Rene Thom came up with the idea of J-equivalence:
Let $M_1$ and $M_2$ be manifolds that are oriented, compact and smooth. Then they are J-equivalent if there is a smooth manifold $X$ with boundary $M_1-M_2$ (we reverse the orientation of $M_2$) and both $M_1$ and $M_2$ are deformation retracts of $X$.
Thus if $M_1$ and $M_2$ are J-equivalent, they are homotopic. Are they related by a simple homotopy?
Note: Asking if they are instead diffeomorphic is, I believe, an open question. Although scattered results exists, eg this theorem of Smale: If $M_1$ and $M_2$ are J-equivalent homotopy spheres of dimension $2m-1$ with $m>2$, then they are diffeomorphic.