]{x1D716},\unicode[STIX]{x1D716})$ is disconnected and the restriction of $v^{\prime }$ to the boundary component $\{-\unicode[STIX]{x1D716}\}\times P$ gives a unit length section. Let $v^{\prime \prime }:N(P)\rightarrow \det (TC)$ be the unique extension of this section as a unit length section then $v^{\prime \prime }$ gives a trivialization of $\det _{\mathbb{C}}(TC)$ over $N(P)$ . The trivializations $v^{\prime }$ and $v^{\prime \prime }$ can be compared along $P\times \{\pm \unicode[STIX]{x1D716}\}$ . Since $s$ has a transverse zero along $P$ we find the following: $v^{\prime }=v^{\prime \prime }$ along $P\times \{-\unicode[STIX]{x1D716}\}$ and $v^{\prime }=-v^{\prime \prime }$ along $P\times \{\unicode[STIX]{x1D716}\}$ . We use this to fix a trivialization $v$ of $\det _{\mathbb{C}}(TC)$ over all of $C$ as follows:

(5.1) $$\begin{eqnarray}v=\left\{\begin{array}{@{}ll@{}}v^{\prime }(q)\quad & \text{ for }q\in C\setminus N(P),\\ e^{i\unicode[STIX]{x1D70B}(t+\unicode[STIX]{x1D716})/2\unicode[STIX]{x1D716}}v^{\prime \prime }(p,t)\quad & \text{ for }(p,t)\in P\times [-\unicode[STIX]{x1D716},\unicode[STIX]{x1D716}]=N(P).\end{array}\right.\end{eqnarray}$$

We will denote the Conley–Zehnder index of a Reeb orbit $\unicode[STIX]{x1D6FE}$ defined with respect to this trivialization $\operatorname{CZ}^{w_{1}}$ . Denote by $\unicode[STIX]{x1D704}(\unicode[STIX]{x1D6FE})$ the Morse index of an oriented closed geodesic, viewed as a critical point of the energy function on the free loop space.

Lemma 5.4. The Conley–Zehnder index of a Reeb orbit $\unicode[STIX]{x1D6FE}$ in $U^{\ast }C$ with respect to the trivialization $v$ of $\det _{\mathbb{C}}(TC)$ , see (5.1), satisfies the following:

(5.2) $$\begin{eqnarray}\operatorname{CZ}^{w_{1}}(\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D704}(\unicode[STIX]{x1D6FE})-W_{1}(\unicode[STIX]{x1D6FE}).\end{eqnarray}$$

To compute dimensions of disks and spheres in the component $\mathbb{R}^{2n}\setminus C$ after stretching, we use the Conley–Zehnder index of Reeb orbits as defined with respect to the standard global trivialization of the tangent bundle of $\mathbb{R}_{C}^{2n}$ . More precisely, if $(x_{1},y_{1},\ldots ,x_{n},y_{n})$ are standard symplectic coordinates on $\mathbb{R}^{2n}$ then we think of $T\mathbb{R}^{2n}$ as a complex vector bundle $T^{\prime }\mathbb{R}^{2n}$ trivialized by $(\unicode[STIX]{x2202}_{x_{1}},\ldots ,\unicode[STIX]{x2202}_{x_{n}})$ . In a neighborhood of $C$ , the complex line bundle $\det _{\mathbb{C}}(TC)$ is isomorphic to $\det (T^{\prime }\mathbb{R}^{2n})$ and

(5.3) $$\begin{eqnarray}v_{0}=\unicode[STIX]{x2202}_{x_{1}}\wedge \cdots \wedge \unicode[STIX]{x2202}_{x_{n}}\end{eqnarray}$$

gives a non-zero section of $\det _{\mathbb{C}}(TC)$ . We denote the Conley–Zehnder index of Reeb orbits defined with respect to the trivialization $v_{0}$ by $\operatorname{CZ}$ . The following result relates $\operatorname{CZ}$ and $\operatorname{CZ}^{w_{1}}$ ; it is closely related to [Reference ViterboVit90, Theorem 3.1].

Lemma 5.5. If $\unicode[STIX]{x1D6FE}$ is a Reeb orbit in $U^{\ast }C$ , if $\operatorname{CZ}$ denotes the Conley–Zehnder index defined with respect to the trivialization $v_{0}$ , and if $\operatorname{CZ}^{w_{1}}$ is as in Lemma 5.4, then

$$\begin{eqnarray}\operatorname{CZ}(\unicode[STIX]{x1D6FE})=\operatorname{CZ}^{w_{1}}(\unicode[STIX]{x1D6FE})+W_{1}(\unicode[STIX]{x1D6FE})-\unicode[STIX]{x1D707}_{C}(\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D704}(\unicode[STIX]{x1D6FE})-\unicode[STIX]{x1D707}_{C}(\unicode[STIX]{x1D6FE}),\end{eqnarray}$$

where $\unicode[STIX]{x1D707}_{C}(\unicode[STIX]{x1D6FE})$ denotes the Maslov index of the projection of $\unicode[STIX]{x1D6FE}$ into $C$ .

Proof. We need to compare the trivializations $v$ in (5.1) and $v_{0}$ in (5.3) of $\det _{\mathbb{C}}(TC)$ along a given Reeb orbit $\unicode[STIX]{x1D6FE}$ . Let $\ell _{v}$ and $\ell _{v_{0}}$ denote the fields of real lines in $\det _{\mathbb{C}}(TC)$ spanned by $v$ and  $v_{0}$ , respectively. To simplify the comparison we use the fact that Conley–Zehnder and Maslov indices are invariant under homotopies and replace $C$ by its image $\unicode[STIX]{x1D70C}_{s}(C)$ under fiber scaling of  $\mathbb{R}^{2n}$ :

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{s}(x_{1},y_{1},\ldots ,x_{n},y_{n})=(s^{-1}x_{1},sy_{1},\ldots ,s^{-1}x_{n},sy_{n}),\quad s>0.\end{eqnarray}$$

Note that $\unicode[STIX]{x1D719}_{s}^{\ast }\unicode[STIX]{x1D714}_{0}=\unicode[STIX]{x1D714}_{0}$ , and that, since $\unicode[STIX]{x1D6FE}$ lies in a compact subset of $C$ , there is a neighborhood $U$ of $\unicode[STIX]{x1D6FE}$ in $C$ such that $\unicode[STIX]{x1D70C}_{s}(U)$ lies in an arbitrarily small neighborhood of the subspace $y_{j}=0$ , $j=1,\ldots ,n$ , provided $s>0$ is sufficiently small.

Let $\text{Gr}_{\text{Lag}}(\mathbb{R}^{2n})$ denote the Lagrangian Grassmannian of $\mathbb{R}^{2n}$ . Consider the Lagrangian subspaces $\unicode[STIX]{x1D6F1}_{y}=\{x_{j}=0,~j=1,\ldots ,n\}$ and $\unicode[STIX]{x1D6F1}_{x}=\{y_{j}=0,~j=1,\ldots ,n\}$ of $\mathbb{R}^{2n}$ , and let $Z\subset \text{Gr}_{\text{Lag}}(\mathbb{R}^{2n})$ denote the Maslov cycle of subspaces that intersect $\unicode[STIX]{x1D6F1}_{y}$ in a subspace of dimension ${>}0$ . After small deformation of $P$ , we may assume that the path of Lagrangian tangent planes $TC|_{\unicode[STIX]{x1D6FE}}$ along $\unicode[STIX]{x1D6FE}$ does not intersect $Z$ in $\unicode[STIX]{x1D6FE}\cap P$ . Assume now that the scaling parameter $s>0$ is sufficiently small so that $\unicode[STIX]{x1D719}_{s}(TC)$ is approximately equal to the subspace $\unicode[STIX]{x1D6F1}_{x}$ outside the caustic $\unicode[STIX]{x1D6F4}$ of $\unicode[STIX]{x1D719}_{s}(U)$ (the locus of tangencies with the subspace $\unicode[STIX]{x1D6F1}_{y}$ ). Then each transverse intersection of $TC|_{\unicode[STIX]{x1D6FE}}$ with $Z$ of sign $\unicode[STIX]{x1D716}=\pm 1$ corresponds to a transverse passage of $\ell _{v}$ through $\ell _{v_{0}}$ of sign  $\unicode[STIX]{x1D716}$ . To see this, note that such an intersection corresponds to a point where the curve $\unicode[STIX]{x1D6FE}$ crosses the caustic $\unicode[STIX]{x1D6F4}$ transversely and the tangent space of $TC$ along $\unicode[STIX]{x1D6FE}$ is approximately constant in the  $(n-1)$ directions tangent to $\unicode[STIX]{x1D6F4}$ and makes a $\unicode[STIX]{x1D70B}$ -rotation in the complex line normal to $\unicode[STIX]{x1D6F4}$ .

Furthermore, the line $\ell _{v^{\prime \prime }}$ , see (5.1), is approximately constant with respect to $\ell _{v_{0}}$ in $N(P)\cap \unicode[STIX]{x1D6FE}$ , since $\unicode[STIX]{x1D6FE}\cap P\cap Z=\varnothing$ , and hence $\ell _{v}$ makes a $\pm \unicode[STIX]{x1D70B}$ -rotation with respect to $\ell _{v_{0}}$ for each intersection $P\cap \unicode[STIX]{x1D6FE}$ . Since $P$ is dual to $W_{1}$ , adding these two contributions we find that

$$\begin{eqnarray}\operatorname{CZ}(\unicode[STIX]{x1D6FE})-\operatorname{CZ}^{w_{1}}(\unicode[STIX]{x1D6FE})=W_{1}(\unicode[STIX]{x1D6FE})-\unicode[STIX]{x1D707}_{C}(\unicode[STIX]{x1D6FE}),\end{eqnarray}$$

as claimed.◻

5.3 Dimension formulae and neck-stretching

Let $C\subset \mathbb{R}_{\text{st}}^{2n}$ be a Lagrangian submanifold with Legendrian boundary $\unicode[STIX]{x1D6E4}$ and let $\unicode[STIX]{x1D719}(S^{n})\subset \mathbb{R}_{\text{st}}^{2n}$ be a Lagrangian sphere with exactly one double point $a$ of grading $|a|=n$ . Since $\unicode[STIX]{x1D719}(S^{n})$ is compact, by choosing a sufficiently large ball, we consider $\unicode[STIX]{x1D719}$ as a Lagrangian immersion into $\mathbb{R}_{C}^{2n}$ , see §5.1. The moduli space ${\mathcal{M}}$ of holomorphic disks in $\mathbb{R}_{C}^{2n}$ with boundary on $\unicode[STIX]{x1D719}(S^{n})$ and one positive puncture at $a$ is a closed manifold of dimension

$$\begin{eqnarray}\dim ({\mathcal{M}})=|a|-1=n-1.\end{eqnarray}$$

Below we will consider the behavior of holomorphic disks in this moduli space under neck-stretching around $C$ . More precisely, we will take limits of curves as we degenerate the almost complex structure near $C$ in such a way that, in the limit, the ambient $\mathbb{R}_{C}^{2n}$ falls apart in two pieces. The upper piece $X^{+}$ has one positive end as in $\mathbb{R}_{C}^{2n}$ and one negative end, the negative half of the symplectization of the unit conormal bundle $U^{\ast }C$ . The lower piece $X^{-}=T^{\ast }C$ . SFT-compactness [Reference Bourgeois, Ekholm and EliashbergBEE12] describes the limits of holomorphic curves in $\mathbb{R}_{C}^{2n}$ in terms of holomorphic buildings with at least one level in $X^{+}$ and perhaps some levels in $X^{-}$ , where the levels are asymptotic to Reeb orbits in $U^{\ast }C$ . (The compactness theorem in [Reference Bourgeois, Ekholm and EliashbergBEE12] also gives additional levels in the symplectization $\mathbb{R}\times U^{\ast }C$ . Here we will simply consider such levels as curves in  $X^{-}$ , which means that the levels in $X^{-}$ in general consist of broken curves. Since the relevant indices and virtual dimensions are additive, this elision will not lose track of any information that we need.)

Since the pieces of the limiting holomorphic building glue to a disk, each piece is a disk or sphere with punctures. We will thus consider dimensions of the following curves:

1. – spheres in $T^{\ast }C$ with positive punctures at Reeb orbits in $U^{\ast }C$ ;

2. – spheres in $X^{+}$ with negative punctures at Reeb orbits in $U^{\ast }C$ ;

3. – disks in $X^{+}$ with a positive boundary puncture at $a$ and negative interior punctures at Reeb orbits in $U^{\ast }C$ .

We consider first a sphere $v$ in $T^{\ast }C$ with positive punctures at Reeb orbits $\unicode[STIX]{x1D6FE}_{1},\ldots ,\unicode[STIX]{x1D6FE}_{k}$ of indices $\operatorname{CZ}^{w_{1}}(\unicode[STIX]{x1D6FE}_{1})=i_{1},\ldots ,\operatorname{CZ}^{w_{1}}(\unicode[STIX]{x1D6FE}_{k})=i_{k}$ . We have [Reference Eliashberg, Givental and HoferEGH00, §1.7]

(5.4) $$\begin{eqnarray}\dim (v)=2(n-3)+\mathop{\sum }_{j=1}^{k}(i_{j}-(n-3)).\end{eqnarray}$$

Consider next the case of a sphere $w$ in $X^{+}$ with no positive puncture and with negative punctures at Reeb orbits $\unicode[STIX]{x1D6FD}_{1},\ldots ,\unicode[STIX]{x1D6FD}_{r}$ . We have

(5.5) $$\begin{eqnarray}\displaystyle \dim (w) & = & \displaystyle 2(n-3)-\mathop{\sum }_{j}(\operatorname{CZ}(\unicode[STIX]{x1D6FD}_{j})+(n-3))\nonumber\\ \displaystyle & = & \displaystyle 2(n-3)-\mathop{\sum }_{j}(\unicode[STIX]{x1D704}(\unicode[STIX]{x1D6FD}_{j})-\unicode[STIX]{x1D707}_{L}(\unicode[STIX]{x1D6FD}_{j})+(n-3)).\end{eqnarray}$$

Similarly, if $u$ in $X^{+}$ is the component with positive puncture at $a$ and negative punctures at $\unicode[STIX]{x1D6FD}_{1},\ldots ,\unicode[STIX]{x1D6FD}_{r}$ then

(5.6) $$\begin{eqnarray}\displaystyle \dim (u) & = & \displaystyle (n-1)-\mathop{\sum }_{j}(\operatorname{CZ}(\unicode[STIX]{x1D6FD}_{j})+(n-3))\nonumber\\ \displaystyle & = & \displaystyle (n-1)-\mathop{\sum }_{j}(\unicode[STIX]{x1D704}(\unicode[STIX]{x1D6FD}_{j})-\unicode[STIX]{x1D707}_{L}(\unicode[STIX]{x1D6FD}_{j})+(n-3)).\end{eqnarray}$$

Any punctured holomorphic curve has an underlying somewhere injective curve; for the latter result in the punctured setting, see e.g. [Reference Dimitroglou-Rizell and EvansDE14]. Standard transversality arguments, perturbing the almost complex structure near an injective point, imply that moduli spaces of somewhere injective holomorphic curves are cut out transversely for generic data. In the proof of Lemma 5.6 below, we use such transversality arguments only for curves where monotonicity of the Lagrangian implies that any multiple cover has higher virtual dimension than that of the underlying simple curve.

Let $\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D70B}_{2}(\mathbb{R}^{2n},\unicode[STIX]{x1D719}(S^{n}))$ denote the generator of positive area. As usual, fix an almost complex structure $J$ on $\mathbb{C}^{n}$ standard in a neighborhood of $\unicode[STIX]{x1D719}(S^{n})$ .

Proof. The dimension of any curve in the limit is non-negative by transversality, and those dimensions sum to $n-1$ , the dimension of the space of disks in class $\unicode[STIX]{x1D6FD}$ , by additivity of dimension in holomorphic buildings. If no component of the limit building lies in $T^{\ast }C$ the conclusion is immediate, so assume some component is contained in $T^{\ast }C$ .

The dimension of any sphere with only one positive puncture in $T^{\ast }C$ is

$$\begin{eqnarray}(n-3)+i>(n-1),\end{eqnarray}$$

where $i$ is the Morse index of the contractible geodesic corresponding to the Reeb orbit at the positive puncture. There must therefore be a sphere with several positive punctures $\unicode[STIX]{x1D6FE}_{0},\ldots ,\unicode[STIX]{x1D6FE}_{m}$ , $m\geqslant 1$ , inside $T^{\ast }C$ ; the dimension $d_{\text{in}}$ of such a sphere is

$$\begin{eqnarray}d_{\text{in}}=2(n-3)+\mathop{\sum }_{j=0}^{m}(i_{j}-(n-3)).\end{eqnarray}$$

At each $\unicode[STIX]{x1D6FE}_{j}$ there is (a possibly broken) outside sphere with a negative puncture at $\unicode[STIX]{x1D6FE}_{j}$ of dimension $d_{\text{out}}^{j}$ given by

$$\begin{eqnarray}d_{\text{out}}^{j}=(n-3)-i_{j}+m_{j},\end{eqnarray}$$

where $m_{j}$ is the Maslov index of the geodesic corresponding to $\unicode[STIX]{x1D6FE}_{j}$ . Gluing the half-cylinder on this geodesic with boundary on the zero-section $C$ to the punctured sphere in $\mathbb{R}_{C}^{2n}-C$ , we obtain a holomorphic disc with boundary on $C$ . Monotonicity and the condition on minimal Maslov number then implies that $m_{j}\geqslant 3$ . Assume now that the dimension $d_{\text{in}}\geqslant n-2$ . Then $0\leqslant d_{\text{out}}^{j}\leqslant 1$ , with right equality for at most one $j$ . In particular,

$$\begin{eqnarray}i_{j}-(n-3)\geqslant 2\quad \text{with}~i_{j}-(n-3)\geqslant 3~\text{except for one }j.\end{eqnarray}$$

Thus

$$\begin{eqnarray}d_{\text{in}}\geqslant (n-3)+i_{0}+\mathop{\sum }_{j>0}(i_{j}-(n-3))\geqslant n,\end{eqnarray}$$

which contradicts $d_{\text{in}}\leqslant n-1$ . We conclude that $d_{\text{in}}<n-2$ , and hence the subset swept by evaluation of holomorphic curves in all moduli spaces of building components inside $T^{\ast }C$ has total dimension ${<}n$ . For generic data, this will then be disjoint from the zero-section $C\subset T^{\ast }C$ .◻

5.4 Proof of Theorem 1.3

Fix $C\subset \mathbb{R}_{\text{st}}^{2n}$ as in the formulation, adjust the Liouville structure near the ideal boundary of $C$ if necessary, and let $X=\mathbb{R}_{C}^{2n}\backslash C$ . From Lemma A.7, and its analogue in the case when $C$ has boundary as discussed in §A.2, we know that $\unicode[STIX]{x1D70B}_{n}(X)$ is determined as follows: