Let $M\_0^4$ and $M\_1^4$ be two closed smooth 4-manifolds and let $M$ be an $h$-cobordism between them (i.e., a compact smooth 5-manifold with boundary the disjoint union of $M\_0$ and $M\_1$ and with the inclusions of $M\_0$ and $M\_1$ into $M$ both being homotopy equivalences). In the case where $M\_0$ and $M\_1$ are simply-connected there is a result due to [Curtis, Hsiang, Freedman, and Stong](https://link.springer.com/article/10.1007/s002220050031) that says that actually there is a manifold $A \subset M$ that is an h-cobordism between $A\_0 := A \cap M\_0$ and $A\_1 := A \cap M\_1$ (here $A$ is a manifold with corners) such that $A\_0$ and $A\_1$ are contractible manifolds and $M - int(A)$ is a product cobordism. Further conditions where later added on (for example, that $A$ may be chosen so that $M-A$ is simply-connected) - see [here](https://math.berkeley.edu/%7Ekirby/papers/Kirby%20-%20Akbulut%27s%20corks%20and%20h-bordisms%20of%20smooth%2C%20simply%20connected%204-manifolds%20-%20MR1392665.pdf) for further discussion.


I am wondering if a similar result exists in the case of $h$-cobordisms between non-simply-connected 4-manifolds. Does anyone know of such a result or have ideas on why such a result should not exist?



h-cobordisms between non-simply-connected 4-manifolds
Let $X$ be a smooth, closed 4-manifold. Every element of $\operatorname{Wh}(\pi\_1(X))$ can be realised, for some $k \in \mathbb{N}$, as the Whitehead torsion $\tau(W,X \#^k S^2 \times S^2) \in \operatorname{Wh}(\pi\_1(X))$ of a smooth $h$-cobordism $(M;X\#^k S^2 \times S^2,Y)$, for some $Y$. The Whitehead group is nontrivial for many fundamental groups, for example $\pi\_1(X) \cong C\_5$.


If an $h$-cobordism $M$ admits a decomposition as in the statement of the CFHS theorem, then its Whitehead torsion lies in the image of $\operatorname{Wh}(\{1\}) \to\operatorname{Wh}(\pi\_1(X))$, and is therefore trivial since $\operatorname{Wh}(\{1\})=0$.


It follows that the $h$-cobordisms with nontrivial Whitehead torsion do not admit such a contractible submanifold $A$ as in the question.


An alternative version of the question would perhaps begin with the hypothesis of an $s$-cobordism.


