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Framed bordism and Lagrangian embeddings of exotic spheres. (English) Zbl 1244.53089
Let \(L\) be a compact exact Lagrangian submanifold of \(T^*S^m\) whose Maslov class vanishes. \(L\) is a rational homology sphere which represents a generator of \(H_m(T^*S^m,\mathbb Z)\). The classical nearby Lagrangian Arnol’d’s conjecture would imply that \(L\) is diffeomorphic to the standard sphere. One approach to proving the diffeomorphism statement which is implied by Arnol’d’s conjecture would be first to prove that \(L\) is a homotopy sphere by establishing the vanishing of its fundamental group, then to use Kervaire’s and Milnor’s classification of exotic spheres to exclude the remaining possibilities. In this paper, the author starts the second part of the program and proves that every homotopy sphere which embeds as a Lagrangian in \(T^*S^{4k+l}\) must bound a compact parallelizable manifold. It implies that if \(\Sigma^{4k+1}\) is an exotic sphere which does not bound a parallelizable manifold, then the cotangent bundles \(T^*\Sigma^{4k+1}\) and \(T^*S^{4k+1}\) are not symplectomorphic. That is, the author proves that such an exotic sphere cannot embed as a Lagrangian in the cotangent bundle of the standard sphere.

53D12 Lagrangian submanifolds; Maslov index
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
57R55 Differentiable structures in differential topology
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