Framed bordism and Lagrangian embeddings of exotic spheres

Details Framed bordism and Lagrangian embeddings of exotic spheres Framed bordism and Lagrangian embeddings of exotic spheres

2008-12-29

arxiv.org/abs/0812.4781v2

Mathematics

Symplectic Geometry

98 pages, 2 figures. Reorganised discussion of the gluing theorem in accordance with referee requests. Version accepted by Ann

Scientific paper

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.

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