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Let, $V$ denote the Stiefel manifold of 2-frames $V_{10,2}$ . Consider the the map $S_\text{diff} (V) \xrightarrow{\eta} N_\text{diff} (V) $ in the surgery exact sequence of a smooth manifold. . There is a bijection, $N_\text{diff}(V) \equiv [V,G/O]$. So, we have a map $S_\text{diff}(V) \xrightarrow{\eta} [V,G/O]$. In my case, from cell structure; $[V,G/O] \cong [S^8,G/O] \oplus [S^9,G/O] \oplus [S^{17},G/O] \cong Z \oplus \text{Torsion}.$ We also know in my case $\eta$ is onto. For each torsion element of $[V,G/O]$ it was possible to find the explicit element in $N_\text{diff}(V)$ and produce an inverse in $S_\text{diff}(V)$. What I want now is to do the exact same thing for the free part $Z$. But, the free part is in $[S^8,G/O]$ and even if I find the generator, I don't know how to get an element in $N_\text{diff}(V)$ corresponding to it, let alone find an inverse. So,-

  1. What could be possible ways to proceed in this situation?
  2. Has any similar kind of work been done before in any paper? I could not find any.

Any kind of help is appreciated. Thank You.

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  • $\begingroup$ It looks like you are assuming that $n=10$ but you do not write it. $\endgroup$
    – John Klein
    Commented Dec 25, 2023 at 15:08
  • $\begingroup$ @JohnKlein Right, I started with n=10 case. But, it will work in general for n even in a lot of cases. That is why, forgot to mention. I will edit. $\endgroup$
    – Sagnik Biswas
    Commented Dec 26, 2023 at 16:19

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