Notes on Exact Lagrangian Immersions with a Single Double Point by Ekholm-Smith

This is a third in a series about direct uses of moduli spaces, in this case, of pseudo-holomorphic curves. The paper in the title is by Tobias Ekholm and Ivan Smith, posted in 2011 and published 2016. We’ll go over some of the main arguments for their theorem but first, I want to outline some classical results.

Suppose we have an exact symplectic manifold $(M,\omega = d\lambda)$; i.e. the symplectic form is an exact form. Then necessarily, $M$ is an open manifold. A Lagrangian submanifold $L\subset M$ is such that the restriction of the symplectic form to $L$ is trivial. We say that $L$ is exact if restricting the primitive $\lambda$ to $L$ gives an exact 1-form; i.e. $\lambda|_L = df$ for some function $f:L\to \mathbb{R}$. Now, there is a standard exact symplectic form on $\mathbb{R}^{2n} \cong \mathbb{C}^n$ given by $\omega = \sum^n_{i=1} dx_i\wedge dy_i$ which is exact. One choice of primitive is simply $\lambda = \sum^n_{i=1}x_i dy_i$.

Theorem (Gromov, 1985): Let $L$ be any closed embedded Lagrangian submanifold in the standard symplectic $\mathbb{C}^n$. Then, there exists a non-constant holomorphic map $u:(D^2,S^1) \to (\mathbb{C}^n,L)$.

Corollary: There are no exact Lagrangian submanifolds in $\mathbb{C}^n$. In particular, if $L$ is any Lagrangian in $\mathbb{C}^n$, then $H^1(L,\mathbb{R}) \neq 0$.

Proof: If there were an exact Lagrangian submanifold $L$, then we can compute the area/energy of the non-constant holomorphic disk: $\int_{D^2}u^*d\lambda = \int_{S^1} \lambda$. But the boundary $S^1$ is on $L$ and exactness means $\lambda|_L = df$, hence $\int_{S^1}\lambda = \int_{S^1}df = 0$ by Stokes theorem. But non-constant holomorphic disks have nonzero energy, a contradiction. We then also see that if $H^1(L,\mathbb{Q})=0$, then $\lambda|_L$ would be exact. So Lagrangians in $\mathbb{C}^n$ are subject to some topological constraints. $\square$

Lagrangian Immersions

On the other hand, Lagrangian immersions are a whole other class of objects. For example, every closed, orientable 3-manifold can be exact Lagrangian immersed into $\mathbb{C}^3$ with only one double point even though there are plenty of 3-manifolds which are rational (or even integral) homology 3-spheres. This is a result of a paper by Ekholm-Eliashberg-Murphy-Smith which came out after the Ekholm-Smith paper. In that paper, they have an h-principle for exact Lagrangian immersions.

Gromov had already studied Lagrangian immersions (without imposing the exactness condition). We can ask how the space of Lagrangian immersions sits in the space of all smooth immersions but this is not quite the right question. Note that for a given $f:L \to \mathbb{C}^n$, there are bundle maps of the form $F:TL \to T\mathbb{C}^n$ that sends tangent spaces to linear Lagrangian spaces in $\mathbb{C}^n$ and moreover, $F$ sits over $f$; i.e. $T_xL$ is sent to $T_{f(x)}\mathbb{C}^n$. Such maps are called Gauss maps and they land in a Grassmannian. One example is $df$ but $F$ doesn’t have to be $df$. So anyways, we can ask, “How does the space of Lagrangian immersions sit in the space of smooth immersions that have abstract Lagrangian Gauss maps?

Theorem (Gromov-Lees): There is an h-principal which tells us that the space of smooth immersions with abstract Lagrangian Gauss map deformation retract onto the space of Lagrangian immersions. Moreover, an $n$-manifold $L$ admits a Lagrangian immersion into $\mathbb{C}^n$ if and only if $TL \otimes \mathbb{C}$ is a trivial complex vector bundle.

Example: Any stably parallelizable manifold can be immersed into $\mathbb{C}^n$. Spheres are stably parallelizable and there is an explicit example: the Whitney sphere which is an exact Lagrangian immersion $w:S^n \to \mathbb{C}^n$. We think of $S^n$ as defined by $|x|^2 +y^2$ for $(x,y)\in \mathbb{R}^n \times \mathbb{R}$. Then $w(x,y) = (1+iy)x \in \mathbb{C}^n$. Note that $w(0,1)=w(0,-1)$ gives the only double point.

Note: These double points can be cancelled in pairs using the Whitney trick and we’ll still be left with a Lagrangian submanifold though it may no longer be exact. So when there’s an even number of double points, then we can obtain an embedding. Also, when two Lagrangians $L_1,L_2$ intersect transversally at a single point $p$, there is a procedure called Polterovich surgery which creates a submanifold $L_1 \#_pL_2$ that is topologically the connected sum of the two at $p$ and agrees with $L_1 \cup L_2$ outside of a neighborhood of $p$. There’s a standard local model for transverse intersections of Lagrangians and the surgery can be viewed as taking place locally, relying on certain choices such as a curve $\gamma$ satisfying some properties. So the result is not a unique Lagrangian but if we scale $s\gamma$, the family of Lagrangians are all Hamiltonian isotopic. The order matters because $L_2 \#_p L_1$ is rarely Lagrangian isotopic to $L_1 \#_p L_2$. But we can also perform the surgery on the transverse double point of an immersed Lagrangian and topologically, what we get is $L \#(S^1 \times S^{n-1})$.

Now, while there are a lot of Lagrangian immersions in symplectic geometry, often with infinitely many nonisotopic classes, interestingly, when we impose the exactness condition, it dramatically constrains the topology. One demonstration of this is the following:

Theorem (Ekholm-Smith, 2011): Let $K$ be a closed, orientable manifold of dimension $2n$ where $n>2$ and suppose the Euler characteristic $\chi(K) \neq -2$. If $K$ admits an exact Lagrangian immersion $K \to \mathbb{C}^{2n}$ with exactly one transverse double point and no other self intersections, then $K$ is diffeomorphic to the standard sphere $S^{2n}$.

Sketch of Proof

Before sketching the proof, a reminder about the definition of monotone Lagrangian $L$. For a holomorphic disk with boundary on $L$, we can find its symplectic area. It also has a Maslov index. If these two numbers are always proportional for all holomorphic disks by a fixed constant $c>0$, then we say that $L$ is monotone. Note that if $L$ is exact, the symplectic area of the disk is 0 and so then $c=0$ in that case. Interestingly, Abouzaid-Kragh proved that closed, exact Lagrangians in cotangent bundles always have vanishing Maslov class so they would also be monotone. This vanishing does not occur in more general Weinstein domains.

The rest of this note is primarily dedicated to sketching the proof of this result. In broad strokes:

1. Performing Lagrangian surgery on $K$ at the double point, we obtain a $K\# (S^1 \times S^{2n-1})$ which is a monotone embedded Lagrangian submanifold of minimal Maslov number $2n$. There are actually two resulting Lagrangians from some surgery choices; we’ll denote the Lagrangian of interest as $K_+$. The heart of the proof is to construct a parallelizable manifold that has $K_+ \cong K\#(S^1 \times S^{2n-1})$ as boundary. The bounding manifold is first constructed as a cobordism which is a moduli space of certain $J$-holomorphic disks. The cobordism is then capped off on one end by using fibered products of moduli spaces and abstract chains. The construction and study of these moduli spaces is intricate so I will save the details for below.
2. We can attach a 2-handle to $S^1 \times S^{2n-1}$ in order to kill the generator of $H^1$ and this also kills the Poincaré duel in $H^{2n-1}$. The result is simply $S^{2n}$. Surgery steps (handle attachments) produce a cobordism so we can surger $K_+ \cong K\#(S^1 \times S^{2n-1})$ which is equivalent to attaching a cobordism to the parallelizable manifold where one end is just $K$ itself. This cobordism is also parallelizable.
3. Next, show that $K$ is an $\mathbb{Z}$-homology sphere. To do this, they study the Legendrian lift of the immersion $f:K \to \mathbb{C}^{2n}$, denote it as $\tilde{f}:K \to \mathbb{C}^{2n}\times \mathbb{R}$ with contact form $d\lambda - \theta$. Since there was only one tranverse double point for $f$, then $\tilde{f}(K)$ has exactly one Reeb chord, call it $a$.
4. The Chekanov-Eliashberg differential graded algebra $\mathcal{A}$ is generated by the Reeb chordw of the lift which, in this case, is just $a$. Since the Euler characteristic of $K$ is assumed to not be $-2$, the grading (Maslov index) of the Reeb chord must be even. On the other hand, the differential $\partial$ is a degree 1 map. Since all the generators of the DGA are even degree, the differential is trivial which guarantees that the DGA is linearizable because it admits an augmentation. Recall that an augmentation $\epsilon:\mathcal{A} \to \mathbb{F}$ is a chain map to the ground field where the ground field is viewed as a trivial chain. In particular, $\epsilon \circ \partial =0$ and $\epsilon(1)=1$.
5. The DGA is a powerful invariant but also infinite dimensional. The augmentation gives a way to produce linearized Legendrian contact homology, a finite dim invariant. There are inequalities for this homology theory, proved by Ekholm-Etyner-Sabloff (see Theorem 1.2 in their paper). In our setting, let $c_m$ be the number of Reeb chords of degree $m$ and $b_m$ the $m$th Betti number of $\tilde{f}(K)$ over $\mathbb{F}$. Then $b_m \leq c_m + c_{n-m}$. If we sum up over all $0 \leq m \leq n$, then $\dim H^*(\tilde{f}(K),\mathbb{F}) \leq 2 \cdot \text{total number of Reeb chords}$. For us, since there is only one Reeb chord and closed manifolds have $\dim H^* \geq 2$, we have that $\dim H^*(\tilde{f}(K),\mathbb{F})=2$ for every field: all the middle cohomology groups vanish over every field. This means there is no torsion or free part for these middle parts and hence, $\dim H^*(\tilde{f}(K),\mathbb{Z})=2$ as well, hence $K$ is a $\mathbb{Z}$-homology sphere.
6. To show that $K$ is a homotopy sphere, they use a result of Damian that uses lifted Floer homology for monotone Lagrangian embeddings and apply it to $K_+$. It is essentially the Floer homology of the embedding equipped with a local system defined by pushing forward the trivial local system from the universal cover and he used it to derive constraints on monotone Lagrangians in $\mathbb{C}^k$ for large Maslov number. In the setting here, when $n > 2$, he showed $K_+$ is a smooth manifold that fibers over a circle with fiber a homotopy $(2n-1)$-sphere. Since $K_+$ is obtained by attaching a 1-handle to $K$, surgery b adding a 2-handle cancels the original 1-handle. And since $K_+$ is a fibration over the circle with fiber being a homotopy sphere, the cancellation shows that $K$ is just a homotopy sphere. This means $K_+ \cong \Sigma \# (S^1 \times S^{2n-1})$ for some homotopy sphere $K=\Sigma$.
7. By Kervaire-Milnor, the only homotopy $2n$-sphere for $n>2$ to bound a parallelizable manifold is the standard sphere.

Constructing the Moduli Space

Similar to Gromov and Abouzaid’s work, Ekholm-Smith also study a 1-parameter family of FLoer equations with Lagrangian boundary condition $L=K\#(S^1 \times S^{2n-1})$. The moduli space $\mathcal{F}(0\beta)$ of Floer holomorphic disks in the trivial relative homotopy class is used to build the bounding manifold mentioned above with stably trivial tangent bundle.

This $\mathcal{F}(0\beta)$ is noncompact because of bubbling but can be compactified using broken curves. In their situation, there is only one possible bubbling configuration which forms the boundary they attach. The bubbling is complicated by having automorphisms so they use ambient hypersurfaces to stabilize. The bubble configurations are a fibered product of two other moduli spaces $\mathcal{N} = \mathcal{F}^*(-\beta) \times_L \mathcal{M}^*(\beta)$. Through some gluing analysis, they then piece together the boundary and create the bounding manifold $\mathcal{B}$ which comes from capping off $\mathcal{F}(0\beta)$. The cap is also a fiber product.

In order to understand the stable tangent bundle of $\mathcal{B}$, they study the index bundles and appropriate Fredholm problem. Interestingly, it’s helpful to view $L$ as a piecewise linear (PL) embedding rather than a smooth one for this analysis. They then find that if the index bundle of an associated manifold $W’$ is stably trivial, so is that of $L$. This comes from finding an isotopy which is covered by a homotopy of (stable) Lagrangian Gauss maps into the Lagrangian Grassmannian.

The question of extending the stable trivialization over the cobordism $\mathcal{F}(0\beta)$ and the cap is an interesting one. They study the energy functional of the standard metric on $S^{2n-1}$ and find a $(6n-7)$-skeleton of the free loop space over which there is an explicit trivialization for the three spaces: $\mathcal{F}(0 \beta), \, \mathcal{F}^*(-\beta),\, \mathcal{M}^*(\beta)$. The problem of extending the stable trivialization onto the cap is reduced to a problem about spin structures which they show is unobstructed and hence, the whole bounding manifold is stably parallelizable.

Unlike Abouzaid’s work in his framed bordism paper and Lagrangian embeddings of exotic spheres where he uses CW approximations, Ekholm-Smith find $C^1$-structures on the moduli spaces that they work with.

Corollaries

As part of their proof, they worked with the stable Lagrangian Gauss map. If that map is trivial, we get some interesting results. For example, if the dimension is divisible by 8, the stable Lagrangian Gauss map is trivial.

Corollary: If $n\geq1$, then a monotone Lagrangian submanifold $L \subset \mathbb{C}^{8n}$ of minimal Maslov number $8n$ bounds a parallelizable manifold. Combined with results of Damian, the previous statement also implies that $L$ would be a product of standard spheres or an exotic sphere bundle.

Also, let $P = S^1 \times S^{8n-1}$ with $n \geq 1$. It’s a fact that $P \# \Sigma$ is diffeomorphic to $P$ if and only if $\Sigma$ is diffeomorphic to standard $S^{8n}$.

Corollary: If $n \geq 1$ and $\Sigma$ is a homotopy $8n$-sphere, then the cotangent bundles $T^*P$ and $T^*(P\# \Sigma)$ are symplectomorphic if and only if $\Sigma \cong S^{8n}$ is the standard sphere.

Remark: It is known that the two cotangent bundles mentioned are diffeomorphic regardless of which homotopy $8n$-sphere $\Sigma$ is. So the corollary is telling us that the symplectic geometry of the cotangent bundles does know something about the smooth structures of the base manifolds. For example, the group of homotopy spheres in dimension 8 and 16 both have order 2.