5 problems

Let $K_m$ be the complete graph on $m$ vertices, let $C_n$ be the cycle graph on $n$ vertices, and let $\Delta_{3}^t(K_m\square C_n)$ denote their total $3$-cut complex. The total…

Let $C_m$ denote the digital cycle with $m$ vertices, and let $H_q(C_m)$ and $\bar H_q(C_m)$ be the homology theories used in the paper. Digital-cycle homology conjecture. For…

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 $X\subset \mathbb{Z}^n$ and $Y\subset \mathbb{Z}^m$ be digital images with $c_1$-adjacency, and let $f:X\to Y$ be continuous. Chain-map conjecture. The induced homomorphism … i…

Let $X$ be a shelf, and let $H_*^{\mathrm{lbo}}(X)$ denote its lbo homology groups. Torsion-freeness conjecture. There is no torsion in $H_*^{\mathrm{lbo}}(X)$. The conjecture is s…