Suppose $X$ and $Y$ are closed 4-manifolds, not necessarily simply connected.
Such manifolds are said to be s-cobordant if there is a 5-manifold $W$ with $\partial W = X \sqcup Y$ such that the inclusion maps $X, Y \hookrightarrow W$ are simple homotopy equivalences.
If $X$ and $Y$ are homotopy equivalent, are they s-cobordant? What if they are simply homotopy equivalent? I am interested in the smooth or PL categories.
Take two homotopy equivalent 3-dimensional lens spaces $L$ and $L'$ that are not homeomorphic. Then $S^1 \times L$ and $S^1 \times L'$ are closed, smooth 4-manifolds that are simple homotopy equivalent but not $s$-cobordant.