11 problems

Filtered-subcomplex splitting conjecture. Over $k=\mathbb{Z}[y^2]$,

Fix a decomposition of the alphabet $X_{(m)}=A\cup B$ and let $C_\bullet^{(m,A,B)}$ be the subchain complex generated by words that begin with letters from $A$ and end with letters…

Equivariant chain-level isomorphism conjecture. The chain complexes $(C^\epsilon,\partial^\epsilon)$ and $(\boldsymbol{C},\boldsymbol{\partial})$ are isomorphic as $\mathbb{Z}[G]$-…

Exactness conjecture. The chain complex obtained by applying the Chain Complex Construction Procedure results in a free resolution of the cokernel of the initial map. That is, it i…

Let $K_i\subset Y_i$ for $i=1,2$ be rationally null-homologous knots. Write $K_1\# K_2$ for their connected sum, and let $B^\pm(K)$ denote the bent complexes associated to a knot…

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…

A sparse chain complex over $\mathbb{F}_2$ has a sparse lift to integer coefficients when its boundary operators can be lifted while remaining sparse. Torsion obstruction conjectur…

A sparse $2$-complex over $\mathbb{F}_2$ is a chain complex in degrees $0$, $1$, and $2$ whose boundary operators are sparse. A stable lift is obtained by adding finitely many cont…

A chain complex over $\mathbb{F}_2$ is sparse when every row and column of every boundary operator has $O(1)$ nonzero entries; a sparse lift to $\mathbb{Z}$ has uniformly bounded r…

Arbitrary finite-field homological distance conjecture. For any pair of bounded chain complexes of vector spaces over a finite field, the homological distances…

Let $M$ be an oriented closed manifold of dimension $n$. Let $C^{\mathrm{sing}}(\Loop_{\Sph{1}}M;\R)$ be the smooth singular chain complex with boundary $\Bdd$, and let…