21 problems

Let $\Gamma$ be a connected ribbon graph. Denote its bigraded $n$-color homology by $CH_n^{i,j}(\Gamma;\mathbb{C})$, where $i$ is the homological grading and $j$ is the quantum gra…

For each $m$, let ${\cal D}^*_m$ be the dual diagram complex with differential $\partial$, and let $H^1_\partial({\cal D}^*_m)$ denote its first cohomology group. The graph-homolog…

For each integer $i\geq1$, let … be the homology class constructed from Kontsevich's graph-homology construction. Morita's conjecture. The classes $\mu_i$ are non-trivial for all…

Merger-operation conjecture. The merger operation provides an almost contraction, at least for linear Poisson structures, and thereby yields a canonical solution of the Maurer–Cart…

Let $QA_\infty$ denote the stable object whose homology groups are being considered, and let $H_k(QA_\infty)$ be its degree-$k$ homology group. Kontsevich's conjecture. The stable…

For each $k=2,3,4,\dots$, let $W_{2k}\in\mathsf{FG}_1^{2k\text{-loop}}(0)$ be the cycle described in the preceding lemma, which states that $W_{2k}$ is non-zero and satisfies…

Morita's conjecture. The Morita homology classes $[\mu_k]\in H\bigl(\mathsf{FG}_0^{2k+2}(0)\bigr)$ are non-trivial for every $k\geq 1$. These classes are part of an infinite family…

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 $G$ be a connected graph and let $\Gamma$ be a ribbon graph associated with $G$. Suppose that $G$ has a bridge. Consider the spectral sequence of $n$-color homologies of…

Let $\Gamma$ be a bridgeless plane graph of a connected graph $G$. Let $\psi(\Gamma)$ be the invariant represented by the all-$x^{n-1}$ tensor in the $n$-color homology, and let…

Generation conjecture. The class generating $H_0(\Gamma_{0,3})$, together with the classes in the virtual cohomological dimension, generates all classes under the modular operad st…

Let $G$ be an undirected graph, and let $\mathcal{H}_n^{\normalfont {\textrm{Cube}}}(G)$ denote its $n$-th cubical homology group. Suppose that $G$ contains no $3$-cycles and no…

For even $n$, Morita constructed cycles in $H_{2n-4}(\operatorname{Out}(F_n);\mathbb{Q})$ and Conant--Vogtmann lifted them to $H_{2n-4}(\operatorname{Aut}(F_n);\mathbb{Q})$. Morita…

For each $k\geq 1$, the construction gives a class … corresponding under Kontsevich's isomorphism to the displayed primitive cohomology class. Non-triviality conjecture. These clas…

Let $H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}$ be the symplectic-invariant homology algebra, let $\mathcal{A}^+(\phi)$ be the subalgebra generated by the positive-degree elements o…

Let $V$ be a symplectic vector space, and let ${\mathfrak{h}}_V$ be the Lie algebra generated by finite planar trivalent trees whose leaves are decorated by elements of $V$. Write…

Let $C_2\Gamma$ be the ordered configuration space of two points on a graph $\Gamma$, let $H_1(C_2\Gamma)$ be its first homology group, and let $\beta_1$ denote first Betti number.…

Let $B_n\Gamma=\pi_1(UC_n\Gamma)$ be the braid group of $n$ points on a graph $\Gamma$, and let $H_1(B_n\Gamma)$ denote its first homology group. Kim–Ko–Park's conjecture. The grou…

Let $G$ be a connected planar graph. A cube complex is a chain complex … indexed by the vertices of an $n$-dimensional hypercube. For a vertex $v$, write $|v|=\sum_i v_i$ and let…

Let $(G,S)$ be a ribbon graph embedded in $S^3$, and let $H_i$ denote its homology groups. Planar graph detection conjecture. The ribbon graph $(G,S)$ is isotopic to a planar graph…

The paper considers the homology of the sheet algebra and graph homology constructed from the categorified chromatic algebra. Structural conjecture. All graph homology is structure…