Suppose we have a cobordism $W$ of manifolds $M\_0$ and $M\_1$ and suppose the inclusion of $M\_0$ into $W$ is a homotopy equivalence. Is the same true for the inclusion of $M\_1$ (ie. is $W$ already an h-cobordism)?

Using Poincare Lefschetz duality one can show that this map induces isomorphisms on homology. 
Hence it suffices to show that the inclusion $M\_1\rightarrow W$ induces an isomorphism on $\pi\_1$.