Framed bordism and Lagrangian embeddings of exotic spheres

TL;DR

This paper demonstrates that certain exotic spheres in specific dimensions cannot be embedded as Lagrangians in standard cotangent bundles, revealing new constraints on symplectic embeddings related to exotic smooth structures.

Contribution

It establishes that exotic spheres not bounding parallelisable manifolds cannot be realized as Lagrangian submanifolds in cotangent bundles of standard spheres in dimensions 1 mod 4.

Findings

Exotic spheres in these dimensions are not symplectomorphic to standard ones.

Lagrangian embeddings of such exotic spheres are obstructed.

The proof uses moduli spaces of solutions to perturbed Cauchy-Riemann equations.

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.