Given a smooth manifold (not closed, maybe with boundary) $M$ in $R^n$, take a section with a hyperplane $H$ of some dimension $d$. Assume that $M$ has $M\cap H$ as deformation retract. For example, a cylinder.
Now we pick the function squared distance to $H$. If it is not a Morse function, deform it a bit so that you have a Morse function. Let $f$ be that function.
Question: Is it true that one can give a deformation retract of $M$ onto $M\cap H$ as the flow of the vector field given by the gradient $\nabla f$?
My idea: If $\nabla f$ does not vanish you are done. Assume that it vanishes. Since you have all the homotopy information in $f^{-1}(-\epsilon,\epsilon)$, the handles that you attach at each critical level are not that important, so (what I expect is that) there is a pairing of critical points that cancel each other: there is a deformation of the Morse function through Morse functions that finishes cancelling pairs of critical points of indexes $(r,r+1)$. The end result is that you have a deformation retract given by the flow of $f$, or a modification of $f$.
I know that the cancellation is at least algebraic, but I wonder if it is geometric as well.