12 problems

Goussarov–Habiro conjecture. Let $d\in\mathbb{N}^*$. Any two homology cylinders over $\Sigma$ are $Y_{d+1}$-equivalent if, and only if, they are not distinguished by finite-type in…

Classification conjecture. Finite-type invariants classify homology cylinders, and in particular homology $3$-spheres.

Levine's conjecture. Every link with vanishing $cbar\mu$-invariants is an $chF$-link in the sense of Levine.

Let $\mathcal{H}_{g,1}$ be the homology cobordism group of homology cylinders over the surface $\Sigma_{g,1}$. Perfectness conjecture. The group $\mathcal{H}_{g,1}$ is perfect. Thi…

Let $\mathcal T_n$ be the degree-$n$ tree group associated to homology cylinders, and let $\eta'_n\colon\mathcal T_n\to {\sf D}'_n$ be Levine's homomorphism. The preceding theorem…

Let ${\sf L}'_n$ be the degree-$n$ quasi-Lie algebra on generators $X_1,\ldots,X_m$, and let ${\sf D}'_n$ be the kernel of the bracketing map … Let $\mathcal T_n$ be the degree-$n$…

For every $k\geq 1$, consider the homomorphism … from the exact sequence associated with the $A_n$-filtration of homology cylinders, where ${\sf K}^{\sf Y}_{4k-3}$ is the kernel ap…

Perfectness conjecture. The monoid ${\mathcal{C}_{g,1}}^\mathrm{irr}$ and the group ${\mathcal{H}_{g,1}}$ do not have non-trivial abelian quotients.

Stable lower central series conjecture. The lower central series of the Torelli group stably coincides with the restriction of the $Y$-filtration:

Residual nilpotence conjecture for the $Y$-filtration. The canonical map

Stable equality conjecture. For any $g\geq 0$ and all $j\geq 1$,

Stable injectivity conjecture. For $g\geq 3$, this map is injective.