Framed bordism and Lagrangian embeddings of exotic spheres

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 CauchyRiemann equation introduced by Gromov.

A symplectic prolegomenon

“…Abouzaid and Kragh [2,61] have proved that L and L must be homotopy equivalent, and in a few cases-L = S 4n+1 or L = (S 1 × S 8n−1 ); see [1,33]-it is known that T * L remembers aspects of the smooth structure on L, i.e., T * L ∼ = ω T * (L#Σ) for certain homotopy spheres Σ. Question 2.3.…”

“…If char(K) = 2, index theory associates to an intersection point (i.e., Hamiltonian chord) x a one-dimensional K-vector space or x generated by two elements (the "coherent orientations" at x) subject to the relation that their sum vanishes. For the differential we choose a generic family J = {J t } t∈ [0,1] of taming almost complex structures on X. Given two intersection points x ± , we denote by M(x − , x + ) the moduli space of Floer trajectories…”

“…For rational curves, a generic choice of J will be regular for simple (non-multiple-cover) curves. For discs, viewed as maps of the strip D 2 \{±1}, which extend smoothly over the boundary punctures, a generic path {J t } t∈ [0,1] of complex structures will again be regular for solutions which have an injective point x, i.e., one where u −1 (u(x)) = {x}.…”

“…Hypersurfaces or complete intersections in P n of sufficiently low total degree n are monotone. The minimal Chern number N min (X) > 0 of a monotone symplectic manifold is defined by the equality { c 1 …”

On the immersion classes of nearby Lagrangians: Table 1.

“…In a different direction, there are special cases, see [1,10], in which we now know that L is diffeomorphic to the zero section. The purpose of this paper is to study a different direction, by finding obstructions to the existence of embedded representatives in a given class of Lagrangian immersions.…”