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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?

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

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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.

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