6 problems

Finite subset-space cell-structure conjecture. The space $\exp_k X$ has a cell structure obtained from $\exp_{k-1} X$ by adding cells of dimensions $k-c\leq i\leq nk$.

Let $k\geq 2$, $d\geq 2k$, and let $\Delta$ be a $(d-1)$-sphere with $g_k(\Delta)=1$. Cell-complex realization conjecture. There exists a $d$-dimensional cell complex $\mathcal{C}$…

The conjecture. Under these hypotheses, the map

Let $X$ be a regular CW complex, and let $X^{(n)}$ denote its $n$-skeleton. Converse of the closed-manifold theorem. If $X=X^{(n)}$, then $X$ is a closed manifold. This would chara…

The hypercellular complex characterization conjecture. For a graph $G$, the following conditions are equivalent:

For an integer $n$, consider $n$-dimensional cell complexes $K$ whose underlying spaces are $(n-1)$-connected and do not embed in $S^{2n}$. An $h$-minor is a cell complex obtained…