Lagrangian exotic spheres

TL;DR

This paper investigates the symplectic properties of cotangent bundles of homotopy spheres, establishing conditions for symplectomorphism and Lagrangian embeddings, and introduces new explicit Lagrangian embeddings and re-parameterizations with distinct Hamiltonian isotopy classes.

Contribution

It provides new criteria for symplectomorphism of cotangent bundles and conditions for Lagrangian embeddings, along with explicit constructions and re-parameterization results.

Findings

Cotangent bundles of homotopy spheres are symplectomorphic only under specific class conditions.

Certain Lagrangian embeddings imply the homotopy sphere bounds a parallelizable manifold.

Existence of non-Hamiltonian isotopic re-parameterizations of the zero-section in cotangent bundles.

Abstract

Let k>2. We prove that the cotangent bundles of oriented homotopy (2k-1)-spheres S and S' are symplectomorphic only if the classes defined by S and S' agree up to sign in the quotient group of oriented homotopy spheres modulo those which bound parallelizable manifolds. We also show that if the connect sum of real projective space of dimension (4k-1) and a homotopy (4k-1)-sphere admits a Lagrangian embedding in complex projective space, then twice the homotopy sphere framed bounds. The proofs build on previous work of Abouzaid and the authors, in combination with a new cut-and-paste argument, which also gives rise to some interesting explicit exact Lagrangian embeddings into plumbings. As another application, we show that there are re-parameterizations of the zero-section in the cotangent bundle of a sphere which are not Hamiltonian isotopic (as maps, rather than as submanifolds) to the original zero-section.