Group Actions on 1-Manifolds — Question 21

Definitions and mathematical statement

Let $S^1={\mathbb{R}}/{\mathbb{Z}}$. Under the usual integer-regularity interpretation, let $r\in{\mathbb{Z}}$ with $r\geq 1$, and let $\mathrm{Diff}_+^r(S^1)$ denote the group of orientation-preserving $C^r$ diffeomorphisms of $S^1$, whose inverses are also $C^r$. Equip this group with the usual $C^r$ topology, namely uniform convergence of maps and their derivatives through order $r$ in local coordinates on the compact circle. Define the commuting-pair space \[ \mathcal C_r=\left\{(f,g)\in\mathrm{Diff}_+^r(S^1)\times\mathrm{Diff}_+^r(S^1):f\circ g=g\circ f\right\}, \] and give it the subspace topology inherited from the product $C^r$ topology.

The question is whether, for every such $r$, the space $\mathcal C_r$ is locally connected. Explicitly, is it true that for every $(f,g)\in\mathcal C_r$ and every relatively open set $U\subseteq\mathcal C_r$ containing $(f,g)$, there exists a relatively open connected set $V\subseteq\mathcal C_r$ such that \[ (f,g)\in V\subseteq U? \]

The supplied statement writes only $r\geq1$ and does not explicitly specify the topology or the allowed regularity parameters. The formulation above makes the standard finite integer $C^r$ interpretation explicit; inclusion of noninteger regularity or $r=\infty$, and the associated topology, remains unresolved by the supplied record.

Short English statement

For every positive integer differentiability order $r$, is the space of commuting pairs of orientation-preserving circle diffeomorphisms locally connected in the usual $C^r$ topology? This makes explicit a conventional interpretation of the source's unspecified regularity range and topology. The supplied registry labels the problem open; that status has not been independently verified.

Sources

• Local UnsolvedMath registry, permanent ID $800304$, source code catalogue entry, titled "Group Actions on 1-Manifolds --- Question 21." The record attributes the problem to Navas, with proposed year 2017, and labels it open. Its supplied creation and update date is 2026-10-16.
• Navas, Group Actions on 1-Manifolds: A List of Very Concrete Open Questions (2017), https://arxiv.org/abs/1712.06462. The supplied provenance identifies the item as "Question 21 (#20)", records access on 2026-10-16, and specifies extraction by arxiv_latex. These are supplied provenance details; neither the source nor the access claim has been independently verified here.