Small-Genus Cohomology of High-Dimensional Diffeomorphism Groups

Definitions and mathematical statement

Let $n\in{\mathbb{Z}}$ satisfy $n\geq 3$. For a nonnegative integer $g$, let \[ W_{g,1}=D^{2n}\mathbin{\#}_g(S^n\times S^n) \] denote the smooth manifold obtained by taking the connected sum of $g$ copies of $S^n\times S^n$ and removing the interior of a smooth $2n$-disk; conventionally $W_{0,1}=D^{2n}$. Its boundary is diffeomorphic to $S^{2n-1}$.

Use the customary convention that $\operatorname{Diff}_{\partial}(W_{g,1})$ is the full group of smooth diffeomorphisms of $W_{g,1}$ that agree with the identity on a neighborhood of the boundary, equipped with the $C^\infty$ topology. Here $BG$ denotes the classifying space of a topological group $G$, and \[ H^*(B\operatorname{Diff}_{\partial}(W_{g,1});{\mathbb{Q}}) =\bigoplus_{k\geq 0}H^k(B\operatorname{Diff}_{\partial}(W_{g,1});{\mathbb{Q}}) \] is singular cohomology with its grading and cup product.

Problem. Determine this graded rational cohomology ring at small genus, particularly determine \[ H^*\!\left(B\operatorname{Diff}_{\partial}\left((S^n\times S^n)\setminus\operatorname{int}(D^{2n})\right);{\mathbb{Q}}\right) \qquad\text{for every integer }n\geq 3. \] The requested computation concerns all cohomological degrees. The source gives no numerical meaning to "small genus," so the genus-one case is an explicit target while the extent of the remaining small-genus request is unresolved. The collar convention for boundary fixing and the $C^\infty$ topology above are customary interpretations of notation that the supplied record does not explicitly define.

Short English statement

Determine the full rational cohomology ring of the classifying space of the boundary-fixing diffeomorphism group of a punctured connected sum of $g$ copies of $S^n\times S^n$, in dimension $2n\geq 6$. The explicit focus is $g=1$; the other genera intended by "small genus" are unspecified.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $30004203$, source problem code catalogue entry, titled Small-Genus Cohomology of High-Dimensional Diffeomorphism Groups, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2019 and names no individual proposer.
• Supplied report citation: Homotopy Theory (2020), Oberwolfach Reports, pp. 2183--2256, DOI 10.4171/owr/2019/36. https://doi.org/10.4171/owr/2019/36. The record describes the report as documenting workshop discussions and open problems.
• Literature-source entry labeled "original Oberwolfach report": https://doi.org/10.4171/OWR/2019/36.
• Supplied literature source: The Cohomology of Torelli Groups Is Algebraic. https://arxiv.org/abs/1908.04724.
• Supplied literature source: A Homological Approach to Pseudoisotopy Theory. https://arxiv.org/abs/2002.04647.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and reports a stable low-degree rational cohomology computation and known unstable classes, while stating that the requested full small-genus cohomology is not determined. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent checking.