Kirby Problem 4.48

Definitions and mathematical statement

Let $\Delta$ be a compact contractible topological $4$-manifold with boundary. Contractibility means that its identity map is homotopic to a constant map. Define \[ \operatorname{Homeo}_{\partial}(\Delta) =\{h:\Delta\to\Delta\text{ homeomorphism}:h|_{\partial\Delta}=\mathrm{id}\}. \] Use the compact-open topology, whose subbasic sets specify that the image of a compact subset lies in a chosen open subset. Since $\Delta$ is compact and metrizable, this is the usual topology of uniform convergence on maps; restricted to the homeomorphism group it gives the standard group topology.

Is this space contractible for every such $\Delta$? Explicitly, does there exist a continuous map \[ F:\operatorname{Homeo}_{\partial}(\Delta)\times[0,1] \longrightarrow\operatorname{Homeo}_{\partial}(\Delta) \] with $F(h,0)=h$ and $F(h,1)=\mathrm{id}$ for every $h$? Every intermediate map must remain a homeomorphism fixing the boundary pointwise. A contraction to any other fixed group element is equivalent by composition with its inverse.

Contractibility of the underlying manifold is a hypothesis; contractibility of its homeomorphism space is the proposed conclusion. Constructing an isotopy separately for each boundary-fixing homeomorphism would show path connectedness only. The stronger request requires those isotopies to vary continuously with the entire homeomorphism parameter. No smoothability of $\Delta$ is assumed, and replacing homeomorphisms by diffeomorphisms would change the question.

Short English statement

Is the compact-open space of boundary-fixing homeomorphisms of every compact contractible four-manifold contractible?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2924 (KP-4.48), 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 4.48. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.