Kirby Problem 4.89

Definitions and mathematical statement

A smooth circle action on a smooth $4$-manifold $X$ is a smooth map $a:S^1\times X\to X$ with $a(1,x)=x$ and $a(uv,x)=a(u,a(v,x))$. It is effective if the only circle element fixing all of $X$ is $1$. An action preserves the boundary. Consider simply connected compact smooth $4$-manifolds with nonempty boundary, in the bounded setting of the source.

Part (a) asks which diffeomorphism types of such manifolds admit an effective circle action. Part (b) asks, for each eligible $X$, to classify its actions up to smooth conjugacy. Conjugacy means that a diffeomorphism $F:X\to X'$ satisfies $F(a(u,x))=a'(u,F(x))$ for every $u\in S^1$ and $x\in X$; the circle parameter is kept fixed.

Part (c) starts with a prescribed smooth circle action on $\partial X$ and asks for necessary and sufficient conditions for an extension to a smooth action on $X$, and for uniqueness of that extension. For a fixed boundary marking, uniqueness is understood up to equivariant diffeomorphism restricting to the identity on the boundary; forgetting the marking gives the corresponding conjugacy question.

The extension must satisfy the circle group law throughout the interior. Extending each boundary diffeomorphism separately, or extending only the loop it represents in a diffeomorphism group, does not necessarily supply a circle action. Fixed points and exceptional orbits in the interior are allowed.

Short English statement

Classify effective smooth circle actions on simply connected compact four-manifolds with boundary, including existence, conjugacy and extension of prescribed boundary actions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2965 (KP-4.89), 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 4.89. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.