# Elementary Morse Cobordism of Diffeomorphic Boundary Components

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.

## 1 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.