20 problems

Let $\Gamma_{n,s}$ be the graph automorphism group with parameters $n$ and $s$, and let $2n+s-3$ be its virtual cohomological dimension. Below-vcd vanishing conjecture. All classes…

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 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

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…

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

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…

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

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

Geometric interpretation conjecture. Under this identification, the operation $\Delta$ on $\mathsf{HGK}_0$ corresponds to pullback under $\pi$, and this pullback induces a well-def…

Let $c_n$ be the coefficients in the factorizing operator … where $w_\Gamma$ is the weight of a Leibniz graph $\Gamma$ and $\#\!\operatorname{Aut}(\Gamma)$ denotes the order of its…

Integral homology conjecture. The map $\mathcal{G}^1_n(X)\to\mathcal{G}_n(X)$ induces an isomorphism on homology in degrees $2*\leq n-3$.

Let $\mathrm{HGC}_{-1,n}$ be the hairy graph complex with hair degree parameter $m=-1$ and arity parameter $n$, equipped with its standard differential $\delta$ and an extra differ…

Let $\phi$ be a gluing and let $A_\phi$ be its assembly map for the homology groups of the graph automorphism groups $\Gamma_{n,s}$. A class is positive-dimensional when it has hom…

Let $\Gamma_{n,s}$ denote the relevant graph automorphism group, and let $A_\phi$ be the assembly map associated to a gluing $\phi$ producing the graph $X_{n,s}$. Write $2n+s-3$ fo…

Let $\mathrm{GC}_3$ be the graph complex, and let $L_k$ denote its loop graphs. The partner classes are those classes killed by the loop graphs in the spectral sequence described i…

Let $\overline{\operatorname{Emb}}_c(\mathbb{R}^m,\mathbb{R}^n)$ denote the homotopy fibre of the inclusion of embeddings into immersions, and let $\pi_*^{(t-\ell,t)}$ denote its c…