Kirby Problem 4.56

Definitions and mathematical statement

A smooth $4$-dimensional cobordism from a closed $3$-manifold $M_0$ to $M_1$ is a compact smooth $4$-manifold $W$ with $\partial W=M_0\sqcup M_1$, with the usual boundary identifications. It is an $h$-cobordism if both inclusions $M_j\hookrightarrow W$ are homotopy equivalences.

It is an $s$-cobordism if these homotopy equivalences are simple. For finite CW complexes, a simple homotopy equivalence is a map homotopic to a composition of elementary expansions and collapses, and cellular isomorphisms. An elementary collapse removes a cell together with a free face attached to no other cell; an expansion is the inverse operation. Smooth compact manifolds have compatible finite CW structures, so this definition applies to the inclusions. Equivalently their Whitehead torsions vanish.

Is every smooth $s$-cobordism $(W;M_0,M_1)$ diffeomorphic to $M_0\times[0,1]$? The product carries its standard product smooth structure. The assertion asks for a diffeomorphism of the entire cobordism, not merely for $M_0$ and $M_1$ to be diffeomorphic.

No restriction on the fundamental group is imposed, and the source does not require the boundary manifolds to be orientable. The smooth category is essential. A topological product decomposition would not establish the proposed smooth product structure. Similarly, a homotopy equivalence $W\simeq M_0$ alone is already part of the hypothesis and falls short of the conclusion.

Short English statement

Is every smooth four-dimensional s-cobordism between closed three-manifolds a smooth product?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2932 (KP-4.56), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.56. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.