Homotopy Types of Four-Manifold Homeomorphism and Diffeomorphism Groups

Definitions and mathematical statement

Let $M$ be the fixed manifold of dimension $4$ appearing in the source. Under the interpretation that boundary-fixing means fixing every boundary point, define \[ \operatorname{Homeo}_{\partial}(M) =\{h\colon M\to M\mid h\text{ is a homeomorphism and }h|_{\partial M}=\mathrm{id}_{\partial M}\}. \] If a smooth structure on $M$ is specified, define \[ \operatorname{Diff}_{\partial}(M) =\{f\colon M\to M\mid f\text{ is a smooth diffeomorphism and }f|_{\partial M}=\mathrm{id}_{\partial M}\}. \] Both groups have composition as their operation. If $\partial M=\varnothing$, these boundary restrictions are vacuous. A homotopy type means an equivalence class of spaces under homotopy equivalence.

Problem. For the fixed $4$-manifold $M$, determine the homotopy types of the topological groups \[ \operatorname{Homeo}_{\partial}(M) \qquad\text{and}\qquad \operatorname{Diff}_{\partial}(M). \] This is a request to determine the spaces up to homotopy equivalence, rather than merely selected homotopy groups.

Unresolved qualifications. The source does not specify which $4$-manifolds are intended, whether $M$ is compact, or which smooth structure is used for the diffeomorphism group. It also does not specify the group topologies. The compact-open topology for homeomorphisms and a $C^{\infty}$ topology for diffeomorphisms are possible conventions, but the intended conventions, especially for noncompact $M$, require confirmation. The definitions above use pointwise boundary fixing; requiring maps to equal the identity on a neighborhood of the boundary is another possible convention that the supplied record does not resolve. Thus the displayed problem retains the source's scope, but a fully specified homotopy-type question requires these choices.

Short English statement

For a fixed four-dimensional manifold, determine the homotopy types of its groups of boundary-fixing homeomorphisms and diffeomorphisms. The intended manifold hypotheses, smooth structure, group topologies, and boundary-fixing convention need clarification. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, permanent ID $30004401$, title: Homotopy Types of Four-Manifold Homeomorphism and Diffeomorphism Groups. Exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection; no individual proposer is supplied. The proposed year is recorded as $2020$.
• Supplied source citation: Low-dimensional Topology (2021), Oberwolfach Reports, pp. 465--516, DOI 10.4171/owr/2020/8. https://doi.org/10.4171/owr/2020/8. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or named contributor is supplied.
• Supplied literature reference, labeled "Watanabe, exotic elements of Diff": https://arxiv.org/abs/1812.02448. No more detailed bibliographic identification or theorem statement is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that these homotopy types in dimension four remain incompletely known, with existing results computing selected homotopy groups and obstructions rather than the general spaces. The listed literature sources are https://doi.org/10.4171/owr/2020/8 and https://arxiv.org/abs/1812.02448. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry gives the statement status as corrected_verified, explaining that mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired. This supplied editorial designation does not constitute independent mathematical verification of this draft.