31 problems

Let $\mathcal{I}_g$ be the Torelli subgroup of the mapping class group of the genus-$g$ oriented surface, consisting of mapping classes acting trivially on…

Let $\Sigma_{g,n}$ be a surface of genus $g$ with $n\leq 1$ boundary components, and let $\mathcal{I}(\Sigma_{g,n})$ denote its Torelli group. Johnson's cubic-growth conjecture. Fo…

Let $\mathcal{I}_{g,1}$ be the Torelli group of a genus-$g$ surface with one boundary component, and let $\operatorname{gr}(\mathcal{I}_{g,1})$ be the graded rational Lie algebra a…

Let $T_g$ be the Torelli group of a closed oriented surface $\Sigma_g$ of genus $g$. The higher FR-torsion classes are cohomology classes associated to the universal surface bundle…

Johnson's conjecture. These homomorphisms are isomorphisms for all $k$ and sufficiently large $g$.

Let $\mathcal{I}_g$ and $\mathcal{I}_g^1$ denote the Torelli groups associated to a closed genus-$g$ surface and a genus-$g$ surface with one boundary component, respectively. Let…

Let $\mathcal{I}_3$ denote the Torelli group of a closed genus $3$ surface, and let $\mathbb{Z}$ denote the integers. Non-finite generation conjecture. The group … is not finitely…

Let $T$ be the subgroup of $\mathrm{IA}_3$ generated by the triple-indexed Magnus generators $K_{123}$, $K_{231}$, and $K_{312}$, and let $N$ be the normal closure in…

Blanchet–Costantino–Geer–Patureau Mirand faithfulness conjecture. The projective representation $L_{r_{\omega}}$ is injective. These representations were introduced to obtain mappi…

Looijenga's conjecture. The smooth Torelli group is trivial:

Folklore conjecture. For each $k$, there exists some $G_k$ such that for $g \geq G_k$, the homology group $\operatorname{H}_k(\operatorname{\mathcal{I}}_g;\mathbb{Q})$ is finite-di…

Let $\mathcal{I}_3$ be the Torelli group of a closed oriented surface of genus $3$, consisting of mapping classes acting trivially on first homology. Its second integral homology g…

Let $\mathcal{I}_{g,1}$ be the Torelli group of a genus-$g$ surface with one boundary component, let $\operatorname{IA}_{2g}$ be the IA-automorphism group of a free group of rank…

Let $\mathcal{I}_g$, $\mathcal{I}_{g,1}$ and $\mathcal{I}_g^1$ be the Torelli groups of a closed surface, a surface with one boundary component, and a surface with one marked point…

Let $\mathcal{I}_g$ be the Torelli group of a closed genus-$g$ surface, and let $e_{2i}$ denote the Mumford–Morita–Miller class of degree $4i$. Kawazumi–Morita's conjecture. Stably…

Let $\mathcal{I}_g$, $\mathcal{I}_{g,1}$ and $\mathcal{I}_g^1$ be the Torelli groups of a closed surface, a surface with one boundary component, and a surface with one marked point…

Let $S_g$ be a closed oriented surface of genus $g$, let $\mathop{\mathrm{Mod}}\nolimits(S_g)$ be its mapping class group, and let $\mathcal{I}_g$ be the Torelli group, consisting…

Let $W_g=\#^g S^n\times S^n$, let $B\operatorname{Tor}(W_g,D^2)$ be the classifying space of the Torelli group relative to a disc, and let…

Let $T_g$ be the Torelli group of a genus-$g$ surface, and let $H^*(T_g;\mathbb{Q})$ denote its rational cohomology. Finite-dimensionality conjecture. The cohomology groups…

Let ${{\mathcal I}}_g^b$ be the Torelli group of a genus-$g$ surface with $b$ boundary components, and let $\operatorname{H}_k({{\mathcal I}}_g^b)$ denote its $k$th homology group.…

Let $S_g$ be a closed orientable surface of genus $g$, and let $\mathcal I(S_g)$ denote its Torelli group. Folk finiteness conjecture. For every fixed $k$, the group…

Let $\Sigma^b_g$ be a surface of genus $g\geq1$ with $b\in\{1,2\}$ boundary components, and let $\mathrm{SMod}(\Sigma^b_g)$ be the image of the embedding of the braid group…

Andreadakis' equality conjecture.

Let $S_g$ be a closed oriented surface, let ${\mathcal{M}}(S_g)$ be its mapping class group, and let ${\mathcal{I}}(S_g)$ be the Torelli group, the kernel of the action of…

Let $S$ be a surface and let $H$ be a nonelementary subgroup of its mapping class group. A cover $\widetilde{S}$ to which $H$ lifts is a cover on which every element of $H$ admits…