6 problems

Let $G$ be a digraph. Write $H_*^{\operatorname{cell}}(G;\mathbb R)$ for its cellular homology and $H_*^c(G;\mathbb R)$ for its singular cubical homology. The cellular–singular cub…

Let $X\subset \mathbb{Z}^n$ be a digital image with $c_1$-adjacency. The groups $\bar H_q(X)$ and $\bar H_q^{c_1}(X)$ are, respectively, the cubical and elementary cubical homology…

Let $G$ be a graph, and let ${\mathcal K}=\{K_i\}$ be a covering of $G$ by subgraphs such that every edge, $3$-cycle, and $4$-cycle of $G$ is contained in some $K_i$. Let…

Let $G$ be a graph. Write $4\mathcal{H}_d^{\normalfont {\textrm{Cube}}}(G)4$ for its discrete singular cubical homology group in dimension $d$. Eventual vanishing conjecture. For e…

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…

Cubical homology conjecture. If $M(E,I)$ is locally finite-dimensional, then for all integers $n\geqslant 0$, the groups $\underrightarrow{\lim}_n^{K(\Sigma)}F$ are isomorphic to t…