通过生成函数的拉格朗日同伦球的框架配边
Framed bordism of Lagrangian homotopy spheres via generating functions
AI总结：
利用Bökstedt-Waldhausen关于管道的导数映射结果和Abouzaid-Courte-Guillermou-Kragh的管道型生成函数存在定理，证明了邻近拉格朗日同伦球的光滑结构限制，即若同伦n-球L拉格朗日嵌入到另一同伦n-球M的余切丛中，则θ_n/bP_{n+1}中的差[L]-[M]是Hopf元素η∈π^1_s的倍数，从而为2-挠元，且当n为偶数时L#L微分同胚于M#M；特别地，若同伦8-球L拉格朗日嵌入T^*S^8，则L微分同胚于S^8。
摘要：
本文结合了Bökstedt和Waldhausen关于管道上所谓导数映射的结果，以及Abouzaid、Courte、Guillermou和Kragh关于邻近拉格朗日同伦球的管道型生成函数的存在定理，得到了对邻近拉格朗日同伦球光滑结构的限制。具体地，如果同伦n-球L拉格朗日嵌入到另一个同伦n-球M的余切丛中，则θ_n/bP_{n+1}中的差[L]-[M]是Hopf元素η∈π^1_s的倍数。特别地，由此得出[L]-[M]在θ_n/bP_{n+1}中是2-挠元，因此如果n是偶数，则L#L微分同胚于M#M。作为另一个应用，如果同伦8-球L拉格朗日嵌入到T^*S^8中，则L微分同胚于S^8。本文给出的结果被与Abouzaid、Courte和Kragh的合作工作所涵盖，该工作处理了M为任意光滑流形的一般情况。当M为同伦球时，情况显著简化，本文的目的是给出这一特殊情况主要结果的简明阐述。
英文摘要：
This note combines a result of Bökstedt and Waldhausen concerning the so-called derivative map on tubes with the existence theorem for generating functions of tube type for nearby Lagrangian homotopy spheres due to Abouzaid, Courte, Guillermou and Kragh to obtain a restriction on the smooth structure of nearby Lagrangian homotopy spheres. Concretely, if a homotopy $n$-sphere $L$ admits a Lagrangian embedding in the cotangent bundle of some other homotopy $n$-sphere $M$, then the difference $[L]-[M]$ in $θ_n/bP_{n+1}$ is a multiple of the Hopf element $η\in π^1_s$. In particular it follows that $[L]-[M]$ is 2-torsion in $θ_n/bP_{n+1}$, hence if $n$ is even then $L\# L$ is diffeomorphic to $M \# M$. As another application, if a homotopy $8$-sphere $L$ admits a Lagrangian embedding in $T^*S^8$, then $L$ is diffeomorphic to $S^8$. The results presented in this note are subsumed by a joint work with Abouzaid, Courte and Kragh which treats the general case in which $M$ is an arbitrary smooth manifold. When $M$ is a homotopy sphere the situation is significantly simpler and the purpose of this note is to give a concise exposition of the main result in this special case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
