Let $(M,V,V')$ be a smooth manifold triads. I would like to find a Morse cobordism which is elementary, i.e. there exists Morse function $f:M\to[0,1]$ such that $f^{-1}(0)=V, f^{-1}(1)=V'$ and of only $1$ critical point in the interior of $M$, such that $V$ is diffeomorphic with $V'$.
Furthermore, $V,V'$ in the example should not be $\mathbb S^1,\mathbb S^3,\mathbb S^7,\mathbb S^{15}$ (since we can use Morse function of $3$ critical points https://en.wikipedia.org/wiki/Eells%E2%80%93Kuiper_manifold).
This actually suffices to find a manifold $V$ that is invariant under some $p$-surgery. I cannot come up with such example yet. Thanks for any help.
Best Answer
A family of solutions can be generated easily, for example,
consider a $1$-surgery that leaves $\mathbb S^3$ invariant (seen from a Morse function of $3$ critical points on $\mathbb{CP}^2$), do a nontrivial surgery to $\mathbb S^3$ (say becoming $\mathbb S^1\times \mathbb S^2$).
Now the surgery is done on the resultant space without the region affected by the second surgery. Hence this give rises to a nontrivial example.