13 problems

Let $K$ be a countable CW complex, and let $AE(X)$ mean that every map from a closed subset of $X$ to $K$ extends over $X$. A metrizable compactification $\widetilde{X}$ is a compa…

Regularity conjecture for tilted Richardson cells. The CW-complex in the preceding stratification is regular.

Positive-part closure conjecture. The closure of the positive part equals the nonnegative part:

Let the Generalized Upper Bound Theorem mean that, for a simplicial $d$-polytope $P$ and the cyclic $d$-polytope $C(d,n)$, the inequality $f_{d-1}(P) \geq f_{d-1}(C(d,n))$ implies…

Finite-test homotopy equivalence conjecture. The map $f$ is a homotopy equivalence.

Generalized Andrews–Curtis conjecture. Any two simple-homotopy equivalent 2-dimensional CW-pairs are 2-equivalent.

CW reconstruction conjecture. There is a non-unique $\infty$-groupoid $X$ such that $X_k=I^X_k$ and, for each $h e I^Xk$, $M(h)$ is equivalent to $llbracket hrrbracket$. Conversely,…

Positroid stratification conjecture. The positroid stratification of $Gr^{\geq 0}(k,n)$ is a regular CW-complex. In particular, the closure $\overline{\Pi}_{\mathcal{M}}$ of each p…

Antieau-Williams' conjecture.

Antieau-Williams' conjecture.

Let $G$ be a finitely presentable group and let $n\geq 0$. For a finite $3$-dimensional CW complex $X$ with cohomological dimension at most $2$ and fundamental group…

Let $X$ be a finite $3$-dimensional CW complex whose cohomological dimension is at most $2$. Wall's D(2) problem. $X$ is homotopy equivalent to a $2$-dimensional CW complex. The pr…

Let $M$ be a smooth compact manifold and let $(M_i)_{i \in I}$ be a finite family of closed subspaces of $M$ that intersects cleanly. A family intersects cleanly when, locally, its…