Mod-p Formality of Little-Disks Cochains

Definitions and mathematical statement

Let $n\in{\mathbb{Z}}$ with $n>2$, let $p$ be a prime, and let ${\mathbb{F}}_p={\mathbb{Z}}/p{\mathbb{Z}}$. Write $D_n$ for the little $n$-disks operad: in positive arity $r$, its space $D_n(r)$ consists of configurations of $r$ ordered, pairwise disjoint little $n$-disks inside the unit $n$-disk, with operadic composition given by insertion. A nonsymmetric cooperad retains the cocomposition maps and their compatibility identities while forgetting symmetric-group actions.

The source denotes the cochain cooperad by $C^*(D_n;{\mathbb{F}}_p)$. For a differential graded nonsymmetric cooperad $A$ over ${\mathbb{F}}_p$, formality means that there is a zigzag of morphisms of differential graded nonsymmetric cooperads \[ A\;\xleftarrow{\simeq}\;A_1\;\xrightarrow{\simeq}\;\cdots\;\xleftarrow{\simeq}\;A_m\;\xrightarrow{\simeq}\;H^*(A), \] where every arrow is a quasi-isomorphism in every arity and $H^*(A)$ has zero differential and the induced cocomposition maps. Arrow directions may vary throughout the zigzag.

Intended question. For every integer $n>2$ and every prime $p$, is the full nonsymmetric cochain cooperad $C^*(D_n;{\mathbb{F}}_p)$ formal? In the intended cochain-cooperad model, this asks for an aritywise quasi-isomorphism zigzag \[ C^*(D_n;{\mathbb{F}}_p)\;\simeq\;H^*(D_n;{\mathbb{F}}_p) \] in the category of nonsymmetric differential graded cooperads, with zero differential on the right. The question concerns all arities, without an arity truncation.

Unresolved qualifications. The supplied record does not specify a cochain model or the construction of its cooperad structure. Contravariance of cochains gives maps into cochains on products of operad spaces; interpreting these as strict cocomposition maps into tensor products requires additional model choices or comparison data. Thus the displayed formulation is conditional on fixing the intended cochain-cooperad model. The record also does not specify whether morphisms must preserve additional multiplicative or homotopy-coherent structure, or whether nullary operations, counits, or coaugmentations are included. These conventions must be resolved before the target is fully precise; formality of the underlying nonsymmetric differential graded cooperad is the interpretation displayed above.

Short English statement

For every integer $n>2$ and every prime $p$, can the mod-$p$ cochains of the little $n$-disks operad, after forgetting symmetric-group actions, be connected to their cohomology by quasi-isomorphisms preserving the intended cooperad structure in all arities? The cochain model and required structure are unspecified. The supplied dated curation labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30003183$, source problem code catalogue entry, title "Mod-p Formality of Little-Disks Cochains," in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year $2016$ and names no individual proposer.
• Supplied report citation: Topologie (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/35, pp. 2009--2068. https://doi.org/10.4171/owr/2016/35. This is also listed among the supplied literature sources as "Oberwolfach Report 35." The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: Boavida de Brito--Horel, https://arxiv.org/abs/1903.09191. No paper title, theorem number, or precise range of applicability is provided in the record.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that higher little-disks formality in positive characteristic is known in finite ranges or truncated arities, while the full nonsymmetric statement for $n>2$ remains unresolved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with an accompanying assertion that mathematical content was preserved while notation, typography, or omitted definitions were repaired. This is source metadata, not independent mathematical verification by this draft.