26 problems

Determinant-class conjecture. Every finite $CW$-complex is of determinant class.

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…

Let $K$ be a countable CW complex. Write $AE(X)$ for the property that every map from a closed subset of $X$ to $K$ extends over $X$. The class of compacta … is said to have a univ…

Deformation conjecture. Every simple homotopy equivalence of 2-complexes can be obtained by a 2-deformation.

Let $d\geq 1$ and let $\ell\geq 2$ be an integer relatively prime to $(d-1)!$. A finite CW complex is a topological space built from finitely many cells, and for a topological Brau…

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:

Wall's conjecture. Every algebraic $\mathbb{Z}G$-complex $A_*$ is admissible.

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

Aldous–Farb–Sageev conjecture. If $K_1$ and $K_2$ have a common cellular covering, then they have a common finite cellular covering.

Let $X$ and $Y$ be smooth CW complexes, and let $f:Y\to X$ be a continuous map. A subset $A\subset Y$ is closed, and $f$ is smooth on $A$ when this condition is relevant. Whitney a…

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,…

CW-complex correspondence conjecture. There is a one-to-one correspondence of presentations of CW-complexes, up to homotopy of the attaching maps, and presented associative $infty$…

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.

Sharpness conjecture. The sharpness of the equation should be achieved by such height-type functions in the general case of CW-complexes with a generalized notion of spanning tree…

Graph quantum mechanics has walk sum formulae and gluing formulae, and Laplacians for CW-complexes provide a more general setting. The CW-complex extension conjecture. The same app…

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…

A finite CW-complex is a CW-complex with finitely many cells, and an H-space is a based space equipped with a multiplication having a homotopy unit. The notation $S^7$ denotes the…

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…

Let $X$ be a finite simply connected $CW$-complex, let $p$ be a prime, and let $\exp_p(X)$ denote the smallest power $p^t$ that annihilates the $p$-primary torsion in $\pi_i(X)$ fo…

For a reduced word $(i_1,i_2,\ldots,i_d)$, let … be the corresponding totally positive space, whose closure is decomposed into cells according to which parameters are nonzero. Fomi…

Miller function-space conjecture. These spaces should not have CW homotopy type.