Kirby Problem 4.74

Definitions and mathematical statement

Let $X$ be a closed connected smooth four-manifold and $f:X\to X$ a diffeomorphism. A smooth pseudo-isotopy from the identity to $f$ is a diffeomorphism $F:X\times[0,1]\to X\times[0,1]$ restricting to the identity on $X\times\{0\}$ and to $(x,1)\mapsto(f(x),1)$ on the other end. It need not preserve intermediate levels. A smooth isotopy instead is a path of diffeomorphisms of $X$ from the identity to $f$.

To stabilize $f$, first isotope it to a representative equal to the identity on an embedded four-ball. Use that ball to form $X\#_N(S^2\times S^2)$ and extend the representative by the identity on the added summands. Call $f$ stably smoothly isotopic to the identity if some finite $N$ and some allowed choice of representative and ball yield an extension smoothly isotopic to the identity.

Does there exist an $X$ and an $f$ smoothly pseudo-isotopic to the identity but not stably smoothly isotopic to it?

The negative stabilization condition is universal over every finite number of summands and every permitted choice used to extend the map. Showing failure of ordinary isotopy on $X$ alone is therefore insufficient. The fundamental group is not restricted. This asks about the endpoint diffeomorphism, not whether a particular chosen pseudo-isotopy can itself be deformed to a level-preserving one; changing the pseudo-isotopy while keeping its endpoint is allowed.

Short English statement

Can a diffeomorphism be smoothly pseudo-isotopic to the identity yet resist smooth isotopy after every finite sphere-product stabilization?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2950 (KP-4.74), 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 4.74. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.