# h-cobordisms between non-simply-connected 4-manifolds

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 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 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?

• cf: Frank Quinn, Dual decompositions of 4-manifolds. III: s-cobordisms. Trans. Am. Math. Soc. 359, No. 4, 1433-1443 (2007). Commented Sep 17, 2021 at 10:19

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.