Framed bordism of Lagrangian homotopy spheres via generating functions

This note combines a result of B\"okstedt 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 $\theta_n/bP_{n+1}$ is a multiple of the Hopf element $\eta \in \pi^1_s$. In particular it follows that $[L]-[M]$ is 2-torsion in $\theta_n/bP_{n+1}$, 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.