[Paper Review] Framed bordism and Lagrangian embeddings of exotic spheres

Mohammed Abouzaid|arXiv (Cornell University)|Dec 29, 2008

Geometric and Algebraic Topology15 references4 citations

TL;DR

This paper establishes that every homotopy sphere embedding as a Lagrangian in the cotangent bundle of the standard sphere $T^*S^{4k+1}$ must bound a compact parallelisable manifold. Using a parametrised family of perturbed Cauchy-Riemann equations and Gromov-Floer compactification with holomorphic disc and sphere bubbles, the authors construct a smooth manifold with corners whose boundary includes the Lagrangian, proving its bounding manifold is parallelisable via $K$-theory and holomorphic curve analysis. The result implies that certain exotic spheres, such as the 28 in dimension 9, cannot embed as Lagrangians in $T^*S^9$, distinguishing symplectic structures beyond diffeomorphism type.

ABSTRACT

In dimensions congruent to 1 modulo 4, we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere. More precisely, we prove that such an exotic sphere cannot embed as a Lagrangian in the cotangent bundle of the standard sphere. The main ingredients of the construction are (1) the fact that the graph of the Hopf fibration embeds the standard sphere, and hence any Lagrangian which embeds in its cotangent bundle, as a displaceable Lagrangian in the product a symplectic vector space of the appropriate dimension with its complex projective space, and (2) a moduli space of solutions to a perturbed Cauchy-Riemann equation introduced by Gromov.

Motivation & Objective

• To prove that any homotopy sphere Lagrangian in $T^*S^{4k+1}$ bounds a compact parallelisable manifold.
• To resolve a key step toward the nearby Lagrangian conjecture by showing that symplectic topology obstructs certain exotic smooth structures.
• To demonstrate that the symplectic structure of $T^*S^{4k+1}$ detects the smooth structure of the base sphere beyond diffeomorphism type.
• To construct a compact, parallelisable bounding manifold via gluing moduli spaces of pseudo-holomorphic curves with disc and sphere bubbles.

Proposed method

• Construct a one-parameter family of perturbed Cauchy-Riemann equations on a symplectic manifold $\mathbb{C}^n \times \mathbb{C}\mathbb{P}^{n-1}$ using a compactly supported Hamiltonian isotopy.
• Use Gromov-Floer compactification to include solutions with holomorphic disc and sphere bubbles, forming a compactified moduli space with boundary diffeomorphic to the Lagrangian.
• Cap off disc bubble strata using a second moduli space of holomorphic discs to form a closed manifold with boundary $L$, while sphere bubble strata are described as bundles over $S^2$.
• Prove the resulting manifold is parallelisable by analyzing the $K$-theory class of the tangent space of the moduli space and showing it lifts from the space of smooth maps.
• Use the fact that $\mathbb{C}\mathbb{P}^{n-1}$ has even first Chern class when $n$ is even to ensure the $K$-theory class is trivial in a way compatible with parallelisability.
• Apply analytic estimates and Sobolev embedding to control the gluing of solutions near higher codimension strata, ensuring smoothness of the resulting manifold with corners.

Experimental results

Research questions

• RQ1 Can exotic spheres that are not boundaries of parallelisable manifolds embed as Lagrangians in $T^*S^{4k+1}$?
• RQ2 Does the symplectic structure of $T^*S^{4k+1}$ detect the smooth structure of the base sphere beyond its diffeomorphism type?
• RQ3 What topological obstructions arise for Lagrangian embeddings of homotopy spheres in cotangent bundles of spheres?
• RQ4 How does the moduli space of pseudo-holomorphic curves with bubbles contribute to constructing a parallelisable bounding manifold?
• RQ5 Why does the mod $4$ value of the dimension (specifically $4k+1$) matter in the parallelisability argument?

Key findings

• Every homotopy sphere that embeds as a Lagrangian in $T^*S^{4k+1}$ must bound a compact parallelisable manifold, as stated in Theorem 1.1.
• In dimension 9, only two of the eight exotic spheres bound parallelisable manifolds, so the remaining six cannot embed as Lagrangians in $T^*S^9$, as per Corollary 1.2.
• The cotangent bundles $T^*Σ^{4k+1}$ and $T^*S^{4k+1}$ are not symplectomorphic if $\Sigma^{4k+1}$ is an exotic sphere not bounding a parallelisable manifold.
• The construction of the bounding manifold relies on gluing moduli spaces of pseudo-holomorphic curves with disc and sphere bubbles, resulting in a smooth manifold with corners.
• The parallelisability of the bounding manifold is established via $K$-theory analysis of the tangent space of the moduli space, relying on the even first Chern class of $\mathbb{C}\mathbb{P}^{n-1}$ when $n$ is even.
• The argument crucially depends on the dimension being $4k+1$, as the mod $4$ value enters in the $K$-theory computation to ensure triviality of the tangent bundle.