53 problems

Drinfeld's conjecture. The inclusion is surjective; equivalently, $\mathfrak{grt}$ is isomorphic to the free graded Lie algebra with one generator in every odd degree at least $3$.…

Let $C_n^r$ denote the $r$th power of the cycle on $n$ vertices, and let $\Delta_d^t(C_n^r)$ denote its total $d$-cut complex. For $d\geq3$ and $n\geq(2d-1)r+1$, Shen et al.'s conj…

Brown's conjecture. The free Lie algebra generated by the $\sigma_{2k+1}$ can be extended to a larger free Lie algebra involving generators of higher degrees as well.

Căldăraru's conjecture. For every genus $g\geq 2$, there is a natural identification

Let $A$ be an associative algebra with product $m$, and let $varphi$ be a $k$-cocycle. The infinitesimal deformation $m_t=m+tvarphi$ determines a cycle in the homology of the graph…

Let $\hat {\cal B}_{m}^{\,u}$ be the diagram space with filtration quotients ${\cal G}_{f,o,e}^{\,}\,\hat {\cal B}_{m+2f}^{\,u}$, where $f,o,e$ are the filtration parameters. Vanis…

Let $\hat {\cal B}_{m}^{\,u}$ be the diagram space. A caterpillar diagram is a diagram of the illustrated type in the source, with an even number of legs on each body segment. Cate…

Let ${\cal B}_{m}^{\,0}$ be the space of connected graph diagrams with zero univalent vertices, and call a diagram planar if it admits a planar realization. Planar spanning conject…

Let $\hat {\cal B}_{m}^{\,u}$ be the diagram space equipped with the filtration whose quotient at $(f,o,e)$ is ${\cal G}_{f,o,e}^{\,}\,\hat {\cal B}_{m}^{\,u}$. Let $T(m,u)$ be the…

Configuration-space integral realization conjecture. There exists a map

Merkulov's low-valence conjecture. Every homology class in $\mathsf{GC}_d$ should admit a representative supported on graphs whose vertices all have valence $3$ or $4$; equivalentl…

Nontriviality conjecture. For every $k=1,2,\dots$, the element $X_{4k+2}$ defines a non-trivial cohomology class in

Let $g$ be the genus, and consider the odd commutative graph complex $\mathcal{GC}_-^g$, whose cohomology is supported in degrees $1\leq k\leq 2g-2$. The top degree is $k=2g-2$. Vo…

Let $\mathrm{G}_n$ be the Kontsevich graph complex, with differential $d$, and let $\mathrm{G}^{\mathrm{Me}}_n=\{\Gamma\in\mathrm{G}_n\mid \pi_5\Gamma=\pi_5d\Gamma=0\}$ be Merkulov…

Let $d$ be an integer, and let $\textsf{OGC}_{2d+1}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ be the oriented graph complex and…

Let $n$, $p$, and $m$ be integers with $m\geq 2$ and $1\leq p\leq n-3$. Let $\mathcal{L}_{m\cdot p}^{n}$ denote the space of long links with $m$ components of dimension $p$ in…

The properads $\mathcal{H}\mathit{olieb}$ and $\mathcal{L}\mathit{ieb}$ control homotopy Lie bialgebra structures, while their wheeled closures…

Let $\mathsf{GC}^{\mathrm{div}}$ be the divergence-free subalgebra of Kontsevich's graph-complex Lie algebra, and let $\mathsf{GC}_{\mathrm{(g)}}$ be the surface graph complex. The…

Let $Z_{(g)}$ and $B_{(g)}$ be the cocycles and coboundaries defining $\mathfrak r_{(g)}=(Z_{(g)}/B_{(g)},\\{\\,\\ \\})$. Let $V_{\alpha,\beta,\gamma}$ be the derivations from the…

Let $\mathsf{GC}^{\text{loop}}_{(1)}$ denote the genus-one graph complex shown in the source, and let $\mathfrak b$ be the Lie subalgebra of…

For $n\geq0$, let $t_{2n}$ be the loop-free graph with $2n+2$ vertices described in the source, and let $\delta_{2n}\in\mathfrak{r}^{\mathrm{non-fr}}_{(1)}$ be Enriquez's derivatio…

Let $\mathfrak{r}_{(g)}$ denote the Grothendieck–Teichmüller Lie algebra of a surface of genus $g$, defined as $(Z_{(g)}/B_{(g)},\\{\\,\\ \\})$, and regard…

Let $g\geq 2$, let $H^*$ be the decorated symplectic space used to define the surface graph complex $\mathsf{GC}_{\mathrm{(g)}}$, and let $\mathfrak{h}'$ be the Lie subalgebra gene…

Let $\Omega^{\bullet}_{\mathrm{can}}$ be the graded space of canonical differential forms, let $\mathbb{L}(\Omega^{\bullet}_{\mathrm{can}})$ be the free Lie algebra on it, and let…

Lower-range hairy graph complex conjecture. The same conclusion should hold whenever