10 problems

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…

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 $\Gamma$ 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…

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…

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 $G$ 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…