Note on Framed Bordism and Lagrangian Embeddings of Exotic Spheres

These are some notes on the 2011 paper mentioned in the title by Mohammed Abouzaid. Recall that every smooth manifold $M$ embeds as an exact Lagrangian submanifold into its own cotangent bundle $T^*M$; just use the zero section. However, an interesting question we may ask is whether a homotopy sphere $\Sigma^n$ embeds as a Lagrangian (may be non-exact) into the cotangent bundle for the standard sphere $T^* S^n$. By the Whitney trick, there is no obstruction to a smooth embedding of exotic $n$-spheres into $T^*S^n$.

Main Theorem: Every homotopy sphere $\Sigma^{4k+1}$ which embeds as a Lagrangian in $T^*S^{4k+1}$ must bound a compact parallelizable manifold.

The work of Smale in dimensions 5 and higher shows that every homotopy sphere is in fact, homeomorphic to the standard sphere. With the work of Freedman and Perelman for dimensions 4 and 3, respectively, we now know that every homotopy $n$-sphere is homeomorphic to the standard $S^n$. On the other hand, with knowledge of Kervaire-Milnor’s work on exotic spheres which bound parallelizable manifolds, we can say something more. Recall that in their paper “On Groups of Homotopy Spheres,” they showed that homotopy spheres form an abelian group under connected sum in dimensions other than 4. Moreover, there is a cyclic subgroup where the elements are homotopy spheres that bound parallelizable manifolds.

Therefore, if $\Sigma$ does not bound a parallelizable manifold, then $\Sigma$ does not embed as a Lagrangian into $T^*S^{4k+1}$ but does embed as a Lagrangian into $T^* \Sigma$. This means that the symplectic topology of $T^*\Sigma$ and $T^* S^{4k+1}$ are different. On the other hand, consider the unit disk bundle $D(T^* \Sigma)$; it has boundary which is some $S^{n-1}$ bundle over $\Sigma$; this is usually written as $S(T^*\Sigma)$. If we cut out a small disk $D^{2n}$ in $D(T^*\Sigma)$ and also a small disk $D^{2n}$ in $D(T^* S^n)$, then we can glue these together and we have a cobordism $W$ from $S(T^* \Sigma)$ to $S(T^*S^n)$. If this cobordism is trivial $W \cong S(T^*\Sigma) \times [0,1]$, we’d be able to conclude that that $D(T^* \Sigma) \cong D(T^*S^n)$ are diffeomorphic as manifolds with boundary. I don’t know if the last step is correct but the stated conclusion is correct: the two cotangent bundles are diffeomorphic as smooth manifolds.

Corollary of the main theorem: If $\Sigma$ does not bound a parallelizable manifold, then it cannot be Lagrangian embedded into $T^* S^{4k+1}$ and hence, $\Sigma$ is not diffeomorphic to the standard sphere (since the standard sphere Lagrangian embeds just by the zero section). Hence, the symplectic topology of $T^*S^{4k+1}$ is able to detect the exotic smooth structure on $\Sigma$ while the smooth topology of $T^*S^{4k+1}$ is not able to.

Sketch of Proof

Consider the Hopf fibration $h:S^{2n-1} \to \mathbb{CP}^{n-1}$ which sends $(z_1,…,z_n) \mapsto [z_1:…:z_n]$ (later, we substitute $n=2k+1$ so that $2n-1 = 4k+1$). Then, the graph of $h$ denoted $\Gamma(h)$ can be viewed as a submanifold of $\mathbb{C}^n \times \mathbb{CP}^{n-1}$ which has symplectic form $\omega:=p_1^* \omega_0 - p_2^* \omega_{FS}$. The graph of $h$ is exactly half the dimension and on charts, the standard symplectic form on $\mathbb{C}^n$ and the Fubini-Study form on $\mathbb{CP}^{n-1}$ are really quite similar. Since the Hopf map is nearly tautological, $p^*_1 \omega_0$ and $p^*_2 \omega_{FS}$ coincide on $\Gamma(h)$. In other words, $\Gamma(h)$ is a Lagrangian sphere. By the Weinstein neighborhood theorem, there is a standard neighborhood of $\Gamma(h)$ in $\mathbb{C}^n \times \mathbb{CP}^{n-1}$ which is symplectomorphic to a neighborhood of the zero section of $T^*S^{2n-1}$ and $\Gamma(h)$ maps to the zero section. Hence, if we have a homeomorphism between a homotopy sphere and the standard sphere, $f:\Sigma \to S^{2n-1}$ and we have a Lagrangian embedding of $\Sigma$ into $T^*S^{2n-1}$, then we may study $\Sigma$ in this Hopf fibration way. We may need the homeomorphism to be smooth (though it needn’t have smooth inverse).

The $\mathbb{C}^n$ factor is noncompact and so we can find a Hamiltonian diffeomorphism with compact support which displaces $\Gamma(h)$; i.e. we can move $\Gamma(h)$ by a Hamiltonian diffeomorphism $\psi$ such that $\psi(\Gamma(h)) \cap \Gamma(h) = \varnothing$. Let’s let $M:=\mathbb{C}^n \times \mathbb{CP}^{n-1}$ and $L = \Gamma(h)$.

With this in hand, Abouzaid begins a construction of a parallelizable manifold $W(L)$ with $\partial W(L) = L$. Part of the construction is to realize that a particular (parametrized) moduli space of curves has $L$ as one of its boundary components. That manifold is not compact but there are ways to compactify it and fill in the other boundary components. Let’s see this in more detail.

In Floer theory (and historically, well-known to Gromov), when one introduces a Hamiltonian to perturb the Cauchy-Riemann equation. Thus, a Hamiltonian isotopy gives us a 1-parameter family of deformed Cauchy-Riemann equations. Letting the parameter be in $[0,\infty)$, we may first study the moduli space at $t = 0$; this is the usual undeformed Cauchy-Riemann equation.

A map $u:(D^2,S^1) \to (M,L)$ gives a homotopy class in $\pi_2(M,L)$; let’s study those in the constant homotopy class. This means $u$ can be deformed to map entirely into $L$ and since disks are contractible, it can be deformed to a constant map on $L$. This means that the moduli space $\mathcal{M}_0(L,0,J)$ of $J$-holomorphic disks with boundary on $L$ of constant homotopy class is diffeomorphic to $L$ itself. Now, letting $t$ vary, we get a manifold $P$ of dimension $2n$ with $L$ as boundary. Of course, we must find some generic path $J_t$ of almost complex structures so that $P$ is a smooth manifold.

This moduli space $P$ is not compact but admits a compactification which corresponds geometrically to considering exceptional solutions to the parametrized deformation of the Cauchy-Riemann equation. These exceptional solutions are holomorphic disc or sphere bubbles. The codim-1 strata corresponding to disc bubbles can be “capped off” by a different moduli space of holomorphic discs whose boundary is this codim-1 strata. This relies on $\Sigma$ being simply connected. The sphere bubbles form a codim-2 strata and can also be dealt with since, recall, the sphere bubbles live in $\mathbb{CP}^{n-1}$. One of the boundary components of this new manifold is diffeomorphic to $L$.

The remaining boundary components correspond to sphere bubbles and have explicit descriptions as bundles over $S^2$. Apparently they are $S^1$ bundles over $\mathbb{CP}^{n-2}$ bundles over $S^2$ (so their dimension is $2n-1$). Abouzaid is able to show they are just $S^{2n-3}$ bundles over $S^2$ with structure group $SO(2n-2)$. These bundles are classified by homotopy classes of maps $S^2 \to BSO(2n-2)$ which is $\pi_1(SO(2n-2))=\mathbb{Z}/2$. Abouzaid showed that if $n=2k+1$ is odd, then we have the trivial bundle $S^2 \times S^{2n-3}$ and if $n$ is even, we have the nontrivial one.

Gluing these pieces into $P$, we get a manifold $\widehat{W}(L)$ which has $L$ and disjoint union of these sphere bundles as boundary. Whenever $n$ is odd, Abouzaid showed that these sphere bundle boundary components of $\widehat{W}(L)$ are stably parallelizable. This is why the main theorem is stated in terms of dimension $4k+1$. To show this stable parallelizability, first, we know that $T(S^2 \times S^{2n-3}) \oplus \underline{\mathbb{R}}^2$ is trivial because of the spheres. Then, he shows that these sphere bundle boundary components are the only obstructions to $\widehat{W}(L)$ being stably parallelizable. But also, picking the component of $\widehat{W}(L)$ which contains the constant disks, we now have a stably parallelizable cobordism from $L$ to some disjoint union of $S^2 \times S^{2n-3}$. Since this cobordism is not closed, there is no top degree cell and the cobordism can be retracted onto a CW skeleton of degree one less than top degree. This degree is below the dimension where stable and unstable orthogonal groups differ so the obstructions to genuine parallelizability vanish.

All that to say, the cobordism is parallelizable. Abouzaid then shows that up to connected sum with a framed standard sphere, every stable framing on $S^2 \times S^{2n-3}$ arise as induced framing on the boundary of a parallelizable $2n$-manifold. So if we just pick such a filling, we then obtain a parallelizable manifold $W(L)$ with $L$ as boundary. This proves the main theorem.

Remark 1: I’ve not been entirely honest. As I outlined how he had to work with all these strata, it can seem incredibly difficult to do the sculpting work to make everything fit together in a coherent smooth way. And indeed, it is very difficult so Abouzaid actually worked with a CW approximation which had all the homotopy theoretic properties needed to make the conclusions; e.g. that it’s parallelizable. The analytic work Abouzaid did to construct the moduli spaces and prove their properties is spectacular.

Remark 2: Dimension 9 is the first time the theorem applies. Kervaire-Milnor showed there are 8 smooth structures on $S^9$ and 2 of them bound parallelizable manifolds. So in particular, 6 of the exotic 9-spheres do not admit Lagrangian embeddings into $T^*S^9$.

Results of Ekholm-Kragh-Smith

There are related results by Ekholm-Kragh-Smith. The first is the following.

Theorem: Let $n=2k-1 > 4$ be an odd integer, $\Theta_{2k-1}$ the group of oriented homotopy $(2k-1)$-spheres under connect sum and $bP_{2k}$ the subgroup of those that bound parallelizable $2k$-manifolds. If the cotangent bundles $T^*\Sigma$ and $T^*\Sigma’$ are symplectomorphic, then $[\Sigma] = \pm[\Sigma’] \in \Theta_{2k-1}/bP_{2k}$.

This extends Abouzaid’s result in the case where the dimension is $4k+1$. If $T^* \Sigma \cong T^* S^{4k+1}$ as symplectic manifolds, then $\Sigma \in bP_{4k+2}$. The converse of this statement is Abouzaid’s result.

Example: Since $\Theta_7 = bP_8$, we cannot say whether Milnor’s exotic 7-spheres have symplectomorphic cotangent bundles from this theorem. For dimensions 15 and 19, there are 16,256 and 523,264 exotic spheres, respectively. In both cases, exactly half bound parallelizable manifolds so the quotient groups are both $\mathbb{Z}/2$. This means that the cotangent bundles fall into at least two symplectomorphism classes. In dimension 13, the quotient group is $\mathbb{Z}/3$ and since $2 \equiv -1 \pmod{3}$, there are also at least two symplectomorphism classes.

Theorem: Let $n = 4k -1$ and $\Sigma \in \Theta_{4k-1}$ be an oriented homotopy sphere. If $\mathbb{RP}^n \#\Sigma$ admits a Lagrangian embedding in $\mathbb{CP}^n$, then $\Sigma \# \Sigma \in bP_{4k}$.

An earlier result by Abouzaid-Smith showed that in the plumbing $T^*S^n \#_{pl} T^*S^n$, every exact Lagrangian submanifold of vanishing Maslov class is a homotopy sphere. This plumbing is symplectomorphic to the Milnor fiber of the $A_2$-singularity defined by $z^3 + \sum^n x^2_j$ inside of $(\mathbb{C}^{n+1},\omega_{st})$ with its symplectic form inherited by restricting the standard form $\omega_{st}$. The cut and paste techniques of Abouzaid-Smith were utilized by EKS to prove:

Proposition: Let $n > 4$ be an odd integer. Every multiple of $\Sigma \in \Theta_n$ admits a Lagrangian embedding into $T^*\Sigma \#_{pl} T^*S^n$. Contrast this with the first theorem of this section where we cannot embed $\Sigma \notin bP_{n+1}$ into $T^*S^n$.

Remark 1: I should point out that these theorems are proved by studying moduli of holomorphic curves.

Remark 2: We can also plumb together $k$ copies of $T^* S^n$ to get the Milnor fiber of $A_k$ which is defined by the equation $z^k + \sum^n x^2_j = 1$; by projecting to the $z \in \mathbb{C}$ coordinate, we get a Lefschetz fibration where the generic fibers of $A_1$-type of one dimension lower; i.e. $T^*S^{n-1}$. There are $k$ singular fibers corresponding to when $z$ is a $k$th root of unity.

Remark 3: The nearby Lagrangian conjecture says that for any closed, exact Lagrangian $L \subset T^* M$, $L$ is Hamiltonian isotopic to the zero section (and hence, diffeomorphic to $M$). An interesting thing to investigate is whether there are distinct homotopy spheres $\Sigma,\Sigma’$ such that there exists an exact symplectomorphism $T^* \Sigma \to T^*\Sigma’$. If there is, then $\Sigma$ embeds as a closed, exact Lagrangian but it cannot be Hamiltonian isotopic to the zero section because $\Sigma$ and $\Sigma’$ are not diffeomorphic. This would give a negative answer to the conjecture. To start this investigation, we should consider pairs $\Sigma,\Sigma’$ such that $[\Sigma] = \pm [\Sigma’] \in \Theta_n/bP_{n+1}$. For example, when $n=7$, then any two homotopy 7-spheres have a chance to have symplectomorphic cotangent bundles (unless there are new results I don’t know about that say something about this). Since all homotopy 7-spheres are the links of singularities defined by integer polynomials and they live in $\mathbb{C}^5 = \mathbb{R}^{10}$, we can view them as cut out by some real polynomial equations. Then, complexify $\mathbb{R}^{10}$ and also these defining polynomial equations to get some affine varieties in $\mathbb{C}^{10}$. Then, the tubular neighborhoods of the links inside these affine varities are symplectomorphic to their cotangent bundles. It’s an interesting question whether these cotangent bundles are smoothly, symplectically, or exact symplectically isotopic within $\mathbb{C}^{10}$.

Addendum: By Cieliebak-Eliashberg (lemma 11.2), any symplectomorphism between Liouville domains of finite type (i.e. have compact skeleton), can be smoothly isotoped to an exact symplectomorphism. In fact, looking at the proof, it seems that it is isotopic through symplectomorphisms. Hence, since cotangent bundles of closed manifolds are finite type, we can change the question above to ask: “Are there distinct homotopy spheres $\Sigma,\Sigma’$ such that their cotangent bundles are symplectomorphic?” If there is, we can make it an exact symplectomorphism.