Homological Stability for Stabilized Four-Manifold Diffeomorphisms

Definitions and mathematical statement

Let $W=S^2\times S^2\setminus\operatorname{int}(D^4)$, where $D^4$ is a smoothly embedded closed ball, and for each integer $n\geq 1$ let \[ W_n=\mathop{\natural}^{n}W \] be the boundary connected sum of $n$ copies of $W$. Thus $W_n$ is a compact smooth four-manifold with boundary diffeomorphic to $S^3$. Use the conventional interpretation that $\operatorname{Diff}_{\partial}(W_n)$ is the full group of smooth diffeomorphisms that are the identity on a neighborhood of the boundary, equipped with its $C^\infty$ topology. Write $BG$ for the classifying space of a topological group $G$, and $H_k(-;{\mathbb{Q}})$ for singular homology with rational coefficients.

Under the usual boundary connected sum stabilization convention, attaching another copy of $W$ and extending diffeomorphisms by the identity gives maps \[ s_n:B\operatorname{Diff}_{\partial}(W_n)\longrightarrow B\operatorname{Diff}_{\partial}(W_{n+1}). \] Define the stable homology in degree $k$ by \[ H_k^{\mathrm{st}}=\varinjlim_{n\geq 1}\left(H_k(B\operatorname{Diff}_{\partial}(W_n);{\mathbb{Q}}),(s_n)_*\right), \] and denote the canonical map to this direct limit by \[ \iota_{n,k}:H_k(B\operatorname{Diff}_{\partial}(W_n);{\mathbb{Q}})\longrightarrow H_k^{\mathrm{st}}. \]

A precise interpretation of the supplied question is: is the following assertion true? \[ \forall k\in{\mathbb{Z}}_{\geq 0}\quad\exists n\in{\mathbb{Z}}_{\geq 1}\quad \iota_{n,k}\text{ is an isomorphism}. \]

The source does not explicitly specify the topology on the diffeomorphism group, the boundary-fixing convention, or the stabilization maps; the definitions above make the conventional interpretation explicit and require confirmation against the original report. The phrase "is equal to the stable-limit homology" is also unresolved: the displayed question interprets it as an isomorphism induced by stabilization, rather than an unspecified abstract isomorphism. The supplied quantifier asks for one finite stage in each degree. It does not explicitly require that every subsequent stage have its canonical map to the limit be an isomorphism, or that all subsequent stabilization maps be isomorphisms; those stronger eventual-stability formulations are not asserted here.

Short English statement

For every nonnegative homological degree, ask whether the rational homology of the classifying space of boundary-fixing diffeomorphisms of some finite boundary connected sum of punctured $S^2\times S^2$ maps isomorphically to its stabilization limit. The intended meaning of equality with the limit, and whether stability must persist at all later stages, require confirmation.

Sources

• Local UnsolvedMath registry, permanent ID 30004402, source problem code catalogue entry, title "Homological Stability for Stabilized Four-Manifold Diffeomorphisms," in the Oberwolfach Reports Open Problems collection. The supplied record lists proposed year 2020 and supplies no individual proposer.
• Supplied source citation: Low-dimensional Topology (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/8, pp. 465--516. https://doi.org/10.4171/owr/2020/8. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: Galatius--Randal-Williams, "Stable moduli spaces." https://arxiv.org/abs/1403.2334.
• Supplied literature assessment, checked 2026-08-22: the record labels the problem open and states that Galatius--Randal-Williams compute the stable rational homology, while finite-stage attainment in each fixed degree remains unresolved in the checked literature. This is supplied dated curation derived from the OWR record, not an independently verified mathematical or literature assessment; the applicability of the cited computation and the time-sensitive status require independent verification.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired. This supplied editorial label does not constitute mathematical review of the present draft.