Foliations and Diffeomorphism Groups — Question 39

Definitions and mathematical statement

The supplied record leaves the topology on the representation space and the definition of the Euler number implicit. The following is a conventional interpretation; these conventions require confirmation against the cited source.

Let $G=\mathrm{Homeo}_0({\mathbb{R}}^2)$ denote the identity component of the homeomorphism group of the plane, equipped with the compact-open topology. Give \[ \mathrm{Hom}({\mathbb{Z}}^2,G) \cong \{(f,g)\in G\times G: f\circ g=g\circ f\} \] the subspace topology inherited from $G\times G$, where the identification uses the ordered standard generators of ${\mathbb{Z}}^2$.

To specify an Euler-number convention, let $e\in H^2(BG;{\mathbb{Z}})$ be the universal Euler class, normalized so that its restriction along the standard rotation subgroup $\mathrm{SO}(2)\hookrightarrow G$ is the usual Euler class of the universal oriented real two-plane bundle. Here $BG$ denotes the classifying space of the topological group $G$. For a representation $\rho:{\mathbb{Z}}^2\to G$, form the associated flat plane bundle \[ E_\rho=({\mathbb{R}}^2\times{\mathbb{R}}^2)/{\mathbb{Z}}^2\longrightarrow T^2={\mathbb{R}}^2/{\mathbb{Z}}^2, \qquad m\cdot(x,y)=(x+m,\rho(m)(y)). \] Let $c_\rho:T^2\to BG$ be its classifying map. Orient $T^2$ using the ordered standard basis of ${\mathbb{Z}}^2$, and define \[ \operatorname{eu}(\rho) =\langle c_\rho^*e,[T^2]\rangle\in{\mathbb{Z}}, \qquad \mathrm{Hom}_k({\mathbb{Z}}^2,G) =\{\rho\in\mathrm{Hom}({\mathbb{Z}}^2,G):\operatorname{eu}(\rho)=k\}, \] with the induced subspace topology.

Under these conventions, the question supplied as Question 39 is: \[ \text{For every }k\in{\mathbb{Z}},\quad \mathrm{Hom}_k\left({\mathbb{Z}}^2,\mathrm{Homeo}_0({\mathbb{R}}^2)\right) \text{ is connected?} \] Here connected means that the space admits no decomposition into two disjoint nonempty relatively open subsets. The question does not assert that every Euler-number level is nonempty. The supplied record labels the problem open; that label is not an independently verified mathematical assessment.

Short English statement

For each integer $k$, consider all actions of ${\mathbb{Z}}^2$ on the plane by homeomorphisms in the identity component whose Euler number is $k$. Is this space connected? The draft uses the compact-open topology and an explicitly normalized Euler class; the supplied record does not specify these conventions.

Sources

• Local UnsolvedMath registry, permanent ID $800098$, exact source problem code catalogue entry, title "Foliations and Diffeomorphism Groups --- Question 39." The supplied record attributes the problem to Nariman and Yazdi, gives proposed year 2025, and labels its status open. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 39, https://arxiv.org/abs/2503.18274. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. This access information is supplied provenance, not a claim of independent source consultation or verification.