  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.
