Framed bordism and Lagrangian embeddings of exotic spheres
Framed bordism and Lagrangian embeddings of exotic spheres
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奇异球体的框架边界和拉格朗日嵌入
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发表时间：
2008
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通讯作者：
M. Abouzaid
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作者：
M. Abouzaid
在同余于1模4的维中，证明了不限制可平行流形的奇异球面的余切丛不与标准球面的余切丛同余同构。更准确地说，我们证明了这样一个奇异球体不能作为拉格朗日函数嵌入到标准球体的余切丛中。构造的主要成分是(1)Hopf模型的图形嵌入了标准球面，因此嵌入到其余切丛中的任何拉格朗日量作为可位移拉格朗日量在乘积中嵌入适当维辛向量空间及其复射影空间，以及(2)由Gromov引入的摄动CauchyRiemann方程的解的模空间。
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 bration 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.