29 problems

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…

Local compactness conjecture. Locally compact fully characterises the connected cofibrant objects: a connected object $\mathbb{X}$ is cofibrant if and only if it is locally compact…

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 $Z$ be the space in the paper, let $\mathfrak{C}_*$ be the filtered chain complex constructed there, and let $C_*(Z)$ denote the cubical singular chain complex of $Z$, equipped…

Let $\mathfrak{C}_*$ and ${\mathscr C}_*$ be the filtered chain complexes constructed in the paper, and let the map $\XiBar$ be the map referred to in the source. Filtered homotopy…

Characteristic-zero equivalence conjecture. These four complexes have isomorphic cohomology groups.

Let a CSS code be represented by a chain complex, and suppose that the $X$- and $Z$-stabilizer sets have even sizes, so that stabilizers can be randomly paired and spliced independ…

Let $A_0,A_1,\dots,A_n$ be real vector spaces of dimensions $a_0,a_1,\dots,a_n$, arranged in a random chain complex … Write $\beta_i$ for the $i$th Betti number and let $\chi$ deno…

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 $\mathbb{Z}[G]$-…

Broken-trajectory homology conjecture. There is an isomorphism

Thom-Smale-Witten quasi-isomorphism conjecture. The Thom-Smale-Witten complex of $(f^\epsilon,g)$ is quasi-isomorphic to $(\check{\mathbb{C}},\check{\partial})$.

Zeng–Pryadko's conjecture. Equality holds in this upper bound for every degree $m$ and all chain complexes $C$ and $D$.

Let $G$ be a group and let $P_*$ be an acyclic complex of projective modules over the group algebra of $G$. Acyclic-complex conjecture. The complex $P_*$ should be totally acyclic,…

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…

A mixed complex is a tuple $\mathbb{M}=(A,B,\rho,\nu)$ consisting of the corresponding $\mathbb{X}$-complexes and structure maps. Mixed complexes are said to admit a classification…

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…

Let the codes of HHO, BE, and LD be viewed as chain complexes over $\mathbb{F}_2$. A sparse lift is a lift of such a complex to a chain complex over $\mathbb{Z}$ whose boundary ope…

Let $\mathcal{C}$ be a family of low-density parity-check codes whose associated chain complexes over $\mathbb{F}_2$ are sparse. A lift is a chain complex over $\mathbb{Z}$ whose b…

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

Binary homological distance conjecture. In the binary case, $q=2$, this identity should apply to products of arbitrary bounded complexes.