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 $\pi_1$.